Question

Q7 Find the Fourier transform of
f(x) = 1/√x

Answer 1

Answer:

Fourier Transform of 1|x|√

complex-analysis fourier-analysis fourier-transform

I want to find the fourier transform of 1|x|√. I checked the table of common fourier transforms in Wikipedia, and I know the answer should be

2π|ω|−−−√

What I can't find out, however, is why that is the answer.

I tried

$f^(ω)=∫∞−∞1|x|−−√e−iωxdx</p><p> \\ =∫∞01x−−√e−iωxdx+∫∞01x−−√eiωx$

but that just gives me two unsolvable exponential integrals.

I also tried finding the answer \\ through residue calculus, as the function has a single singularity at 0, which yields

$f^(ω)=2πi Resz=0e−iωz|z|−−√=2πilimz→0(e−iωz)=2πi</p><p>What am I doing wrong? Or a$

✍ Hope it's helpful to you ✍

Answer 2

Step-by-step explanation:

Fourier Transform of 1|x|√

complex-analysis fourier-analysis fourier-transform

I want to find the fourier transform of 1|x|√. I checked the table of common fourier transforms in Wikipedia, and I know the answer should be

2π|ω|−−−√

What I can't find out, however, is why that is the answer.

I tried

\begin{gathered}f^(ω)=∫∞−∞1|x|−−√e−iωxdx < /p > < p > \\ =∫∞01x−−√e−iωxdx+∫∞01x−−√eiωx\end{gathered}

f

(

ω)=∫∞−∞1∣x∣−−√e−iωxdx</p><p>

=∫∞01x−−√e−iωxdx+∫∞01x−−√eiωx

but that just gives me two unsolvable exponential integrals.

I also tried finding the answer \\ through residue calculus, as the function has a single singularity at 0, which yields

f^(ω)=2πi Resz=0e−iωz|z|−−√=2πilimz→0(e−iωz)=2πi < /p > < p > What am I doing wrong? Or af

(

ω)=2πiResz=0e−iωz∣z∣−−√=2πilimz→0(e−iωz)=2πi</p><p>WhatamIdoingwrong?Ora

✍ Hope it's helpful to you ✍