# How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?

We were told today by our teacher (I suppose to scare us) that in certain schools for physics in Soviet Russia there was as an entry examination the following integral given

$$\int\limits_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x\,,$$

where $a \in \mathbb{R}$, $b \in [0,2)$, and $\mathfrak{i}$ is the imaginary unit. And since we are doing complex analysis at the moment, it can, according to my teacher, be calculated using complex methods.

I was wondering how this could work? It seems hard to me to find a good curve to apply the residue theorem for this object, I suppose. Is there a trick to compute this integral?

• A keyhole contour. On $\mathbb{C}\setminus [0,+\infty)$, write $z^{-ia} = \exp(-ia \log z)$. Then $\lvert z^{-ia}\rvert \leqslant \exp (2\pi\lvert a\rvert)$. Nov 20, 2017 at 19:14
• @ThePirateBay: Mathematica has to be trained to consider the right constraints and perform the correct simplifications. The answer is elementary, but it is not that easy to find. Nov 20, 2017 at 19:21
• @MathematicianByMistake: I totally agree. With some experience in converting series into integrals (and viceversa) via $\mathcal{L}/\mathcal{L}^{-1}$ such integral is a rather innocent-looking one, but do they teach the Mellin transform in Russian high schools? Nov 20, 2017 at 19:36
• @MathematicianByMistake: In Soviet Russia, integral tackles you! Nov 21, 2017 at 0:22
• Yes I showed this question to my mother. She chuckled and a tear of nostalgia came to her eye. Thanks for this. Nov 21, 2017 at 0:37

$\phantom{}$ Dear MSE users, this is the new episode of Mister Feynman and Monsieur Laplace versus contour integration.

Tonight we have a scary integral, but we may immediately notice that $$\mathcal{L}(x^{-ia})(s) = s^{ia-1}\Gamma(1-ia),\qquad \mathcal{L}^{-1}\left(\frac{1}{x^2+bx+1}\right)(s) =\frac{e^{Bs}-e^{\overline{B}s}}{\sqrt{b^2-4}}$$ where $B$ is the root of $x^2+bx+1$ with a positive imaginary part. By the properties of the Laplace transform, the original integral is converted into $$\frac{\Gamma(1-ia)}{\sqrt{b^2-4}}\int_{0}^{+\infty}s^{ia-1}\left(e^{Bs}-e^{\overline{B}s}\right)\,ds$$ which can be evaluated in terms of the $\Gamma$ function.
Due to the reflection formula, the final outcome simplifies into $$\frac{\left(B^{-i a}-\overline{B}^{-i a}\right) \pi }{\sinh(\pi a)\sqrt{4-b^2}}$$ and we may notice that $B=\exp\left[i\arccos\frac{b}{2}\right]$ allows a further simplification.

Outro: poor children.

• This is a truly impressive answer. Are you sure that you're human? Nov 21, 2017 at 4:35
• @BobaFret Do you have evidence than any nonhumans could produce an answer of this quality? Nov 21, 2017 at 5:25
• +1 for a great opening line. And the rest of the answer is remarkable. Nov 21, 2017 at 8:36
• This is exactly the approach I take with most daunting integrals, in the hopes it one day works. Seems one does get lucky sometimes :)
– JPhy
Nov 21, 2017 at 14:49
• @ThePirateBay: just done :) Nov 21, 2017 at 16:58

$\newcommand{\Res}{\text{Res}}$ Firstly define: \begin{align} f(z) = \frac{1}{z^2+bz+1} \end{align} Secondly define: \begin{align} g(z)= (-z)^{-ia}f(z) \end{align} We use the Principal Log to define $(-z)^{-ia}$. The main reason for the minus sign is that I want to work with the principal Log. Consider now the keyhole contour $K_R$ consisting of $C_R \cup C_R^+ \cup C_R^-$. The circle part is $C_R$ with radius $R$, and $C_R^+$ is the segment that connects $0$ to $R$ on the right half plane from above and $C_R^-$ the same as $C_R^+$ but from below.

So our contour looks like the one below:

Note that $\int_{C_R} g(z)dz \to 0$ as $R\to \infty$ (why?). Now the poles of this function $g(z)$ is at $z_1= -\frac{1}{2}b - i\ \sqrt[]{1-\frac{1}{4}b^2}$ and $z_2 = -\frac{1}{2}b + i\ \sqrt[]{1-\frac{1}{4}b^2}$. Note that for large enough $R$ both poles will be in the area enclosed by our contour.

On $C_R^+$ we have: \begin{align} \lim_{R\to \infty} \int_{C_R^+} g(z) dz = e^{-a\pi}\int^{\infty}_0\frac{x^{-ia}}{x^2+bx+1} dx \end{align} Similarly on $C_R^-$ we have: \begin{align} \lim_{R\to \infty} \int_{C_R^-} g(z) dz = -e^{a\pi}\int^{\infty}_0\frac{x^{-ia}}{x^2+bx+1} dx \end{align} So: \begin{align} \lim_{R\to\infty} \int_{K_R} g(z) dz &= (-e^{a\pi} + e^{-a\pi} )\int^\infty_0 \frac{x^{-ia}}{x^2+bx+1} dx \\ &=-2\sinh(a\pi) \int^\infty_0 \frac{x^{-ia}}{x^2+bx+1} dx \end{align} On the other hand we have by the Residue theorem: \begin{align} \lim_{R\to\infty} \int_{K_R} g(z) dz = 2\pi i\left( \Res_{z=z_1}g(z)+\Res_{z=z_2}g(z)\right) \end{align} Let's calculate the residues: \begin{align} \Res_{z=z_1} g(z) = \frac{\left(\frac{1}{2}b + i\ \sqrt[]{1-\frac{1}{4}b^2}\ \right)^{-ia}}{-2i \ \sqrt[]{1-\frac{1}{4}b^2} } \end{align} The other one: \begin{align} \Res_{z=z_2} g(z) = \frac{\left(\frac{1}{2}b - i\ \sqrt[]{1-\frac{1}{4}b^2}\ \right)^{-ia}}{2i \ \sqrt[]{1-\frac{1}{4}b^2} } \end{align} Define $\beta:=\frac{1}{2}b + i\ \sqrt[]{1-\frac{1}{4}b^2}$. So: \begin{align} \lim_{R\to\infty} \int_{K_R} g(z) dz = \pi \frac{\bar\beta^{-ia}-\beta^{-ia}}{\sqrt[]{1-\frac{1}{4}b^2}} = 2\pi \frac{\bar\beta^{-ia}-\beta^{-ia}}{ \sqrt[]{4-b^2}} \end{align} This means: \begin{align} \int^{\infty}_0 \frac{x^{-ia}}{x^2+bx+1}dx &= \frac{2\pi}{-2\sinh(a\pi)} \frac{\bar\beta^{-ia}-\beta^{-ia}}{ \sqrt[]{4-b^2}} \end{align} Note that $\beta = \exp(i\arccos(b/2))$. So $\beta^{-ia}=\exp(a\arccos(b/2))$ and $\bar\beta^{-ia}=\exp(a\arccos(b/2)).$ Substituting $\beta$ gives us the final result:

\begin{align} \int^{\infty}_0 \frac{x^{-ia}}{x^2+bx+1}dx = \color{red}{\frac{2\pi\sinh(a\cdot\arccos(b/2))}{\sinh(a\pi) \ \sqrt[]{4-b^2}} } \end{align}

This integral is real.

Remark

It may be interesting to note the following. For positive $x$ one has: $x^{-ia}=e^{-ia\text{Log}(x)}=e^{-ia\ln(x)}=\cos(a\ln(x))-i\sin(a\ln(x))$. So we have: \begin{align} \int^\infty_0 \frac{x^{-ia}}{x^2+bx+1} dx = \int^\infty_0 \frac{\cos(a\ln(x))}{x^2+bx+1} dx - i\int^\infty_0 \frac{\sin(a\ln(x))}{x^2+bx+1} dx \end{align} Since our integral is real we get the following identities for free: \begin{align} \color{blue}{\int^\infty_0 \frac{\cos(a\ln(x))}{x^2+bx+1} dx = \frac{2\pi\sinh(a\cdot\arccos(b/2))}{\sinh(a\pi) \ \sqrt[]{4-b^2}}} \end{align} And: \begin{align} \color{blue}{\int^\infty_0 \frac{\sin(a\ln(x))}{x^2+bx+1} dx = 0 } \end{align}

• The last integral is $0$ and it can be proved by splitting the integral over $[0,1]$ and $[1,\infty)$ and putting $t=1/x$ in second integral. Nov 21, 2017 at 8:39
• I'd assume this was the expected answer. Nov 21, 2017 at 10:57
• @nbubis yeah it seems that I went for the answer on "I was wondering how this (complex method) could work?" - OP and Jack for the answer on "Is there a trick to compute this integral?" - OP. What a fantastic man! Nov 21, 2017 at 11:27
• Although I love Jack's answer, this answer is exactly what the OP was asking for. I'm "voting" for this one! Nov 24, 2017 at 16:28

In a complex analysis class, you’re more likely to come across the integral $$\int_{0}^{\infty} \frac{x^{\alpha}}{1+2x \cos \beta +x^{2}} \, dx = \frac{\pi \sin (\alpha \beta)}{\sin(\alpha \pi) \sin(\beta)}, \quad (-1<\alpha <1, \ 0 < \beta < \pi).$$

See this answer, for example, which uses a semicircle in the upper half-plane that is indented at the origin.

Alternatively, you could also use the keyhole contour that Shashi used.

Also see this question about the peculiar symmetry of this integral when it's written in a slightly different form.

The function on the right is not defined at $\alpha =0$. But you could show separately (by using the same contour or by completing the square) that the value of the integral at $\alpha =0$ is $$\lim_{a \to 0} \frac{\pi \sin (\alpha \beta)}{\sin(\alpha \pi) \sin(\beta)} = \frac{\beta}{\sin \beta}.$$

Now if we assign the function on the right the value $\frac{\beta}{\sin \beta}$ at $\alpha =0$, then right side of the equation is a holomorphic function for $-1 <\operatorname{Re}(\alpha) <1$ with $\beta$ fixed.

And since the integral on the left is absolutely convergent in the strip $-1 < \operatorname{Re}(\alpha) < 1$, we can use a property of the Mellin transform that states that the integral defines an holomorphic function in that strip.

(This is very similar to property of the Laplace transform mentioned here, and can be proved in essentially the same manner.)

So by the identity theorem, the formula holds for $-1 <\operatorname{Re}(\alpha) <1$.

Your integral is the case $\alpha=-ia$ and $\cos(\beta) = \frac{b}{2}$.

If $b\in [0, 2)$, then $\beta$ falls between means $0$ and $\pi$, and we get

\begin{align} \int_{0}^{\infty} \frac{x^{-ia}}{x^{2}+bx+1} \, dx &= \frac{\pi \sin\left(-ia \arccos\left(\frac{b}{2} \right)\right)}{\sin(-ia \pi) \sin \left(\arccos \left(\frac{b}{2} \right) \right)} \\ &= \frac{\pi \sinh \left(a \arccos\left(\frac{b}{2} \right) \right)}{\sinh(a \pi)\frac{\sqrt{4-b^{2}}}{2}}. \end{align}