C.K. Pithawala College of Engineering and Technology

Subject: Advanced Engineering Mathematics (ALA)Topic : Fourier Series By Group no.: Rizwan

Fourier Series Sub-Topic• Periodic Function• Fourier series• Fourier series for Discontinuous function• Change of interval• Even & Odd functions and their fourier series

form• Half- Range Fourier series

Periodic functionDefinition : In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians.A function f(x) is said to be periodic function of x, if there exists a positive real number T such that f(x+T) = f(x)The smallest value of T is called the period of the function.Note:The positive T should be independent of x for f(x) to be periodic. In case T is not independent of x, f(x) is not a periodic function.

For example: Graph of f(x) = A sin x repeats itself after an interval of 2π, so f(x) = A sin x is periodic with period 2π

   Function     Period

 1.         

 sinn x, cosn x,secn x, cosecn x

 π, if n is even2π if n is odd

 2.  tann x, cotn x  π, n is even or odd

 3. | sin x| , | cos x| 

| tan x | , | cot x | | secx| , | cosec x|

 π

 4. x-   = {x}  1

 5.  x1/2 , x2 , x3 +2 , etc

 Period does not exist.

Standard results on periodic functions:

Fourier Series• Let x(t) be a CT periodic signal with period T, i.e.,

• Example: the rectangular pulse train

• Then, x(t) can be expressed as

   where                                                                                                                                                Is the fundamental frequency (rad/sec) of the signal and 

0( ) ,jk tk

kx t c e t

0 2 /T

is called the constant or dc component of x(t)

Discontinues Function• We state Dirichlet's theorem assuming f is a periodic function of period 2π with Fourier

series expansion where

• Th the analogous statement holds irrespective of what the period of f is, or which version of the Fourier expansion is chosen (see Fourier series).

• Dirichlet's theorem: If f satisfies Dirichlet conditions, then for all x, we have that the series obtained by plugging x into the Fourier series is convergent, and is given by

denotes the right/left limits of f.

• A function satisfying Dirichlet's conditions must have right and left limits at each point of discontinuity, or else the function would need to oscillate at that point, violating the condition on maxima/minima. Note that at any point where f is continuous,

• Thus Dirichlet's theorem says in particular that under the Dirichlet conditions the Fourier series for f converges and is equal to f wherever f is continuous

Change Of Interval• In many engineering problems, it is required to expand a function in a Fourier series over an

interval of length 2l instead of 2 . • The transformation from the function of period p = 2  to those of period p = 2l is quite

simple. • This can be achieved by transformation of the variable.• Consider a periodic function f(x) defined in the interval c ≤ x ≤ c + 2l.• To change the interval into length 2 .• Put z = So that when x = c, z = = d and when x = c + 2l, z = = + 2 = d + 2 • Thus the function f(x) of period 2l in c to c + 2l is transformed to the function.• f() = f(z) of the period 2  in d to d + 2  and f(z) can be expressed as the Fourier series

PROOF:• F(z) = ---------- (1)• Where, , n= 1,2,3… , n= 1,2,3…Now making the inverse substitution z = , dz = dxWhen, z = d, x = c• and when, z = d + 2 , x = c + 2lThe expression 1 becomesf(z) = f() = f(x) = • Thus the Fourier series for f(x) in the interval c to c + 2l is given by, f(x) = ------ (2)Where, , n= 1,2,3… , n= 1,2,3…

Example: Obtain Fourier series for the function f(x) = x, 0 ≤ x ≤ 1 p = 2l = 2 = (2 - x) 1 ≤ x ≤ 2

• Solution: Let f(x) = Where,

= = + = =

= = + • Since sin n = sin 2n = 0, cos 2n = 1 for all n = 1,2,3…. = = = 0 if n is even = if n is odd

= = = = 0 ANS: f(x) =

Even and Odd function• This section can make our lives a lot easier because it reduces the work required.

• In some of the problems that we encounter, the Fourier coefficients ao, an or bn become zero after integration.

Finding zero coefficients in such problems is time consuming and can be avoided. With knowledge of even and odd functions, a zero coefficient may be predicted without performing the integration.

• Even FunctionsA function `y = f(t)` is said to be even if `f(-t) = f(t)` for all values of `t`. The graph of an even function is always symmetrical about the y-axis (i.e. it is a mirror image).• Fourier Series for Odd Functions

A function `y = f(t)` is said to be odd if `f(-t) = - f(t)` for all values of t. The graph of an odd function is always symmetrical about the origin.

HALF RANGE FOURIER SERIES

• Suppose we have a function f(x) defined on (0, L). It can not be periodic (any periodic function, by definition, must be defined for all x).

• Then we can always construct a function F(x) such that:

F(x) is periodic with period p = 2L, and F(x) = f(x) on (0, L).

Half range Fourier sine series (cont.)

• Expanding the odd-periodic extrapolation F(x) of a function f(x) into a Fourier series, we find :

Where

Half range Fourier sine series (cont.)

• So that the half range Fourier sine series representation of f(x) is :

Where

• NB: integration is done on the interval 0 < x < L, i.e. where function f(x) is defined.

Half range Fourier cosine series

• Expanding the even-periodic extrapolation F(x) of a function f(x) into a Fourier series,

We find :

With

Half range Fourier cosine series (cont.)

• so that the half range Fourier cosine series representation of f(x) is:

with