(Q1) find the Fourier series for the periodic function(B) f(x)= x³(C) ƒ(x) = [1 + x-1

Answered: (Q1) find the Fourier series for the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(Q1) find the Fourier series for the periodic function
(B) f(x)= x³
(C) ƒ(x) = [1 + x
-1<x<1
-1<x<0
0<x<
<i]

Expert Solution

Step 1: We give the standard form of Fourier series of a function.

Note : '' As per guidelines we will solve the first question. If you want any specific question to be solved please specify that question or post only that question.''

$left parenthesis B right parenthesis space space space space f left parenthesis x right parenthesis equals x cubed space space semicolon space space space space minus 1 less or equal than x less or equal than 1$

(.) Fourier series of a function $f left parenthesis x right parenthesis$ such that $negative c less or equal than x less or equal than c$ is given by,

$f left parenthesis x right parenthesis space equals space a subscript 0 over 2 space plus space sum from n equals 1 to infinity of a subscript n cos open parentheses fraction numerator n pi x over denominator c end fraction close parentheses plus sum from n equals 1 to infinity of b subscript n sin open parentheses fraction numerator n pi x over denominator c end fraction close parentheses$

where $a subscript 0 space comma space a subscript n space space & space space space b subscript n$ are Fourier coefficients given by,

$a subscript 0 equals 1 over c integral subscript negative c end subscript superscript c f left parenthesis x right parenthesis d x space space space space comma space space a subscript n equals 1 over c integral subscript negative c end subscript superscript c f left parenthesis x right parenthesis cos open parentheses fraction numerator n pi x over denominator c end fraction close parentheses d x space space space & space space space b subscript n equals 1 over c integral subscript negative c end subscript superscript c f left parenthesis x right parenthesis sin open parentheses fraction numerator n pi x over denominator c end fraction close parentheses d x$

(*) If $f left parenthesis x right parenthesis$ is an odd function then $integral subscript negative a end subscript superscript a f left parenthesis x right parenthesis d x equals 0$ .

(*) If $f left parenthesis x right parenthesis$ is an even function then $integral subscript negative a end subscript superscript a f left parenthesis x right parenthesis d x space equals space 2 integral subscript 0 superscript a f left parenthesis x right parenthesis d x$ .

(.) A function $f left parenthesis x right parenthesis$ is said to be an even function if $f left parenthesis negative x right parenthesis equals f left parenthesis x right parenthesis$ .

(.) A function $f left parenthesis x right parenthesis$ is said to be an odd function if $f left parenthesis negative x right parenthesis equals negative f left parenthesis x right parenthesis$ .

(.) $cos left parenthesis x right parenthesis$ is an even function and $sin left parenthesis x right parenthesis$ is an odd function.