Find the coefficients of the complex exponential FS for the function:[ 0 if x = [-T, 0]x if x = (0, π]f(x) =

Given function is Therefore the complex form of fouruer series is where fourier coefficients…

Answered: Find the coefficients of the complex…

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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Question

Find the coefficients of the complex exponential FS for the function:
[ 0 if x = [-T, 0]
x if x = (0, π]
f(x) =

Expert Solution

Step 1: Fourier series coefficients find out

Given function is $f left parenthesis x right parenthesis equals open curly brackets table row 0 cell i f space x space element of open square brackets negative pi comma 0 close square brackets end cell row x cell i f space x element of left parenthesis 0 comma pi right square bracket space end cell end table close$

Therefore the complex form of fouruer series is $f left parenthesis x right parenthesis equals sum from n equals negative infinity to infinity of c subscript n e to the power of i n x end exponent$

where fourier coefficients obtained from following :

$f o r space n space not equal to 0 comma space c subscript n equals fraction numerator 1 over denominator 2 pi end fraction integral subscript negative pi end subscript superscript pi f open parentheses x close parentheses e to the power of negative i n x end exponent d x comma n space element of straight integer numbers set minus open curly brackets 0 close curly brackets space a n d space c subscript 0 equals fraction numerator 1 over denominator 2 pi end fraction integral subscript negative pi end subscript superscript pi f left parenthesis x right parenthesis d x$

$equals fraction numerator 1 over denominator 2 pi end fraction open parentheses 0 plus integral subscript 0 superscript pi x e to the power of negative i n x end exponent d x close parentheses$

$equals fraction numerator 1 over denominator 2 pi end fraction open parentheses open square brackets fraction numerator x e to the power of negative i n x end exponent over denominator negative i n end fraction close square brackets subscript 0 superscript pi minus 1 over open parentheses negative i n close parentheses squared space open square brackets e to the power of negative i n x end exponent close square brackets subscript 0 superscript pi close parentheses$