Complex form of fourier series

COMPLEX FORM OF FOURIER SERIES

Let f(x) be a periodic function of period 2л in the interval (c, c + 2л). The Fourier series of f (x) is given by

This is called the complex form of the Fourier series.

Note: If f(x) is a periodic function of period 2l then the complex form of the Fourier series in c < x < c + 2l is given by

Problems based on Complex form of Fourier Series

FORMULA

Example 2.4.1. Find the complex form of the Fourier series of eax, where 'a' is a constant in −l < x < l

Solution : 

Example 2.4.2. Find the complex form of the Fourier series of f(x) = e-xin-1 < x ≤ 1

Solution : 

Example 2.4.3. Find the Complex form of Fourier series of

Solution:

Example 2.4.4. Derive the complex form of Fourier series for f(x) = eax, л < x < л, given that a is a real constant. Deduce that

Solution:

Example 2.4.5 Find the complex form of the Fourier series of the function f(x) = ex when - л < x < л and f(x+2 л) = f (x)

Solution: 

EXERCISE 2.4

1. Find the complex form of the Fourier series of the periodic function f (x) = sinx, 0<x<π,

2. Find the complex form of the Fourier series of f (x) = cos ax in -л<x<л, where a is not an integer.

3. Obtain the complex form of the Fourier series for ƒ (x) = x2, − x < x < л

4. Find the complex form of the Fourier series of ƒ (x) = sin ax where a is not an integer in π < x < π.