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A:
Please try to use this.
First you need to get Grep Command-Line Tool
Open Command Prompt by right-click My Computer –> Open Command Prompt. In the opened window type cd. hit enter.
Go to
C:\Program Files\Grep\grep.exe
Right-click on grep.exe –> Send To –> Desktop
Now run your code which is working fine.
Q:
How to prove that $L^{\frac{2n}{n-1}}=\int^\infty_{ -\infty}\left(1+\frac{|x|^2}{\pi^2}\right)^{ -\frac{n}{2}}dx$?
As a part of this question, how to prove that $L^{\frac{2n}{n-1}}=\int^\infty_{ -\infty}\left(1+\frac{|x|^2}{\pi^2}\right)^{ -\frac{n}{2}}dx$?
The problem is not so hard but I can not find a method to prove it.
I really appreciate any helps!
A:
First, let me explain why $\sqrt{\pi}$ is involved. By definition, the integral of a function $f$ is equal to $$\int^\infty_{ -\infty} f(x) dx = \frac{1}{\sqrt{\pi}} \int^\infty_0 x f(x) dx.$$
So, using this, we have: $$\int^\infty_{ -\infty} \left(1 + \frac{|x|^2}{\pi^2} \right)^{ -\frac{n}{2}} dx = \frac{1}{\sqrt{\pi}} \frac{2
Endryely – 5th February 2022.
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