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
Here is a link to the free book street mathematics pdf file. and the question is in page 15 under the chapter "easy cases".
https://drive.google.com/file/d/15zDYRIy5W9xpcKfkWhv1i-ipRP_fz606/view
It's easy to show that
$$I=\int_{-\infty}^{\infty}\mathrm{e}^{-\alpha x^2}\mathrm{d}x=\sqrt{\frac{\pi}{\alpha}}.$$
Now notice that,
$$\alpha=0 \to I=\infty $$
$$\alpha=1 \to I=\sqrt{\pi} \tag{1}$$
$$\alpha=\infty \to I=0 $$
Among the three choices $$A)~\sqrt{\pi}/\alpha$$ $$B)~1+(\sqrt{\pi}-1)/\alpha$$ $$C)~1/\alpha^2+(\sqrt{\pi}-1)/\alpha$$
only in $A$ and $C$ the relations $(1)$ are valid.