Once when lecturing in class Lord Kelvin used
the word ‘mathematician’ and then interrupting himself asked his class:
’Do you know what a mathematician is?’
Stepping to his blackboard he
wrote upon it the above equation.
Then putting his finger on what he had written, he turned to
his class and said, ‘A mathematician is one to whom that is as obvious
as that twice two makes four is to you.’
** Two interesting ways to arrive at the Gaussian Integral
Woah… The backlash that Lord Kelvin got after this post was just phenomenal.
There are many ways to obtain this integral (click here to know about other methods) , but here are two interesting ways to arrive at the Gaussian Integral which you may/may not have seen and may/may not be easy to follow.
Gamma Function to the rescue
If you know about factorials (5!= 188.8.131.52.1), you know that they make sense only for integers.
But Gamma function
extends this to non-integers values. This integral form allows you to
calculate factorial values such as (½)!, (¾)! and so on.
The same can be used to evaluate the Gaussian Integral as follows:
Differentiating under the Integral sign
In this technique known as ‘Differentiating under the integral sign’, you choose an integral whose boundary values are easy integrals to evaluate.
Here I(0) and I(∞); and differentiate with respect to a parameter β instead of the variable x to obtain the result.