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It is quite a well known fact that: $$\int_0^{+\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}$$ also the value of related series is very similiar: $$\sum_{n = 1}^{+\infty} \frac{\sin n}{n} = \frac{\pi - 1}{2}$$ Combining these two identities and using ${\rm sinc}$ function we get: $$\int_{-\infty}^{+\infty} {\rm sinc}\, x \, dx = \sum_{n = -\infty}^{+\infty} {\rm sinc}\, n = \pi$$ What is more interesting is the fact that the equality: $$\int_{-\infty}^{+\infty} {\rm sinc}^k\, x \, dx = \sum_{n = -\infty}^{+\infty} {\rm sinc}^k\, n$$ holds for $k = 1,2,\ldots, 6$. There are some other nice identities with ${\rm sinc}$ where sum equals integral but moving on to other functions we have e.g.: $$\sum_{n = -\infty}^{+\infty} \binom{\alpha}{n} e^{int} = \int_{-\infty}^{+\infty} \binom{\alpha}{x} e^{itx} \, dx = (1+e^{it})^\alpha, \; \alpha >-1$$which is due to Pollard & Shisha.

And finally the identity which is related to the famous Sophomore's Dream: $$\int_0^1 \frac{dx}{x^x} = \sum_{n = 1}^{+\infty} \frac{1}{n^n}$$ Unfortunately in this case the summation range is not even close to the interval of integration.

Do you know any other interesting identities which show that "sum = integral"?

Max Muller
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qoqosz
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    In http://math.stackexchange.com/questions/13543/are-there-other-analytic-functions-with-this-property-of-sinc-function/13566#13566 I showed an approximation to a function that the sum (1 to $\infty$), the sum of its square, the integral ($-\infty$ to $\infty$), and the integral of its square all agree – Ross Millikan Jul 14 '12 at 14:30
  • @RossMillikan If summation of a series would range also from $-\infty$ to $+\infty$ than ${\rm sinc}$ is a nice example. – qoqosz Jul 14 '12 at 18:00
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    I think one should be able (though it's not obvious to me how except for trivial cases) to generate a wide range such identities by finding functions that kill off the sum and remainder terms in the Euler Maclaurin integration formula: http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula – Alex R. Jul 14 '12 at 20:39
  • @Sam: I think any four functions that are properly summable and integrable can make one by taking a proper linear combination. You just sum an integrate the combination and have four equations in four unknowns. I just used the ones I did because the sums and integrals were tractable. – Ross Millikan Jul 14 '12 at 21:50
  • One can imagine that Poisson summation is a formula of this sort, although the reality is more complicated (the sum of Fourier transforms resist being expressed as a single integral). http://en.wikipedia.org/wiki/Poisson_summation_formula –  Jul 15 '12 at 03:32
  • Actually I would be more interested in neat examples rather than general methods for finding such functions. – qoqosz Jul 15 '12 at 07:39
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    Perhaps related: Factorial and exponential dual identities. – Mike Spivey Jul 26 '12 at 02:55

1 Answers1

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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.

Max Muller
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Argon
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    Thank you for your answer. These references are very helpful. – qoqosz Jul 26 '12 at 18:51
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    Excellent answer! But would you be so kind as to update the links to the references most of which are broken? Thank you. – Hans Dec 28 '15 at 18:13