Fourier Transform of integral of a signal

[maverick280857]

Hi. I have a question regarding the continuous time Fourier Transform of an input signal:

x(t) \rightarrow X(j\omega)

then

\int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{X(j\omega)}{j\omega} + \pi X(0)\delta(\omega)

but if I want to write it in terms of f = \frac{\omega}{2\pi}, should it be:

\int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{X(j\omega)}{j\omega} + \frac{1}{2}X(0)\delta(f)

How does the \pi get replaced by \frac{1}{2} here?

 

[George Jones]

How does the \pi get replaced by \frac{1}{2} here?[/QUOTE]

For a>0, what does \delta (ax ) equal? Why?

 

[maverick280857]

Oh ok, so

\delta(\omega) = \delta(2\pi f) = \frac{1}{2\pi}\delta(f)

Thanks