# Tagged Questions

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### Showing that $\tan(\pi z) = z$ has exactly three solutions in the strip $|\Re(z)| < 1$

We can't use Rouche's theorem here directly, so we have to apply the argument principle. If $f(z) = \tan(\pi z) - z$ , then $f'(z) = \pi \sec^2(\pi z) - 1$. Choose the rectangle $\Gamma$ with ...
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### What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$\int_{0}^\infty dx\, R(x),$$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
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### Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
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### Scary contour integral, but is also an integral representation for $\Gamma$-function

This problem is supposed to be from an old Acta Mathematica volume I circa 1880's, and is attributed to Bourguet. By using a parabola with its focus on the origin as a contour, show that: ...
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### Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$\int ^1 _0 dx \frac{\ln(x-a)}{x-b}$$ by turning it into something in terms of dilogarithms. But for certain values of $a$ ...
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### Choose appropriate contour for a complex integral

I have a problem to solve integral $$I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}}$$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
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### Integral $\displaystyle \int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$

Inspired by the user @Integrals, I thought I'd find some nice integrals! Especially interesting are those involving $\log \pi$. From Borwein and Devlin's "The Computer as Crucible", pg. 58 - show that ...
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### Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$ [duplicate]

Hi I'm trying to show that $$I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}.$$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
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### Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta.$$ The ...
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### Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
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### LogSine Generating Fn $\int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$\int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta.$$ ...