Problem:
Consider the functions $\{\exp[\frac{-x^2}{2}]x^n\}$ on the full line $(-\inf, +\inf)$, show that the functions are normalizable.
What I Have Done:
A function is normalizable if its norm is finite. From pg. 8 and pg. 9 of "Mathematics for Quantum Chemistry (2005, J.M. Anderson)", the norm of a function is
\begin{equation} N(f) = \langle f | f \rangle = \int_a^b f^*f \end{equation}
Therefore, I must show that $N(f_n)$ for $f_n=\exp[\frac{-x^2}{2}]x^n$ is finite $\forall n$
I first set up the integral. Since, $f^* = f$
\begin{align} N(f_n) &= \int_{-\infty}^{+\infty}{(\exp[\frac{-x^2}{2}]x^n)(\exp[\frac{-x^2}{2}]x^n) \ dx} \\ &= \int_{-\infty}^{+\infty}{\exp[-x^2]x^{2n} \ dx} &\text{Using } e^a e^b=e^{a+b} \text{ and } x^a x^b=x^{a+b} \end{align}
Then, I can use a property of even functions to help simplify this integral further. If $f(x) = f(-x)$ a function is even. If $-f(x) = f(-x)$ then a function is odd (from Even and Odd Functions).
\begin{align} u(x) &= \exp[-x^{2}] & u(x) &= u(-x) &\text{even exponent} \\ g(x) &= x^{2n} & g(x) &= g(-x) &\text{even exponent } \forall n \end{align}
Then the product of two even functions is an even function, therefore the integral simplifies to
\begin{equation} N(f_n) = 2\int_{0}^{+\infty}{\exp[-x^2]x^{2n} \ dx} \tag{My Final Integral} \end{equation}
My book (pg. 30 of "Mathematics for Quantum Chemistry") suggests changing My Final Integral to
\begin{equation} \int_0^{+\infty}{\exp[-y]y^{n-\frac{1}{2}} \ dy} \tag{Book Suggestion} \end{equation}
presumably by letting $y = x^2$, but I am not sure how this helps.
For clarity, the substitution to achieve the equation the book suggests occurs like this:
\begin{align} y &= x^2 \\ y^{1/2} &= x \\ \frac{dy}{dx} &= 2x \\ \frac{dy}{dx} &= 2y^{1/2} \\ \frac{dy}{2y^{1/2}} &= dx \\ 2\int_{0}^{+\infty}{\exp[-x^2]x^{2n} \ dx} &= 2\int_0^{+\infty}{\exp[-y]y^{n} \ \frac{dy}{2y^{1/2}}} \\ &= \int_0^{+\infty}{\exp[-y]y^{n}y^{-1/2} \ dy} \\ &= \int_0^{+\infty}{\exp[-y]y^{n-\frac{1}{2}} \ dy} \end{align}
Question:
How can I simplify My Final Integral further and then evaluate it with the given bounds? I think I can use the Gaussian Integral and/or Euler's Integral, but I am not sure how to get the integrand into this form. My intuition is integration by parts, but the $n$ term is throwing me off.
\begin{align} \int_{-\infty}^{+\infty}{\exp[-a(x + b)^2] \ dx} &= \sqrt{\frac{\pi}{a}} \tag{Gaussian Integral} \\ \int_{0}^{+\infty}{\exp[-x^2] \ dx} &= \sqrt{\pi} \tag{Euler's Integral} \end{align}