# Definite integral calculation with poles at $0$ and $\pm i\sqrt{3}$

$$\int_0^\infty \frac{\sin(2\pi x)}{x(x^2+3)} \, dx$$

I looked at $$\frac{e^{2\pi i z}}{z^{3}+3z}$$, also calculated the residues, but they don't get me the right answer. I used that $$\int_{-\infty}^\infty f(z)\,dz = 2\pi i (\sum \operatorname{Res} z_r) + \pi i \operatorname{Res}_0$$, but my answer turns out wrong when I check with wolframalpha.

Residue for $$0$$ is $$1$$, for $$z=\sqrt{3}i$$ it's $$-\frac{e^{-2\pi}}{2}$$ . . .

In a worse attempt I forgot $$2\pi$$ and used $$z$$ only (i.e. $$\frac{e^{iz}}{z^{3}+3z}$$) and the result was a little closer, but missing a factor of 2 and and $$i$$.

Can anyone see the right way? Please do tell.

• How do you get the residue at 0 to be 1? Nov 20, 2011 at 4:12
• Once you calculated the residue of $f(z) = \frac{e^{2\pi i z}}{z(z^2+3)}$ at $z = 0$ and $z = i\sqrt{3}$, you will get $$\int_{-\infty}^{\infty} = 2\pi i \operatorname{Res}_{z=i\sqrt{3}} f(z) + \pi i \operatorname{Res}_{z=0} f(z).$$ Then the actual answer must be $\frac{1}{2i} \int_{-\infty}^{\infty} f(z) \; dz$. I guess this is what you have missed. Nov 20, 2011 at 5:23
• Hey sos440, I do get the right answer with $\frac{1}{2i}$, the $\frac{1}{2}$ comes from the integration boundaries, and the $\frac{1}{i}$ because sine is the imaginary part of $e^{iz}$, right?
– VVV
Nov 20, 2011 at 13:05
• @VVV , what contour are you taking for your calculations?? Also, after the limit when $\,R\to\infty\,$ (or whatever) is done, we still remain with the real part of the complex function ($\,cos 2\pi x\,$ instead of $\,\sin 2\pi x\,$, which does not converge on $\,(0,\infty)\,$... Also, you seem to be taking a contour that contains only the positive imaginary pole and not the negative one. Jul 7, 2012 at 0:48
• @sos440 , what contour are you taking for your calculations? and why the integral is multiplied by $\,1/2i\,$ and not by $\,1/2\pi i\,$ ? Aug 20, 2012 at 2:39

We make the following lemma:

Lemma. Suppose $$f(z)$$ is holomorphic near $$z = z_0$$. Fix $$\theta_0 \in (0, 2\pi)$$. If $$\gamma_\varepsilon$$ denotes a counter-clockwise oriented arc of angle $$\theta_0$$ on the circle of radius $$\varepsilon$$ centered at $$z_0$$, then $$\lim_{\varepsilon\to0} \int_{\gamma_\varepsilon} \frac{f(\zeta)}{\zeta-z_0}\;d\zeta=i\theta_0 f(z_0).$$

Proof. By the substitution $$\zeta = z_0 + \varepsilon e^{i\theta}$$, we have \begin{align*} \left| \int_{\gamma_\varepsilon} \frac{f(\zeta)}{\zeta-z_0}\;d\zeta - i\theta_0 f(z_0)\right| &= \left| i \int_{\theta_1}^{\theta_1+\theta_0} f(z_0 + \varepsilon e^{i\theta})\;d\theta - i\theta_0 f(z_0)\right| \\ & \leq \int_{\theta_1}^{\theta_1+\theta_0} \left| f(z_0 + \varepsilon e^{i\theta}) - f(z_0) \right| \;d\theta, \end{align*} which clearly goes to zero when $$\varepsilon \to 0$$.

As a corollary, if $$f(z)$$ has a simple pole at $$z = z_0$$ then with the same notation as in the Lemma, we have

$$\lim_{\varepsilon\to0} \int_{\gamma_\varepsilon} f(\zeta) \;d\zeta=i\theta_0 \operatorname{Res} \{ f(z), z_0 \}.$$

Now let $$C$$ be the upper-semicircular contour of radius $$R \gg 1$$ with a small semicircular indent of radius $$\varepsilon \ll 1$$ at the origin. Let us write $$C$$ as $$C = \Gamma_{R} + L_{\varepsilon,R} - \gamma_\varepsilon,$$ where $$\Gamma_R$$ and $$\gamma_\varepsilon$$ denote counter-clockwise oriented arcs corresponding to the outer circle and the inner circle of $$C$$, respectively, and $$L_{\varepsilon,R}$$ denote the remaining union of two lines on $$C \cap \Bbb{R}$$. Now for

$$f(z) = \frac{e^{2\pi i z}}{z(z^2 + 3)},$$

the integral in question, which we denote as $$I$$, is equal to

$$I = \frac{1}{2i} \lim_{{\varepsilon \to 0 \atop R \to \infty}} \int_{L_{\varepsilon, R}} f(z) \; dz.$$

Now, by Cauchy integration formula,

$$\oint_{C} f(z) \; dz = 2\pi i \operatorname{Res} \{ f(z), i\sqrt{3} \}.$$

This means that

$$\int_{L_{\varepsilon, R}} f(z) \; dz = 2\pi i \operatorname{Res} \{ f(z), i\sqrt{3} \} + \int_{\gamma_\varepsilon} f(z) \; dz - \int_{\Gamma_R} f(z) \; dz.$$

Taking limit $$\varepsilon \to 0$$ and $$R \to \infty$$, the integral $$\int_{\Gamma_R} f(z) \; dz$$ vanishes by Jordan's lemma. Thus by our lemma,

$$\lim_{{var\epsilon \to 0 \atop R \to \infty}} \int_{L_{\varepsilon, R}} f(z) \; dz = 2\pi i \operatorname{Res} \{ f(z), i\sqrt{3} \} + \pi i \operatorname{Res} \{ f(z), 0 \}.$$

But since

$$\operatorname{Res} \{ f(z), i\sqrt{3} \} = \left. (z-i\sqrt{3})f(z) \right|_{z=i\sqrt{3}} = -\frac{1}{6}e^{-2\pi \sqrt{3}}$$

and

$$\operatorname{Res} \{ f(z), 0 \} = \left. z f(z) \right|_{z=0} = \frac{1}{3},$$

we have

$$I = \frac{1}{2i} \left[ 2\pi i \left(-\frac{1}{6}e^{-2\pi \sqrt{3}}\right) + \pi i \left(\frac{1}{3} \right) \right] = \frac{\pi}{6}\left(1 - e^{-2\pi \sqrt{3}} \right).$$

• +1 Very nice answer! Not only was I answered but I also learned about a rather beautiful, and hopefully pretty useful, generalization of Cauchy's Integral Theorem. Thanks a lot, @sos440 Aug 22, 2012 at 1:55
• Could someone help me with the proof of the corollary? Jan 13, 2019 at 0:01

\begin{align} \int_0^\infty\frac{\sin(2\pi x)}{x(x^2+3)}\,\mathrm{d}x &=\frac12\int_{-\infty}^\infty\frac{\sin(2\pi x)}{x(x^2+3)}\,\mathrm{d}x\tag{1}\\ &=\frac12\int_{i-\infty}^{i+\infty}\frac{\sin(2\pi z)}{z(z^2+3)}\,\mathrm{d}z\tag{2}\\ &=\frac12\int_{i-\infty}^{i+\infty}\frac{e^{i2\pi z}-e^{-i2\pi z}}{2iz(z^2+3)}\,\mathrm{d}z\tag{3}\\ &=\frac1{4i}\int_{\gamma_+}\frac{e^{i2\pi z}}{z(z^2+3)}\mathrm{d}z -\frac1{4i}\int_{\gamma_-}\frac{e^{-i2\pi z}}{z(z^2+3)}\mathrm{d}z\tag{4}\\ &=\frac{2\pi i}{4i}\frac{e^{-2\pi\sqrt3}}{i\sqrt3(i\sqrt3+i\sqrt3)} +\frac{2\pi i}{4i}\left(\frac13+\frac{e^{-2\pi\sqrt3}}{-i\sqrt3(-i\sqrt3-i\sqrt3)}\right)\tag{5}\\ &=\frac\pi6\left(1-e^{-2\pi\sqrt3}\right)\tag{6} \end{align} where $\gamma_+$ passes from $i-R$ to $i+R$ then circles back counterclockwise around the upper half-plane, and where $\gamma_-$ passes from $i-R$ to $i+R$ then circles back clockwise around the lower half-plane.

$\gamma_+$ contains the singularity at $i\sqrt3$.

$\gamma_-$ contains the singularities at $0$ and $-i\sqrt3$.

Explanation of steps

$(1)$ integrand is an even function.

$(2)$ there are no singularities for the integrand in the rectangle with corners $i-R,i+R,R,-R$ and the integral over the ends of the rectangle vanishes as $R\to\infty$.

$(3)$ write $\sin(2\pi z)=\dfrac{e^{i2\pi z}-e^{-i2\pi z}}{2i}$

$(4)$ use contour $\gamma_+$ for $e^{i2\pi z}$ and $\gamma_-$ for $e^{-i2\pi z}$ so that the integrand vanishes over the large circles in the upper and lower half-planes.

$(5)$ use residues to compute the integrals over $\gamma_+$ and $\gamma_-$.

$(6)$ simplification.

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ \begin{align} &\color{#00f}{\large% \int_{0}^{\infty}{\sin\pars{2\pi x} \over x\pars{x^{2} + 3}}\,\dd x}= \half\int_{-\infty}^{\infty}{\sin\pars{2\pi x} \over x\pars{x^{2} + 3}}\,\dd x =\pi\int_{-\infty}^{\infty} {1 \over x^{2} + 3}\,\pars{\half\int_{-1}^{1}\expo{2\pi\ic k x}\,\dd k}\,\dd x \\[3mm]&={\pi \over 2}\int_{-1}^{1}\dd k \int_{-\infty}^{\infty}{\expo{2\pi\ic kx} \over x^{2} + 3}\,\dd x \\[3mm]&={\pi \over 2}\int_{-1}^{1}\bracks{% \Theta\pars{-k}\pars{-2\pi\ic}\, {\expo{2\pi\ic k\pars{-\ic\root{3}}} \over -2\ic\root{3}} +\Theta\pars{k}\pars{2\pi\ic}\, {\expo{2\pi\ic k\pars{\ic\root{3}}} \over 2\ic\root{3}}}\,\dd k \\[3mm]&={\pi \over 2}\int_{-1}^{1}{\pi \over \root{3}}\, \expo{-2\pi\root{3}\verts{k}}\,\dd k ={\root{3} \over 3}\,\pi^{2}\int_{0}^{1}\expo{-2\pi\root{3}k}\,\dd k \\[3mm]&={\root{3} \over 3}\,\pi^{2}\,{\expo{-2\pi\root{3}} - 1 \over -2\pi\root{3}} =\color{#00f}{\large{1 \over 6}\,\pi\pars{1 - \expo{-2\pi\root{3}}}} \end{align}

• What is $\Theta$ in this context? Apr 3, 2022 at 18:25
• @user170231 Heaviside Step Function. Apr 4, 2022 at 11:44

Given that to solve Integral

$$\int_{0}^{\infty}\frac{\sin(2\pi x) }{x(x^2+3)}dx$$

Assume that $$\mathcal{I}(a)=\int_{0}^{\infty}\frac{\sin(ax)}{x(x²+1)}dx \to (1)$$ here $$a$$ is positive real

Differentiating $$(1)$$ we get

$$\mathcal{I}'=\int_{0}^{\infty}\frac{\cos(ax) }{x^2+3}dx$$

Now it is easy to use complex analysis by taking function $$f(z) =\frac{e^{\iota az}}{z^2+3}$$ and using Jordan inequality we will get $$\mathcal{I}'(a) =\frac{\pi}{2\sqrt{3}}e^{-\sqrt{3}a}$$

Now integrate the expression

$$\mathcal{I}(a) =-\frac{\pi}{6}e^{-a\sqrt{3}}+C$$

Where $$C$$ is constant of integration Since $$\mathcal{I}(0) =0$$ then $$C=\frac{\pi}{6}$$ Combine every thing put the value of a and we are done

Resolving $$\frac{1}{x(x^2+1)}$$ into partial fractions yields \begin{aligned} \int_{0}^{\infty} \frac{\sin (2 \pi x)}{x\left(x^{2}+3\right)} d x &=\frac{1}{3}\left( \int_{0}^{\infty} \frac{\sin (2 \pi x)}{x} dx-\int_{0}^{\infty} \frac{x^{2} \sin (2 \pi x)}{x^{2}+3} dx \right)\\ &=\frac{1}{6}\left(\pi-\int_{-\infty}^{\infty} \frac{x \sin (2 \pi x)}{x^{2}+3} d x\right) \end{aligned}

By Jordan’s Lemma, \begin{aligned} & \int_{-\infty}^{\infty} \frac{x \cos (2 \pi x)}{x^{2}+3} d x+i \int_{-\infty}^{\infty} \frac{x \sin (2 \pi x)}{x^{2}+3} d x \\ =& 2 \pi i \operatorname{Res}\left(\frac{z e^{2 \pi zi}}{z^{2}+3}, z=\sqrt 3 i\right)\\ =& 2 \pi i \frac{\sqrt{3} i e^{-2 \sqrt{3} \pi}}{2 \sqrt{3} i} \\ =& \pi ie^{-2 \sqrt{3} \pi}, \end{aligned}

using contour integration along anti-clockwise direction of the path $$\gamma=\gamma_{1} \cup \gamma_{2} \textrm{ where } \gamma_{1}(t)=t+i 0(-R \leq t \leq R) \textrm{ and } \gamma_{2}(t)=R e^{i t} (0

Comparing the imaginary parts of both sides gives $$\int_{-\infty}^{\infty} \frac{x \sin (2 \pi x)}{x^{2}+3} d x = \pi e^{-2 \sqrt{3} \pi}$$

Hence $$\int_{0}^{\infty} \frac{\sin (2 \pi x)}{x\left(x^{2}+3\right)} d x = \frac{\pi}{6}\left(1-e^{-2 \sqrt{3} \pi}\right)$$