Fourier Transform Properties: Deriving TF{∫f(x)dx} = -iF(k)/k + πF(0)δ(k)

[xoureo]

Homework Statement

If [tex]F(k)=TF\{f(x)\},k\neq 0[/tex] where TF is the Fourier transform ,and

[tex]F(0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(u)du\neq 0[/tex] ,

show that

[tex]TF\{\int_{-\infty}^{x}f(u)du\}=-i \frac{F(k)}{k} +\pi F(0)\delta(k)[/tex]

Homework Equations

The Attempt at a Solution

I attempt the following:

[tex]TF\{\int_{-\infty}^{x}f(u)du\}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \{\int_{-\infty}^{x}f(u)du\}e^{-ikx}dx[/tex]

and , integration by parts, with

[tex]w=\int_{-\infty}^{x}f(u)du[/tex]

[tex]dw=f(x)dx[/tex]

give

[tex]\frac{1}{\sqrt{2\pi}}\{\int_{-\infty}^{x}f(u)du \frac{i}{k} e^{-ikx}\}|_{-\infty}^{\infty} -\frac{1}{\sqrt{2\pi}} \frac{i}{k}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx[/tex]

The last term is

[tex] -\frac{i}{k}F(k)[/tex]

What should i do with the first?. I do this, but it probably be bad:

[tex]\frac{i}{k} e^{-ikx}\}|_{-\infty}^{\infty}=\int_{-\infty}^{\infty}e^{-ikx}dx = 2\pi\delta(k)[/tex]

and

[tex]\frac{1}{\sqrt{2\pi}}\left[\lim_{x\rightarrow\infty}\int_{-\infty}^xf(z)dz-\lim_{x\rightarrow -\infty}\int_{-\infty}^xf(z)dz\right] = F(0) [/tex]

Then,

[tex]
\int_{-\infty}^{\infty}\frac{dx}{\sqrt{2\pi}}\int_{-\infty}^xf(z)dze^{-ikx}=-i\frac{F(k)}{k}+2\pi F(0)\delta(k)
[/tex]

where is an extra 2.

I tried to do this problem using convolution, but again i can't achieve the desired result:

[tex](f*g)(x)=\int_{-\infty}^{\infty}f(u)g(x-u)du[/tex]

If [tex]g(x-u)[/tex] is the Heaviside function, and with [tex]x-u>0[/tex] , the convolution is

[tex](f*g)(x)=\int_{-\infty}^{\infty}f(u)g(x-u)du=\int_{-\infty}^{x}f(u)du[/tex]

Then,

[tex]TF\{\int_{-\infty}^{x}f(u)du\}=TF\{(f*g)(x)\}=\sqrt{2\pi}F(k)G(k)[/tex]

But, the Fourier transform of the Heaviside function is:

[tex]TF\{g(x)\}=\frac{-i}{k\sqrt{2\pi}}+\sqrt{\frac{\pi}{2}}\delta(k)[/tex]

Hence,


[tex]TF\{\int_{-\infty}^{x}f(u)du\}=\frac{-iF(k)}{k}+\pi F(k)\delta(k)[/tex]


But, in the last term appears [tex]F(k)[/tex] , and i want to get [tex]F(0)[/tex]. What is wrong??


Thanks

 

[Dick]

There's nothing wrong with your convolution argument. Think what happens if you integrate a test function against F(k)*delta(k). You get the same thing as if you integrate it against F(0)*delta(k) if F is continuous.F(k)*delta(k) and F(0)*delta(k) are the same distribution.

 

[xoureo]

Ok,i understand it now, thanks for the help