Tani Nevins-Klein

I woke up this morning thinking about vector spaces. If that puts you off, this post is not for you. Sorry, Lin. Hello, Stocke.

I had read some Dummit and Foote from a dubiously sourced PDF the night prior, and my mind was spinning. After stumbling to my shelf and getting a copy of Axler, I start messing with an integral that defines an inner product on the vector space of polynomials, $\mathcal{P}(x)$. $$\langle p, q\rangle = \int_0^\infty p(x)q(x) e^{-x} dx,\quad p,q\in \mathcal{P}(x)$$ By the linearity of the integration operator: $$\begin{align*} &\int_0^\infty (a_nx^n + \cdots + a_0x^0) e^{-x} dx \\= a_n &\int_0^\infty x^n e^{-x} dx + \cdots + a_0 \int_0^\infty x^0 e^{-x} dx \end{align*}$$

Something about that $\int_0^\infty x^n e^{-x}$ seems kind of familiar, no? I didn't think so. So I proceeded, taking care to integrate by parts and evaluate my limits properly,

$$\begin{align*}\int_0^\infty x^n e^{-x} dx = -x^ne^{-x} \Bigr]_0^\infty + n \int_0^\infty x^{n-1} e^{-x}dx\\\implies \int_0^\infty x^{n+1} e^{-x} dx = n \int_0^\infty x^n e^{-x} dx \\\end{align*}$$

What a fascinating relationship! I wonder what happens when we consider that integral as a function of $n$? Maybe we should give it a name? (spoiler: $\Gamma(n)$ seems like good a name as any).

I then proceeded to spend the next 20 minutes checking my work and investigating the properties of this strange new function. It was when I began an induction proof that this mysterious function was equal to $n!$ for integers that it dawned on me what I was doing.

Lesson learned: don't do math sleep deprived. One connection trumps a dozen messy computations.