(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is an example provided by my lecturer in his notes. He puts practically zero working in.

When i work the problem through i do not get the same answer as he does.

In this section i have copied the exact text from the problem:

Find the Fourier transform of cos(x). Your answer will include delta functions

We shall use the deﬁnition [tex]\delta(k)=\int\limits_\infty^\infty dx \ e^{ikx}[/tex]

and

[tex]\delta(k)=\delta(-k)[/tex]

he then simply writes:

[tex]Fourier[cos(x)]=\frac{1}{\sqrt{2\pi}}\int\limits_\infty^\infty dx \ e^{-ikx}\ \frac{e^{ix}+e^{-ix}}{2}=\sqrt{\frac{\pi}{2}}\left[\delta(k-1)+\delta(k+1)\right][/tex]

As you can imagine this makes me very irritated so I start to solve it myself

2. Relevant equations

[tex]Fourier[f(x)]=\frac{1}{\sqrt{2\pi}}\int\limits_\infty^\infty dx \ e^{-ikx}\ f(x)[/tex]

[tex]\delta(k)=\int\limits_\infty^\infty dx \ e^{ikx}[/tex]

3. The attempt at a solution

So i work it through line by line:

[tex]Fourier[cos(x)]=\frac{1}{\sqrt{2\pi}}\int\limits_\infty^\infty dx \ e^{-ikx}\ \frac{e^{ix}+e^{-ix}}{2}[/tex]

[tex]Fourier[cos(x)]=\frac{1}{2\sqrt{2\pi}}\int\limits_\infty^\infty dx \ e^{-ikx}\left(\ e^{ix}+e^{-ix}\right)[/tex]

[tex]Fourier[cos(x)]=\frac{1}{2\sqrt{2\pi}}\int\limits_\infty^\infty dx \ e^{-ikx+ix}+e^{-ikx-ix}[/tex]

[tex]Fourier[cos(x)]=\frac{1}{2\sqrt{2\pi}}\int\limits_\infty^\infty dx \ e^{ix(-k+1)}+e^{ix(-k-1)}[/tex]

[tex]Fourier[cos(x)]=\frac{1}{2\sqrt{2\pi}}\int\limits_\infty^\infty dx \ e^{ix(-k+1)}+\frac{1}{2\sqrt{2\pi}}\int\limits_\infty^\infty dx \e^{ix(-k-1)}[/tex]

now recognise

[tex]\delta(k)=\int\limits_\infty^\infty dx \ e^{ikx}[/tex]

can be used. However the definition for the fourier transform shows that the constant should be:

[tex]\frac{1}{2\pi}[/tex]

therefore we need a constant such that [tex]\frac{1}{2\sqrt{2\pi}}*x=\frac{1}{2\pi}[/tex]

this implies x = [tex]\frac{2}{\sqrt{2\pi}}[/tex]

this leads to:

[tex]Fourier[cos(x)]=\frac{1}{2\sqrt{2\pi}}x\frac{2}{\sqrt{2\pi}}\ \delta(-k+1)+\frac{1}{2\sqrt{2\pi}}x\frac{2}{\sqrt{2\pi}}\ \delta(-k-1)[/tex]

[tex]Fourier[cos(x)]=\frac{1}{2\pi}\ \delta(-k+1)+\frac{1}{2\pi}}\ \delta(-k-1)[/tex]

then recognise [tex]\delta(k)=\delta(-k)[/tex]

[tex]Fourier[cos(x)]=\frac{1}{2\pi}\ \delta(k-1)+\frac{1}{2\pi}}\ \delta(k+1)[/tex]

This is of course not:

[tex]

Fourier[cos(x)]=\sqrt{\frac{\pi}{2}}\left[\delta(k-1)+\delta(k+1)\right]

[/tex]

The constant is incorrect.

I am certain that he is correct and that I am wrong.

Why have I got it wrong?

Thanks very much for your time.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Fourier transform of cos x the answer involves δ functions

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