the 3rd half range FS is Periodic For the function ƒ(x) =I € (0, 1)€ [₁,1)0 if x Esin (27x) if x =coefficients) of all the three half-range FS.calculate the general terms (the

The function on is defined as follows.We need to find all half range Fourier series.We know that…

Answered: For the function f(x) = T € [0, 1 ) sin…

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

the 3rd half range FS is Periodic

For the function ƒ(x) =
I € (0, 1)
€ [₁,1)
0 if x E
sin (27x) if x =
coefficients) of all the three half-range FS.
calculate the general terms (the

Expert Solution

Step 1: Introduction

The function $f open parentheses x close parentheses$ on $stretchy left square bracket 0 comma 1 stretchy right parenthesis$ is defined as follows.

$f open parentheses x close parentheses equals open curly brackets table row 0 cell text if end text space x element of stretchy left square bracket 0 comma 1 half stretchy right parenthesis end cell row cell sin open parentheses 2 straight pi straight x close parentheses end cell cell text if end text space x element of stretchy left square bracket 1 half comma 1 stretchy right parenthesis end cell end table close$

We need to find all half range Fourier series.

We know that the half range Fourier sine series of a function $f open parentheses x close parentheses$ defined on $open square brackets 0 comma L close square brackets$ is written as follows.

$f open parentheses x close parentheses equals sum from n equals 1 to infinity of b subscript n sin open parentheses fraction numerator n straight pi straight x over denominator L end fraction close parentheses$ , where $b subscript n equals 2 over L integral subscript 0 superscript L f open parentheses x close parentheses s in open parentheses fraction numerator n straight pi straight x over denominator L end fraction close parentheses d x$ for all $n greater or equal than 1$ .

We know that the half range Fourier cosine series of a function $f open parentheses x close parentheses$ defined on $open square brackets 0 comma L close square brackets$ is written as follows.

$f open parentheses x close parentheses equals a subscript 0 over 2 plus sum from n equals 1 to infinity of a subscript n cos open parentheses fraction numerator n straight pi straight x over denominator L end fraction close parentheses$ , where $a subscript 0 equals 2 over L integral subscript 0 superscript L f open parentheses x close parentheses d x$ and $a subscript n equals 2 over L integral subscript 0 superscript L f open parentheses x close parentheses cos open parentheses fraction numerator n straight pi straight x over denominator L end fraction close parentheses d x$ for all $n greater or equal than 1$ .