Testing several alternatives
For the Gaussian integral $$\int_{-\infty}^{\infty}{e^{-\alpha x^2}}dx,$$ use the three easy-cases tests to evaluate the following candidates for its value.
(a) $\sqrt{\pi}/\alpha\quad$ (b) $1+(\sqrt{\pi} - 1)/\alpha\quad$ (c) $1/\alpha^2 + (\sqrt{\pi} - 1)/\alpha$.
I am using the the book called street mathematics to learn more about dimensional analysis. I am trying to understand a problem in the book. The question is to use dimensional analysis to find the solutions for Gaussian integral.
I tried to understand the question and how to best tackle it but I did not succeed.
The three easy cases referred in the question are:
- when a = ∞
- when a = 0
- when a = 1
what I am trying to understand: is all I have to do put the value (a) in the equation and solve for the integral?
Hint
I think you don't have tools to compute this integral. But if you replace $\alpha \in\{0,\infty ,1\}$ in the solutions you have, which solution could make sense ? For example, for $\alpha =\infty $ the integrand is null. What could be a possible answer ? Then continue like this and try to deduce the correct solution.