4. Exercise §5.2 #9. Let p(x) be a function of period . If (x) = ansin(ne) for all , find theodd coefficients.n=1

Let be a function of period Let We have to calculate the odd coefficients in the Fourier…

Answered: 4. Exercise §5.2 # 9. Let p(x) be a…

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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4. Exercise §5.2 #9. Let p(x) be a function of period . If (x) = ansin(ne) for all , find the
odd coefficients.
n=1

Expert Solution

Step 1: ''Introduction to the solution''

Let $ϕ left parenthesis x right parenthesis space$ be a function of period $straight pi.$

Let $ϕ left parenthesis x right parenthesis equals sum from n equals 1 to infinity of space a subscript n sin left parenthesis n x right parenthesis......... left parenthesis 1 right parenthesis comma for all x element of straight real numbers$

We have to calculate the odd coefficients in the Fourier series(1).

Then, $integral subscript 0 superscript straight pi ϕ left parenthesis x right parenthesis sin left parenthesis m x right parenthesis equals sum from n equals 1 to infinity of space a subscript n space end subscript sin left parenthesis n x right parenthesis sin left parenthesis m x right parenthesis$

$rightwards double arrow integral subscript 0 superscript straight pi ϕ left parenthesis x right parenthesis sin left parenthesis m x right parenthesis equals 1 half sum from n equals 1 to infinity of space open parentheses 2 sin left parenthesis m x right parenthesis sin left parenthesis n x right parenthesis close parentheses........ left parenthesis 2 right parenthesis$