Reference textbook –D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis … · 2012-11-16 · International College, KMITL 13016103 Mathematics Mathematics 3 #11 Analytic

International College, KMITL

1301610313016103Mathematics Mathematics 33#11 Analytic Functions and Integration in the Complex Plane

Reference textbook – D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis with Applications, 2nd ed., The Jones and Bartlett Publisher, 2009

Dr. UkritWatchareeruetai, International College, KMITL

International College, KMITLInternational College, KMITL

Content of this lectureContent of this lecture

� Differentiability

� Analyticity

� Cauchy-Riemann equations

� Integration in the complex plane� Integration in the complex plane

Reference textbook – D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis with Applications, 2nd ed., The Jones and Bartlett Publisher, 2009

Complex Analysis 2


DIFFERENTIABILITYDIFFERENTIABILITY

Complex Analysis 3


DifferentiabilityDifferentiability

� Suppose z = x + iy and z0 = x0 + iy0;

� then the change in z0 is the difference

∆z = z – z0 or

∆z = (x – x ) + i(y – y ) = ∆x + i ∆y∆z = (x – x0) + i(y – y0) = ∆x + i ∆y

� If a complex number w = f(z) is defined at zand z0, then the corresponding change in function is the difference

∆w = f(z0 + ∆z) – f(z0).

Complex Analysis 4



� The derivative of the function f is defined in terms of a limit of the difference quotient ∆w/∆z as ∆z � 0.

Complex Analysis 5



� If the limit in (1) exists, then the function fis said to be differentiable at z0.

� Two other symbols denoting the derivative � Two other symbols denoting the derivative of w = f(z) are w' and dw/dz.

� If the latter notation is used, then the value of a derivative at a specified point z0 is written .

Complex Analysis 6

0zzdz

dw

=


Example Example 11

� Use the definition 3.1 to find the derivative of f(z) = z2 – 5z.

Complex Analysis 7


Rules of differentiationRules of differentiation

Complex Analysis 8


Rules of differentiationRules of differentiation

� The power rule for differentiation of powers of z is also valid:

� Combining (6) and (7) gives the power rule for functions:

Complex Analysis 9


Example Example 22

� Differentiate:

a) f(z) = 3z4 – 5z3 +2z

b) f(z) = z2/(4z + 1)b) f(z) = z2/(4z + 1)

c) f(z) = (iz2 + 3z)5

Complex Analysis 10


A function that is nowhere A function that is nowhere differentiabledifferentiable� For a complex function f to be differentiable at a point z0, we know from the preceding lecture that the limit

zfzzf −∆+ )()(lim 00

must exist and equal the same complex number from any direction.

� The limit must exist regardless how ∆zapproaches 0.

Complex Analysis 11

z

zfzzfz ∆

−∆+→∆

)()(lim 00

0


A function that is nowhere A function that is nowhere differentiabledifferentiable� This means that in complex analysis, the requirement of differentiability of a function f(z) at a point z0 is a far greater demand than in real calculus of function demand than in real calculus of function f(x) where we can approach a real number x0 on the number line from only two directions.

� An example of a complex function that is not differentiable is f(z) = x + 4iy.

Complex Analysis 12


Example Example 33

� Show that the function f(z) = x + 4iy is not differentiable at any point z.

Complex Analysis 13


ANALYTICITYANALYTICITY

Complex Analysis 14


Analytic functionsAnalytic functions

� Even though the requirement of differentiability is a stringent demand, there is a class of functions that is of great importance whose members satisfy even importance whose members satisfy even more severe requirements.

� These functions are called analytic functions.

Complex Analysis 15



� A function f is analytic in a domain D if it is � A function f is analytic in a domain D if it is analytic at every point in D.

� The phrase “analytic on a domain D” is also used but we shall call a function f that is analytic throughout a domain D holomorphicor regular.

Complex Analysis 16



� Note: analyticity at a point is not the same � Note: analyticity at a point is not the same as differentiability at a point!

� Analyticity at a point is a neighborhood property; in other words, analyticity is a property that is defined over an open set.

Complex Analysis 17



� As an example, the function f(z) = |z|2 is � As an example, the function f(z) = |z| is differentiable at z = 0 but is not differentiable anywhere else.

� Even though f(z) = |z|2 is differentiable at z = 0, it is not analytic at that point because there exists no neighborhood of z = 0 throughout which f is differentiable; hence, f(z) = |z|2 is nowhere analytic!

Complex Analysis 18


ExampleExample

� Show that the function f(z) = |z|2 is differentiable only at z = 0.

Complex Analysis 19


Entire functionsEntire functions

� A function that is analytic at every point z in the complex plane is said to be an entire function.

� According to the differentiation rules (2), � According to the differentiation rules (2), (3), (5), and (7), we can conclude that polynomial functions are differentiable at every point z in the complex plane and rational functions are analytic throughout any domain D that contain no points at which the denominator is zero.

Complex Analysis 20


Entire functionsEntire functions

Complex Analysis 21


Differentiability & continuityDifferentiability & continuity

� As in real analysis, if a function f is differentiableat a point, the function is necessarily continuous at at a point, the function is necessarily continuous at the point.

� Of course the converse is not true; continuity of a function f at a point does not guarantee that f is differentiable at the point.

� Example, f(z) = x + 4iy is continuous everywhere but is nowhere differentiable.

Complex Analysis 22


L’Hopital’s ruleL’Hopital’s rule

� L’Hopitals’rule for computing limits of the indeterminate form 0/0, carries over to complex analysis.

Complex Analysis 23


ExampleExample

� Using L’Hopital’s rule to compute

izz

zziz 10

54lim

3

2

2 −−+−

+→

Complex Analysis 24


CAUCHYCAUCHY--RIEMANN RIEMANN EQUATIONSEQUATIONSEQUATIONSEQUATIONS

Complex Analysis 25


CauchyCauchy--Riemann equationsRiemann equations

� Here, we sahll learn a test for analyticity of a complex function f(z) = u(x, y) + iv(x, y) that is based on partial derivatives of its real and imaginary parts u and v.its real and imaginary parts u and v.

Complex Analysis 26



Complex Analysis 27



Complex Analysis 28



Complex Analysis 29


ExampleExample

� Using Cauchy-Riemann equation to show that f(z) = x + 4iy is not analytic.

Complex Analysis 30


ExampleExample

� Verify that the polynomial function f(z) = z2 + z satisfies Cauchy-Riemann equations for all z.

Complex Analysis 31


ExampleExample

� Show that the complex function f(z) = 2x2

+ y + i(y2 – x) is not analytic at any point.

Complex Analysis 32


Criterion for analyticityCriterion for analyticity� By themselves, the Cauchy-Riemann equations do not ensure analyticity of a function f(z) = u(x, y) + iv(x, y) at a point z = x + iy.

It is possible for the Cauchy-Riemann � It is possible for the Cauchy-Riemann equations to be satified at z and yet f(z)may not be differentiable at z, or

� f(z) may be differentiable at z but nowhere else.

� In either case, f is not analytic at z.Complex Analysis 33


Criterion for analyticityCriterion for analyticity

� However, when we add the condition of continuity to u and v and to the four partial derivatives , , and , it can be shown that the

xu ∂∂ yu ∂∂ xv ∂∂yv ∂∂and , it can be shown that the

Cauchy-Riemann equations are not only necessary but also sufficient to guarantee analyticity of f(z) at z.

Complex Analysis 34

yv ∂∂


Criterion for analyticityCriterion for analyticity

Complex Analysis 35


ExampleExample

� Use Theorem 3.5 to check the analyticity of

2222)(

yx

yi

yx

xzf

+−

+=

Complex Analysis 36


CauchyCauchy--Riemann equations in polar Riemann equations in polar coordinatecoordinate� Indeed, the form f(z) = u(r, ɵ) + iv(r, ɵ) is often more convenient to use.

� In polar coordinate, the Cauchy-Riemann equations becomesequations becomes

and

Complex Analysis 37


INTEGRATION IN THE INTEGRATION IN THE COMPLEX PLANECOMPLEX PLANECOMPLEX PLANECOMPLEX PLANE

Complex Analysis 38


Curve revisitedCurve revisited

� Suppose the continuous real-valued functions x = x(t), y = y(t), a ≤ t ≤ b, are parametric equations of a curve Cin the complex plane.

If we use these equations as the real � If we use these equations as the real and imaginary parts in z = x + iy, we can describe the point z on C by means of a complex-valued function of a real variable t called parametrization of C:

Complex Analysis 39


Complex integralComplex integral

� An integral of a function f of a complex variable z that is defined on a contour (piecewise smooth curve) C is denoted by (piecewise smooth curve) C is denoted by

and is called a complex integral or contour integral.

Complex Analysis 40

∫C dzzf )(



� To evaluate a contour integral , let us write (2) in an abbreviated form.

∫C dzzf )(

� The intepretation is

Complex Analysis 41



� Specifically, the right-hand side becomes

Complex Analysis 42


Complex integralComplex integral� If we use the complex-valued function (1) to describe the contour C, then (10) is the same as when the integrand is∫ ′

b

adttztzf )())((

� Thus we arrive at a practical means of evaluating a contour integral.

Complex Analysis 43


ExampleExample

� Evaluate , where C is given by x = 3t, y = t2, -1 ≤ t ≤ 4.

∫C dzz

Complex Analysis 44


ExampleExample

� Evaluating , where C is the circle x = cos t, y = sin t, 0 ≤ t ≤2π.

∫C dzz

Complex Analysis 45


Properties of contour integralProperties of contour integral

Complex Analysis 46


Bounding theoremBounding theorem

� Sometimes, it may be useful to find an upper bound for the modulus or absolute value of a contour integral.

� The following theorem is proved by using � The following theorem is proved by using the form of the triangle inequality.

Complex Analysis 47


ExampleExample

� Find an upper bound for the absolute value of where C is the circle |z| = 4.

∫ +C

z

dzz

e

1

Complex Analysis 48

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Reference textbook –D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis … · 2012-11-16 · International College, KMITL 13016103 Mathematics Mathematics 3 #11 Analytic