Why do we need integrals in Computer Science?

A comment on previous post asked why we need integrals for computer science. One reason is that combinatorial expressions often have representation in terms of integrals. Consider binomial coefficients. We have the following

$${n \choose \frac{n+d}{2}}=\frac{1}{2 \pi} \int_{-\pi}^\pi e^{-\mathbb{i} q d} (2\cos q)^n dq$$


To see where this comes from, consider that $2 \cos x=e^{-\mathbb{i}\pi}+e^{\mathbb{i}\pi}$ is the generating function for number of one step walks starting at 0. Coefficients give number of those walks ending at 1 and -1. Raise it to $n$ and apply inverse Fourier transform to get number of $n$ step walks ending at at $d$, which has an alternative representation in terms of binomial coefficient.

Here's what the result looks like for arbitrary $d$ with $n=10$




Another way to get binomial coefficients is to note that they form coefficients of power of $x$ in expansion of $(1+x)^n$. Use Cauchy's integral formula to recover the coefficient.

We get the following

$${n \choose k} = \frac{1}{2\pi \mathbb{i}}\oint_c \frac{(1+x)^n}{x^{k+1}}$$

Contour integral is counter-clockwise around some contour that encircles the origin.
For instance, you can use the following contour.



Mathematica notebook