1. ## fourier transform

Hello all,

I have a question regarding fourier transforms. I am considering the function $\displaystyle f_{1}$ defined as:

$\displaystyle f_{1}(x)=e^{-x}\chi_{[0,1]}$

Now I would like to find the transform of $\displaystyle f_{1}$. I have done the following:

$\displaystyle (\digamma f_{1})(\gamma)=\digamma(f_{1}(\gamma))=\int_{0}^{1 }e^{-x}\chi_{[0,1]}e^{-2\pi ix\gamma}dx=\int_{0}^{1}e^{-x}\cdot 1\cdot e^{-2\pi ix\gamma}dx=\int_{0}^{1}e^{x}e^{-1-2\pi i\gamma}$

Treating $\displaystyle e^{-1-2\pi i\gamma}$ as a konstant we get:

$\displaystyle \int_{0}^{1}e^{x} e^{-1-2\pi ix\gamma}dx=e^{-1-2\pi ix\gamma} \int_{0}^{1}e^{x}dx=\underline{e^{-1-2\pi ix\gamma}(e-1)}$

Could someone please run through the above and see if it is correct?

Thank you very much.

2. Originally Posted by surjective
Hello all,

I have a question regarding fourier transforms. I am considering the function $\displaystyle f_{1}$ defined as:

$\displaystyle f_{1}(x)=e^{-x}\chi_{[0,1]}$

Now I would like to find the transform of $\displaystyle f_{1}$. I have done the following:

$\displaystyle (\digamma f_{1})(\gamma)=\digamma(f_{1}(\gamma))=\int_{0}^{1 }e^{-x}\chi_{[0,1]}e^{-2\pi ix\gamma}dx=\int_{0}^{1}e^{-x}\cdot 1\cdot e^{-2\pi ix\gamma}dx=\int_{0}^{1}e^{x}e^{-1-2\pi i\gamma}$
Certainly not: $\displaystyle e^{-x}e^{-2\pi i x\gamma}=e^{-x(1+2i\pi\gamma)}$. Go on from there.

3. Hello,

Yes of course. Stupid mistake. Thanks.