Deriving the fourier transform

[iScience]

Homework Statement

derive the Fourier sine and cosine transforms of $$f(x) = e^{-cx}$$ by using $$e^{iax}=cos(ax)+isin(ax)$$ and computing the integral $$\int_0 ^{\infty} e^{-cx}e^{iax}dx$$.

Homework Equations

The Attempt at a Solution

i'm completely clueless, all i did was evaluate what they told me to.

$$\int_0 ^{\infty} e^{-cx}e^{iax}dx = \int_0 ^{\infty} e^{(ia-c)x}dx$$
$$= \frac{e^{(ia-c)x}}{ia-c}\Big|_0^{\infty} = \frac{cos(ax)+isin(ax)}{ia-c}\Big|_0^{\infty} =-\frac{1}{ia-c}$$

 

[mfb]

If you split your last result in imaginary and real part, you can relate it to the integrals in your other thread.