Fourier Series Time Multiplication

[er0ck]

I am stuck at one point in this problem (which is the main step):

The original problem is to find the Fourier Transfer of t^(n-1)e^(-$$\alpha$$*t)u_h

and I know that e^(-$$\alpha$$t)u_h(t) = 1/($$\alpha$$+j$$\omega$$)

I plug that into the general Time multiplication property and I get:

d^(n-1)/d$$\omega$$^(n-1)*[1/($$\alpha$$+j$$\omega$$)]
This is where I'm stuck.

I have the answer is equal to (n-1)!/($$\alpha$$+j$$\omega$$^2)^n
I apologize for the bad formating, this is my first time using physics forum. =/ Thanks in advance!

 

[lippyka]

To solve this, you will need to use the definition of the Fourier Transform. Specifically, you will need to apply the integration by parts technique to the integral: \int_{-\infty}^{\infty} t^{n-1}e^{-\alpha t}u_h(t) e^{-j\omega t} dtThe integration by parts technique will help isolate the t^(n-1) term in the integrand, which can then be combined with the 1/(\alpha + j\omega) factor. The details of this technique can be found in any textbook on Fourier Analysis.