4.1 fourier series

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Engineering Analysis 3 rd year Students

4-Fourier Series The Fourier series provide a method of analysis periodic function in to their constituent component.J.B. Fourier (1768-1830) was among the first to investigate this problem. In his book ``Théorie Analytique de la Chaleur'', written in 1822, he introduced the concept of Fourier series which he used extensively. Recall that a Fourier series is any expression of the form

Periodic Function:A Function f(x) is said to be periodic if F( x+p)=F(x) for all values of x, where p is same positive number P is the interval between two successive repetitions and is called the period of the function.

Example:-

Example:-

Example:Of periodic functions are sin x and cos x of period equal to 2π.

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Example: The function sin(x) has a period 2π, since sin(x+2π)=sin(x) The period of sin (nx) or cos (nx), where n is a positive integer, is 2π/n.

Example

Example

Example

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Finding the smallest positive period of the following function:

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Fourier series:The basic of the Fourier series is that all function of practical significance which are defined in the Interval can be expressed in terms of the convergent trigonometric series of the form:

0

T

f

0

T

f

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Problem: Find the Fourier series corresponding to the function

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Example

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Fourier Coefficients of Even Functions:

Fourier Coefficients of Odd Functions:

Some useful properties of even and odd functions are as follows.

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The product of two odd functions is an even function. The product of two even functions is an even function. The product of an odd function and an even function is an odd function.

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Exercise 1 Classify each of the following function as even, odd or neither even nor odd.

The Fourier series of an even function f(x) is expressed in terms of a cosine series

, bn=0 The Fourier series of an odd function f(x) is expressed in terms of a sine

series

, a0=0 , an=0

Example: Find the Fourier series of the following periodic function.

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Exercise 2

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Expansion of non-periodic functions

If a function f(x) is not periodic then it cannot be expanded in a Fourier series.Given a non-periodic function, a new function may be constructed by taking the values of f(x) in the given range and then repeating them outside of the given range at intervals of 2π For example, the function

f(x)=x is not a periodic function

However, if a Fourier series for f(x)=x is required then the function is constructed outside of this range so that it is periodic with period 2π . For determining a Fourier series of a non-periodic function over a range 2π, exactly the same formulae for the Fourier coefficients are used

Example;Determine the Fourier series to represent the function f(x) =2x in the range −π to+π.The function f(x) =2x is not periodic. The function is shown in the range −π to π of period 2π

Solution:

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Example: Obtain a Fourier series for the function defined as:

Problems

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Half range Fourier series

When a function is defined over the range 0 to π instead of from 0 to 2π it may be expanded in a series of sine terms only or of cosine terms only. The series produced is called a half-range Fourier series.

(a) If a half-range cosine series is required for the function f(x)=x in the range 0 to π then an even periodic function is required.f(x)=x is shown plotted from x=0 to x=π. Since an even function is symmetrical about the y- axis the line AB is constructed as shown. When a half-range cosine series is required then the Fourier coefficients a0 and an are calculated

(b) If a half-range sine series is required for the function

f(x)=x in the range 0 to π then an odd periodic function is required. f(x)=x is shown plotted from x=0 to x=π.Since an odd function is symmetrical about the origin the line CD is constructed as shown. When a half-range sine series is required then the Fourier coefficient bn is calculated.

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Example: 1. Determine the half-range Fourier cosine series to represent the function

f(x) =3x in the range 0 ≤ x ≤ π.2. Determine the half-range Fourier sine series to represent the function

f(x) =3x in the range 0 ≤ x ≤ π.

1. Fourier cosine series

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2. Fourier sine series