# Tagged Questions

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### Solving an integral (using Cauchy contour integral?)

I need to solve this integral: $$f(t)=\int_0^\infty x^2 \sqrt x \left( e^{a x} -1\right)^{-1/2} \frac{e^{i(b-x)t}-1}{b-x} dx$$ where $a$ and $b$ are real, positive ...
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### Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma$ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
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### $\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
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### Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
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### And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...
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### Contour Integral $\int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$\int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x$$ I am not sure as to how to work with the branch ...
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### Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
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### If $\gamma$ is a path from $0$ to $1$, what do we know about $\displaystyle\int_\gamma\frac{1}{z\pm i}dz$?

Let $\gamma$ denote a path from $0$ to $1$ which doesn't cross $\pm i$. What can we say about $$\int_\gamma\frac{1}{z\pm i}dz$$
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### Calculate $\int_0^\infty\frac{\sin x}xdx$ by integration of a suitable function along given paths [duplicate]

How can I calculate $$\int_0^\infty\frac{\sin x}xdx$$ by integration of a suitable function along the following paths: where $R$ and $\varepsilon$ are the radius of the shown outer and inner ...
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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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### How to compute the following integral involving hyperbolic functions?

I'm thinking about contour integrals, but I'm not sure. Thanks for your attention :) $$\int_{0}^{\infty} \frac {\sinh ax \sinh bx}{\cosh cx} dx \\a,b,c \in \Re$$
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### Manipulation of Cauchy's Integral Formula

$\quad$ Using Cauchy's integral theorem, write down the value of a holomorphic function $f(z)$ where $|z|\lt1$ in terms of a contour integral around the unit circle, $\zeta=e^{i\theta}$. $\quad$ ...
### Show that for $|f(z)| \leq C (|z| + 1)\log(|z| + 1)$, there is an $a$ such that $f(z) = az$
Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and suppose a $C \geq 0$ exists such that \begin{align*} |f(z)| \leq C(|z| + 1) \log(|z| + 1) \end{align*} for all $z \in \mathbb{C}$, where \$\log: ...