"Sum equals integral" identities similar to $\int_0^1 \frac{dx}{x^x} = \sum_{n = 1}^{\infty} \frac{1}{n^n}$

Solution 1:

Several papers are dedicated to the subject of integrals of functions that equal the sum of the same function, primarily for estimation purposes.

Boas and Pollard (1973) has some interesting sum-integral equalities:

$$\pi/\alpha=\sum_{n=-\infty}^\infty \frac{\sin^2 (c+n)\alpha}{(c+n)^2}=\int_{-\infty}^\infty \frac{\sin^2 (c+n)\alpha}{(c+n)^2}\, \text{d}n$$

$$\pi\operatorname{sgn} a=\sum_{n=-\infty}^\infty \frac{\sin (n+c)\alpha}{n+c}=\int_{-\infty}^\infty \frac{\sin (n+c)\alpha}{n+c}\, \text{d}n$$

It also gives several general formulae for functions that suffice:

$$\sum_{n=-\infty}^\infty f(n)=\int_{-\infty}^\infty f(n) \, \text{d}n$$

mainly with Fourier analysis.

This paper gives an equality with the Bessel J function:

$$\int_{-\infty}^\infty \frac{J_y (at) J_y(bt)}{t}\, \text{d}t=\sum_{t=-\infty}^\infty \frac{J_y (at) J_y(bt)}{t}$$

and some more references:

There have been a number of studies of this kind of sum-integral equality by various groups, for example, Krishnan & Bhatia in the 1940s (Bhatia & Krishnan 1948; Krishnan 1948a,b; Simon 2002) and Boas, Pollard & Shisha in the 1970s (Boas & Stutz 1971; Pollard & Shisha 1972; Boas & Pollard 1973).

See also Surprising Sinc Sums and Integrals which has some other equalities. This paper also states that (paraphrasing)

If $G$ is of bounded variation on $[−\delta, \delta]$, vanishes outside $(−α, α)$, is Lebesgue integrable over $(−α, α)$ with $0 < α < 2\pi$ and has a Fourier transform of $g$, then

$$\sum_{n=-\infty}^\infty g(n)=\int_{-\infty}^\infty g(x)\, \text{d}x+\sqrt{\frac{\pi}{2}}(G(0-)+G(0+))$$

Ramanujan's second lost notebook contains some sums of functions that equal the integral of their functions (Chapter 14, entries 5(i), 5(ii), 16(i), 16(ii)).

If you want, even more references with examples are in the papers I have mentioned.