# tough integral involving $\sin(x^2)$ and $\sinh^2 (x)$

I ran across this integral I get no where with. Can someone suggest a method of attack?.

$$\int_0^{\infty}\frac{\sin(\pi x^2)}{\sinh^2 (\pi x)}\mathrm dx=\frac{2-\sqrt{2}}{4}$$

I tried series, imaginary parts, and so forth, but have made no progress.

Thanks very much.

• If it's any consolation, Mathematica can't solve "Integrate[Sin[Pix^2]/Sinh[Pix]^2, {x,0,Infinity}]", which means it's not easy.
– user2469
Commented Sep 3, 2011 at 16:53
• @barrycarter: Almost two years later, Mathematica still can't get this one.
– robjohn
Commented Jul 30, 2013 at 22:04
• I love it when a brilliant human mind evaluates an integral the fancy math engines can not.:)
– Cody
Commented Jul 31, 2013 at 11:15
• @Turing $22nov2020$: No yet. Commented Nov 22, 2020 at 5:01
• @FelixMarin This is why I love the (human) brain! Commented Nov 22, 2020 at 17:57

It can be done using contour integration and the calculus of residues.

Sketch: Integrate $$f(z) = \frac{e^{i\pi z^2} e^{\pi z}}{\sinh^2 (\pi z) \cosh(\pi z)}$$ around a rectangular contour with corners at $\pm R$ and $\pm R + i$ and with semicircular indentations of radius $\epsilon$ to avoid the poles at $0$ and $i$, take imaginary parts and let $R\to\infty$, $\epsilon\to 0^+$.

You'll need to use $$f(x)-f(x+i)=\frac{2 e^{i \pi x^2}}{\sinh^2(\pi x)}$$ together with $$\operatorname*{res}_{z=0} \, f(z) = \operatorname*{res}_{z=i} \, f(z) = \frac{1}{\pi}$$ (since these will each contribute $-i \pi$ times the residue in the limit $\epsilon \to 0^+$) and $$\operatorname*{res}_{z=i/2} \, f(z) = \frac{-1+i}{\pi\sqrt{2}}.$$

• Thanks everyone for your contributions and help.
– Cody
Commented Sep 4, 2011 at 11:42
• I wonder if something involving the differentiation of an integral could be done... Commented Sep 4, 2011 at 14:55
• @J. M.: I half-heartedly tried something like that first, but I didn't get anywhere. I'm not Feynman... ;-) Commented Sep 4, 2011 at 17:30

Although this question is two years old, the integral was mentioned in chat recently, I evaluated it, and then found this question. Since there is no complete solution, although Hans Lundmark's suggestion is excellent and similar in nature, I am posting what I have done.

Contours

Since the integrand is even, \begin{align} \int_0^\infty\frac{\sin(\pi x^2)}{\sinh^2(\pi x)}\,\mathrm{d}x &=\frac12\int_{-\infty}^\infty\frac{\sin(\pi x^2)}{\sinh^2(\pi x)}\,\mathrm{d}x \end{align} Define $$f(z)=\frac{\cos\left(\pi z^2\right)}{\sinh(2\pi z)\sinh^2(\pi z)}$$ Note that because $$f(x\pm i) =\frac{-\cos\left(\pi x^2\right)\cosh(2\pi x)\pm i\sin\left(\pi x^2\right)\sinh(2\pi x)}{\sinh(2\pi x)\sinh^2(\pi x)}\\$$ we have \begin{align} \int_\gamma f(z)\,\mathrm{d}z &=\int_{-\infty}^\infty\big[f(x-i)-f(x+i)\big]\,\mathrm{d}x\\ &=-2i\int_{-\infty}^\infty\frac{\sin(\pi x^2)}{\sinh^2(\pi x)}\,\mathrm{d}x\\ &=2\pi i\times\begin{array}{}\text{the sum of the residues}\\\text{inside the contour}\end{array} \end{align} where $\gamma$ is the contour

$\hspace{3.2cm}$

Therefore, $$\int_0^\infty\frac{\sin(\pi x^2)}{\sinh^2(\pi x)}\,\mathrm{d}x =-\frac\pi2\times\begin{array}{}\text{the sum of the residues}\\\text{inside the contour}\end{array}$$ Residues

near $0$ : \begin{align} f(z) &=\frac{\cos\left(\pi z^2\right)}{\sinh(2\pi z)\sinh^2(\pi z)}\\ &=\frac{1-\frac12\pi^2z^4+O(z^8)}{2\pi z\left(1+\frac23\pi^2z^2+O(z^4)\right)\pi^2 z^2\left(1+\frac13\pi^2z^2+O(z^4)\right)}\\ &=\frac{1-\pi^2z^2}{2\pi^3z^3}+O(z)\\[10pt] &\implies\text{residue}=-\frac1{2\pi} \end{align} at $\pm i/2$, use L'Hosptal : \begin{align} \text{residue} &=\lim_{z\to\pm i/2}\frac{(z\mp i/2)\cos\left(\pi z^2\right)}{\sinh(2\pi z)\sinh^2(\pi z)}\\ &=\frac1{2\pi\cosh(\pm\pi i)}\frac{\cos(-\pi/4)}{\sinh^2(\pm\pi i/2)}\\ &=\frac1{2\pi\cos(\pm\pi)}\frac{\sqrt2/2}{-\sin^2(\pm\pi/2)}\\[4pt] &=\frac{\sqrt2}{4\pi} \end{align} near $\pm i$ : \begin{align} f(z\pm i) &=\frac{-\cos\left(\pi z^2\right)\cosh(2\pi z)\pm i\sin\left(\pi z^2\right)\sinh(2\pi z)}{\sinh(2\pi z)\sinh^2(\pi z)}\\ &=\frac{-\left(1-\frac12\pi^2z^4+O(z^8)\right)\left(1+2\pi^2z^2+O(z^4)\right)+O(z^3)}{2\pi z\left(1+\frac23\pi^2z^2+O(z^4)\right)\pi^2 z^2\left(1+\frac13\pi^2z^2+O(z^4)\right)}\\ &=-\frac{1+\pi^2z^2}{2\pi^3z^3}+O(1)\\[10pt] &\implies\text{residue}=-\frac1{2\pi} \end{align} Result

Thus, \begin{align} \int_0^\infty\frac{\sin(\pi x^2)}{\sinh^2(\pi x)}\,\mathrm{d}x &=-\frac\pi2\left(-\frac1{2\pi}-\frac1{2\pi}+\frac{\sqrt2}{4\pi}+\frac{\sqrt2}{4\pi}\right)\\[6pt] &=\frac{2-\sqrt2}{4} \end{align}

• Wow, that's nice robjohn. It sure doesn't bother me that it is two years old. I always enjoy seeing your clever solutions. I can give an upvote. I hate to take Hand greenie away :)
– Cody
Commented Jul 30, 2013 at 21:17
• No, I wouldn't want to do that. Now, I may change my mind if I find a real-only solution :-)
– robjohn
Commented Jul 30, 2013 at 22:02
• While it is true that $\sinh(z)\approx z$, one has to be careful since $\frac1{z^3}$ has a residue of $0$ at $z=0$, but $\frac1{\sinh^3(z)}$ has a residue of $-\frac12$ at $z=0$. Rather than looking at $\epsilon e^{it}$, look at the power series near the singularity; e.g. near $0$, \begin{align} \frac{e^{i\pi z^2}e^{\pi z}}{\sinh^2(\pi z)\cosh(\pi z)} &=\frac{\left(1+O\left(z^2\right)\right) \left(1+\pi z+O\left(z^2\right)\right)}{\pi^2z^2\left(1+O\left(z^2\right)\right) \left(1+O\left(z^2\right)\right)}\\ &=\frac1{\pi^2z^2}+\frac1{\pi z}+O(1) \end{align}
– robjohn
Commented Aug 2, 2013 at 18:49
• Thus, that function has a residue of $\frac1\pi$ at $z=0$. If the contour looks similar to mine, but encompasses $0$ and misses $i$, then we only need worry about the sum of the residues at $0$ and $i/2$. However, you are right, and he shouldn't be considering half the residue at points where the singularity is greater than degree 1.
– robjohn
Commented Aug 2, 2013 at 18:51
• \begin{align} &\lim_{\epsilon\to0}\mathrm{Im}\int_\pi^0\left(\color{#C00000} {\frac1{\pi^2\epsilon^2} e^{-2it}}\color{#00A000} {+\frac1{\pi\epsilon} e^{-it}}\right)\epsilon ie^{it}\,\mathrm{d}t\\ =&\lim_{\epsilon\to0}\mathrm{Im}\int_\pi^0\left(\color{#C00000} {\frac1{\pi^2\epsilon} e^{-it}}\color{#00A000} {+\frac1{\pi}}\right)i\,\mathrm{d}t\\ =&\lim_{\epsilon\to0}\mathrm{Im}\left(\color{#C00000}{\frac{-2}{\pi^2\epsilon}} \color{#00A000}{-i}\right)\\ =&\color{#C00000}{0}\color{#00A000}{-i} \end{align}
– robjohn
Commented Aug 2, 2013 at 20:11

Not sure if anyone is interested in another solution after all those years, but anyway: I managed to evaluate the integral in a slightly different manner that doesn't depend on me coming up with a cleverly constructed function to integrate. However, cleverness is needed elsewhere in the process, so it's not much simpler than the other two solutions.

Define $$\displaystyle f(z) = \frac{\mathrm{e}^{\mathrm{i}\pi z^2}}{\sinh^2 \pi z}$$ and consider the two contours shown on the following sketch:

$$f(z)$$ has poles in all $$\mathrm{i}n$$ for $$n \in \mathbb Z$$; some of them are shown with red asterisks. Here the two rectangles extend to $$\pm\infty$$ on the sides, the little circles around the poles have radius $$r > 0$$ and the three horizontal lines are exactly the same, just shifted by $$\pm \mathrm{i}$$.

The residue of $$f(z)$$ in zero is obviously zero, because the Laurent series will contain only even powers. To calculate the residue in $$z = \mathrm i$$, write $$w = z-\mathrm i$$ and consider the fact that $$\sinh^2 z$$ is $$\mathrm{i}$$-periodic to obtain

$$\frac{\mathrm{e}^{\mathrm{i} \pi (w^2 + 2\mathrm{i} w - 1)}}{\sinh^2 \pi w} = - \frac{1}{\pi^2 w^2} \frac{\mathrm{e}^{\mathrm{i} \pi w^2}}{(1 + \text{even powers of w})^2} (1 - 2\pi w + \cdots)$$

The only way to get the minus first power is to take the $$-2\pi w$$ in the last parentheses and 1's in the expansions of the other two functions. Hence the residue is $$2/\pi$$.

Now denote $$\displaystyle I = \int_K f(x)\,\mathrm{d} x$$, where $$K$$ is the horizontal line with a little circle around 0. Writing the residue theorem for the top and the bottom contour on the sketch, we get

\begin{align*} \int_K f(x)\,\mathrm{d} x - \int_K f(x+\mathrm{i})\,\mathrm{d} x &= 2\pi\mathrm{i} \times \frac{2}{\pi} &\implies&& \int_K \frac{\mathrm{e}^{\mathrm{i} \pi x^2} \mathrm{e}^{-2\pi x}}{\sinh^2 \pi x}\,\mathrm{d} x &= 4\mathrm{i} - I \\ \int_K f(x)\,\mathrm{d} x - \int_K f(x-\mathrm{i})\,\mathrm{d} x &= 0 &\implies&& \int_K \frac{\mathrm{e}^{\mathrm{i} \pi x^2} \mathrm{e}^{2\pi x}}{\sinh^2 \pi x}\,\mathrm{d} x &= - I \\ \end{align*}

Add the two equations together and subtract $$\displaystyle 2 \int_K \frac{\mathrm{e}^{\mathrm{i}\pi x^2}}{\sinh^2 \pi x}\,\mathrm{d} x = 2I$$ to get $$\int_K \frac{\mathrm{e}^{\mathrm{i} \pi x^2} (\mathrm{e}^{2\pi x} - 2 + \mathrm{e}^{-2\pi x})}{\sinh^2 \pi x}\,\mathrm{d} x = 4\mathrm{i} - 4I.$$ However, $$\sinh^2 \pi x = \frac14 (\mathrm{e}^{\pi x} - \mathrm{e}^{-\pi x})^2$$, so the parenthesis in the numerator is $$4\sinh^2 \pi x$$. The sinh's cancel and we get $$4 \int_K \mathrm{e}^{\mathrm{i} \pi x^2}\,\mathrm{d} x = 4\mathrm{i} - 4I$$.

Taking the imaginary part of the last equality, sending $$r \to 0$$ to get rid of the small circle around the pole in $$0$$ (we couldn't do this earlier because the real part of the integral over the little circle actually diverges; however, the residue is 0, so, once we get rid of the divergence, there will be no contribution) and dividing by 8, we get $$\frac12 \int_{-\infty}^\infty \sin \pi x^2\,\mathrm{d} x = \frac12 - \frac12 \int_{-\infty}^\infty \frac{\sin \pi x^2}{\sinh^2 \pi x}\,\mathrm{d} x.$$ Both integrands are even, so we can write $$\int_{-\infty}^\infty = 2 \int_0^\infty$$. The integral on the left is one of the well-known Fresnel integrals and it is equal to $$\frac{1}{2\sqrt2}$$, so we truly obtain the result

$$\int_0^\infty \frac{\sin \pi x^2}{\sinh^2 \pi x}\,\mathrm{d}{x} = \frac12 - \frac{1}{2\sqrt2} = \frac{2-\sqrt2}{4}.$$

• $\displaystyle +1$. Pretty fine !!!. Commented May 8 at 19:53

I would write the $\sin(x^2)$ as $(e^{ix^2}-e^{-ix^2})/2i$ and the sinh as $(e^{ x}-e^{-x})/2$. Then I'd maybe put the integrand in the form of $(e^{p_1(x)}+e^{p_2(x)}+\cdots)^{-1}+(e^{p_3(x)}+e^{p_4(x)}+\cdots)^{-1}+\cdots$ where $p_i(x)$ are polynomes with complex coefficients. I have no clue if that helps, to be honest.

Another idea would be partial integration after multiplying with 1, like: $\int\mathrm dx 1\cdot f(x)= xf(x)-\int\mathrm dx \; x\cdot f'(x)$ Sometimes this helps to handle a $x^2$ in the argument of a complicated function.

Numeric answer would be possible to get using the following tools:

1. Riemann integration
2. root-finding algorithm for the equation
3. some limit sequence for the infinity giving better and better approximations

Riemann integration is needed to calculate F(x). Basically you'll need a root-finding algorithm that works with G : R->R functions, and gives a single x as solution. Just move the constant to the other side to get F(x)-F(0)-c=0. with G(x)=F(x)-F(0)-c. The infinity will break the riemann integration, so you'll need a sequence like { G(a_1)=0, G(a_2)=0, G(a_3)=0, ... } to get better and better approximations with a_1,a_2,a_3, ... sequence increasing towards infinity. The result then looks like {x_1, x_2,x_3,...} sequence which contains the values of x coming from root-finding algorithm.

But there could be better ways to solve this problem...

EDIT: there is problems with this solution. Namely, the a_i is a constant, not a variable, so root-finding might not be needed after all. All I get is approximation of 0=0.