# Fourier coefficients

Compute the sine coefficients for $f(x)=e^{-x^{2}}$ on the interval $[0,2\pi]$. Does this mean $f(x+2\pi k)=f(x)$, $k\in\mathbb{Z}$? Can $x\in[0,\infty)$?

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LCKurtz
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Compute the sine coefficients for $f(x)=e^{-x^{2}}$ on the interval $[0,2\pi]$. Does this mean $f(x+2\pi k)=f(x)$, $k\in\mathbb{Z}$? Can $x\in[0,\infty)$?

You have to know what periodic function you are expanding in a FS before you can calculate the Fourier coefficients. Do you mean to calculate the coefficients for the odd periodic extension of ##f(x)##? Do you know what the graph of the odd periodic extension would look like? Does that answer your last question?

It says on the interval $[0,2\pi]$. Does this mean $f(x+2\pi k)=f(x)$, $k\in\mathbb{Z}$?

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. A real-valued function $f(x)$ of a real variable is called periodic of period $T>0$ if $f(x+T) = f(x)$ for all $x\in\mathbb{R}$.

So, can $x\in[0,\infty)$?

LCKurtz
Homework Helper
Gold Member
It says on the interval $[0,2\pi]$. Does this mean $f(x+2\pi k)=f(x)$, $k\in\mathbb{Z}$?

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. A real-valued function $f(x)$ of a real variable is called periodic of period $T>0$ if $f(x+T) = f(x)$ for all $x\in\mathbb{R}$.

So, can $x\in[0,\infty)$?

I think you have answered your own question. Of course if a function is defined on ##[0,2\pi]## and periodic with period ##2\pi## it is defined for all x. But if you want to calculate the coefficients for your f(x) you need to know what periodic extension you are using. That's why I asked you if you mean to use the odd periodic extension of ##e^{-x^2}## to calculate the coefficients.

I need to use the the half-range sine expansion. Correct?

I think you have answered your own question. Of course if a function is defined on ##[0,2\pi]## and periodic with period ##2\pi## it is defined for all x. But if you want to calculate the coefficients for your f(x) you need to know what periodic extension you are using. That's why I asked you if you mean to use the odd periodic extension of ##e^{-x^2}## to calculate the coefficients.

I need to use the the half-range sine expansion. Correct? However, the problem does not state that the function is periodic, nor that it is defined on $[0,2\pi]$. After all, the Gaussian function is not periodic. My instructor said that I should only consider $[0,\infty)$, but then this would not satisfy the definition of a periodic function.

LCKurtz
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I need to use the the half-range sine expansion. Correct? However, the problem does not state that the function is periodic, nor that it is defined on $[0,2\pi]$. After all, the Gaussian function is not periodic. My instructor said that I should only consider $[0,\infty)$, but then this would not satisfy the definition of a periodic function.

That's right. But if you are going to find a FS that represents ##e^{-x^2}## on some interval, you must decide what interval. You must know that to use the appropriate half range formulas. I would suggest you ask your instructor what interval he wants. It can't be ##(0,\infty)## unless you are talking about a Fourier transform, not a FS.

In this case $T=\pi$, but he said that I should work on the positive x axis only. That's what bothers me. You can't do that, right? $f(x)$ would not be periodic and it would not work because a Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.

LCKurtz
In this case $T=\pi$, but he said that I should work on the positive x axis only. That's what bothers me. You can't do that, right? $f(x)$ would not be periodic and it would not work because a Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.