Exponential and Logarithmic Functions - Home - Louisiana Tech

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Exponential and Logarithmic Functions

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Introduction

1. It is now time to view exponentials and logarithms as functionsof a complex variable.

2. For the exponential function, there will be no surprises.3. The logarithm (and with it complex roots) turn out to be more

subtle.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Introduction1. It is now time to view exponentials and logarithms as functions

of a complex variable.

2. For the exponential function, there will be no surprises.3. The logarithm (and with it complex roots) turn out to be more

subtle.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Introduction1. It is now time to view exponentials and logarithms as functions

of a complex variable.2. For the exponential function, there will be no surprises.

3. The logarithm (and with it complex roots) turn out to be moresubtle.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Introduction1. It is now time to view exponentials and logarithms as functions

of a complex variable.2. For the exponential function, there will be no surprises.3. The logarithm (and with it complex roots) turn out to be more

subtle.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition.

For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem.

The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof.

With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y)

=∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y)

=−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez

=∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y)

= ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y)

= ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = x+ iy ∈ C define ez := ex(

cos(y)+ isin(y))

andcall it the exponential function.

Theorem. The exponential function is entire withddz

ez = ez.

Proof. With ez = ex cos(y)+ iex sin(y) we have

∂u∂x

= ex cos(y) =∂v∂y

∂u∂y

= −ex sin(y) =−∂v∂x

Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and

ddz

ez =∂

∂xex cos(y)+ i

∂

∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2

= ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)

= ex1ex2[

cos(y1)cos(y2)− sin(y1)sin(y2)++icos(y1)sin(y2)+ isin(y1)cos(y2)

]= ex1ex2

[cos(y1 + y2)+ isin(y1 + y2)

]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]

= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2)

= ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2

= ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez

= ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))

6= 0ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi

= eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi

= ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Some Properties

ez1ez2 = ex1(

cos(y1)+ isin(y1))ex2

(cos(y2)+ isin(y2)

)= ex1ex2

[cos(y1)cos(y2)− sin(y1)sin(y2)+

+icos(y1)sin(y2)+ isin(y1)cos(y2)]

= ex1ex2[

cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)

= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2

ez = ex(cos(y)+ isin(y))6= 0

ez+2πi = eze2πi = ez

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition.

For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof.

From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ

= eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ

= eln(r)+iθ . So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ .

So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.

Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition.

For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z)

= eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn)

= eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn

= reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ

= z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez)

= log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy)

= ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn)

= x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as

log(z) := ln(r)+ i(θ +2πn)

elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z

log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn

= z+ i2πn

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition.

For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1) = Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1) = Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1) = Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1)

= log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1) = Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)

= 0+ i(π +2πn) = (2n+1)πiLog(−1) = Log

(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn)

= (2n+1)πiLog(−1) = Log

(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1) = Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1)

= Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1) = Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as

Log(z) := ln(r)+ iθ

log(−1) = log(e0+iπ

)= 0+ i(π +2πn) = (2n+1)πi

Log(−1) = Log(e0+iπ)

= πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem.

On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is

analytic with derivativeddz

log(z) =1z

.

Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is

analytic with derivativeddz

log(z) =1z

.

Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is

analytic with derivativeddz

log(z) =1z

.

Proof.

Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is

analytic with derivativeddz

log(z) =1z

.

Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is

analytic with derivativeddz

log(z) =1z

.

Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition.

A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example.

The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition.

A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function.

Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example.

Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.

The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).

Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.

Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.

Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

We Must Be Careful When Dealing With Branches ofMultivalued Functions

Log(i3

)= Log(−i)

= −π

2i

6= 3π

2i

= 3Log(i)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

We Must Be Careful When Dealing With Branches ofMultivalued Functions

Log(i3

)

= Log(−i)

= −π

2i

6= 3π

2i

= 3Log(i)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

We Must Be Careful When Dealing With Branches ofMultivalued Functions

Log(i3

)= Log(−i)

= −π

2i

6= 3π

2i

= 3Log(i)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

We Must Be Careful When Dealing With Branches ofMultivalued Functions

Log(i3

)= Log(−i)

= −π

2i

6= 3π

2i

= 3Log(i)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

We Must Be Careful When Dealing With Branches ofMultivalued Functions

Log(i3

)= Log(−i)

= −π

2i

6= 3π

2i

= 3Log(i)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

We Must Be Careful When Dealing With Branches ofMultivalued Functions

Log(i3

)= Log(−i)

= −π

2i

6= 3π

2i

= 3Log(i)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we have

log(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2)

in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2)

= log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)

= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)

= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.

0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0

= log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1)

= log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1))

= log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1)

6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi

0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0

= log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1)

= log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1))

= log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1)

=−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.

Proof.

log(z1z2) = log(

r1eiθ1r2eiθ2)

= log(

r1r2ei(θ1+θ2))

= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)

Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition.

For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z)

and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof.

Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.

Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1:

Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.

Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1

= znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz

= en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z)

= en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z)

= e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z).

This assignment has n possible values forz = reiθ with 0≤ θ < 2π: z

1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π:

z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.

Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):

zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).

So we set z1n := e

1n log(z). This assignment has n possible values for

z = reiθ with 0≤ θ < 2π: z1n = r

1n ei( θ

n + 2πkn ) with k = 0, . . . ,n−1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition.

For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.

Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z 6= 0 and c a complex number we define zc := ec log(z)

,which is a multiple valued function.

Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.

Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.

Note.

This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.

Note. This definition is consistent with the definition of real powersfor real numbers

(it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.

Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension)

as well as with the definition ofinteger powers and roots of complex numbers.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.

Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Example.

(1+ i)4i = e4i log(1+i) = e4i log(√

2ei π4)

= e4i(ln(√

2)+i π

4 +i2πn) = e−π−8πn+2i ln(2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Example.

(1+ i)4i

= e4i log(1+i) = e4i log(√

2ei π4)

= e4i(ln(√

2)+i π

4 +i2πn) = e−π−8πn+2i ln(2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Example.

(1+ i)4i = e4i log(1+i)

= e4i log(√

2ei π4)

= e4i(ln(√

2)+i π

4 +i2πn) = e−π−8πn+2i ln(2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Example.

(1+ i)4i = e4i log(1+i) = e4i log(√

2ei π4)

= e4i(ln(√

2)+i π

4 +i2πn) = e−π−8πn+2i ln(2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Example.

(1+ i)4i = e4i log(1+i) = e4i log(√

2ei π4)

= e4i(ln(√

2)+i π

4 +i2πn)

= e−π−8πn+2i ln(2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Example.

(1+ i)4i = e4i log(1+i) = e4i log(√

2ei π4)

= e4i(ln(√

2)+i π

4 +i2πn) = e−π−8πn+2i ln(2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem.

For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have

1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof.

zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c

= ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z)

= ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z)

= e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0

= 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number we have1zc = z−c

Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem.

For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is

zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z)

where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.

The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain

(r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative is

ddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.

ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc

=ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z)

= ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,

−π < θ < π) and its derivative isddz

zc = czc−1.

Proof.ddz

zc =ddz

ecLog(z) = ecLog(z)c1z

= czc 1z

= czc−1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i = ei(iπ) = e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i

= (−1)i = ei(iπ) = e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i

= ei(iπ) = e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i = ei(iπ)

= e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i = ei(iπ) = e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i = ei(iπ) = e−π

(−i)i(−i)i

= ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i = ei(iπ) = e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 )

= eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i = ei(iπ) = e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2

= eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions

logo1

Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents

Even When Principal Values are Used, We Must Be CarefulWith Identities

(ii)i = (−1)i = ei(iπ) = e−π

(−i)i(−i)i = ei(−i π

2 )ei(−i π

2 ) = eπ

2 eπ

2 = eπ

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Exponential and Logarithmic Functions