# Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

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### How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (residue theorem)

How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (although we may use normal real integral to solve, I wonder if contour analysis can also help?) The question offers a hint ...
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### Construct an analytic function $f(z)$ whose real part is $e^x\cos y$.

Construct an analytic function $f(z)$ whose real part is $e^x\cos y$.
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### How do I calculate the definite integral $\int_{-1}^1\frac{\sqrt{1-x^2}}{1+x^2}dx$ using complex variables?

I have tried solving the integral $\oint_{C}\frac{\sqrt{1-z^2}}{1+z^2}dz$ using the upper semi-circle contour; I am getting the poles $z=\pm i$. Only $z=i$ exists within the contour and I have ...
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### Integral Value through Complex Integration (residue theorem)

I'd like to know how to evaluate the integral $$I=\int_0^\infty\frac{e^{-s^2}\sin(s)}{s}\,ds=\frac{\pi}{2}\text{erf}(1/2)$$ through the residue theorem. My first steps were to expand $\sin$ as ...
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### How to calculate this complex integral (with poles on the contour)

I've come up with calculating this complex integral: Compute $\displaystyle\oint_C \dfrac{z}{z^2+4z+3} \mathrm{d}z$, where $C$ is the circle with center -1 and radius 2. The function has a pole, ...
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### Establish an upper bound for $\zeta'(s)\over\zeta(s)$?

Using Perron's formula, I am able to show that $$\psi(x)={1\over2\pi i}\int_{a-i\infty}^{a+i\infty}\left[-{\zeta'(s)\over\zeta(s)}\right]{x^s\over s}\mathrm ds$$ where $\psi(x)$ is the Chebyshev's ...
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### Abel-Plana formula applied to $\frac{1}{(x^2+a^2)^2}$
This is a follow-up on a previous post, where from the comments I realized the remainder term in the Euler-Maclaurin formula applied to $f(x)=\frac{1}{(x^2+a^2)^2}$ does not vanish, as also a ...