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One of my favorite functions is $S(x)=\frac{\sin(x)}{x},$ where we set $S(0)=1$ (the continuous extension). This function solves the Basel Problem- with some assumptions- and does other cool stuff.

I noticed it has the following property:

$$\sum_{n=-\infty}^{\infty}S(n)=\int_{-\infty}^{\infty} S(x)dx=\pi.$$

You can prove the LHS using Fourier series and you can prove the RHS using the Residue Theorem, or some very clever methods.

My question: Are there other nontrivial functions with this property? That is, aside from linear combinations of $S$, are there other elementary functions $f$ that satisfy $$\sum_{n=-\infty}^{\infty}f(n)=\int_{-\infty}^{\infty} f(x)dx\;?$$

Integrand
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2 Answers2

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If $g$ and $h$ are functions such that these sums and integrals exist, say $\sum_n g(n) = a$, $\sum_n h(n) = b$, $\int g(x)\; dx = c$, $\int h(x)\; dx = d$, but $a \ne c$ and $b \ne d$, then you might try linear combinations $f(x) = s g(x) + t h(x)$.
This satisfies your equation if $s a + t b = s c + t d$, i.e. $$\frac{s}{t} = \frac{d-b}{a-c}$$

Robert Israel
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$S(ax)$ works for any $a$. So any of their linear combinations. You may add some odd functions of course.

I've checked for functions of form $exp(-a x^2)$, but the integral is always larger than the sum. Something interesting here, especially as $a \searrow 0$.

I wonder what happens for functions of form $exp(- a t^2 + b t)$, with $a$, $b$ complex, $a$ positive real part. The sums are "theta series", and the integrals are easy. You might have some equalities there.

orangeskid
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