# Tagged Questions

For questions about integration methods that use results from complex analysis and their applications

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### Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$ Any help will be much appreciated.
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### Soving a Complex Integral along a circle

I have a complex integral $$\int_{|z|=r}x \, dz$$ for the positive portion of the circle. I know that this integral seems easy enough but I am having trouble with it, and I'm fairly certain my answer ...
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### How to solve $\int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2}$

I am trying to solve this integral, I think that it could be solve using the complex. $$\int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2}$$
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### question related with cauch'ys inequality

We have a statement that if f(z) is analytic in a domain D and $C = \{z : |z-a| = R\}$ is contained in D. Then, $|f^n(a)|\leq \frac{n!M_R}{R^n}$ where $M_R = \max|f(z)|$ on C. What if the given D ...
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### Differentiating a contour integral

Let $P(z, t)$ be a cubic with the parameter $t$, and consider $$\mathcal{I} = \int_{\gamma(t)} \frac{dz}{\sqrt{P(z, t)}}.$$ Here, $\gamma(t)$ is a contour in the complex plane that encloses any two of ...
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### Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0,$?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0,$$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
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### Evaluate $\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$ where $r>0$

I tried to evaluate $$\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$$ when $r>0$. Clearly $\log z$ is not continuous at $z=-r$. So is this integral meaningful then? Since the function is continuous and ...
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### Function holomorphic except for real line and continuous everywhere is entire

1) I've already shown that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is holomorphic everywhere except for a single point and if it continuous on whole $\mathbb{C}$ then it is entire. It was quite easy. ...
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### Difficulty evaluating complex integral

The integral along the path $\gamma(t)=e^{2ti},\;t\in[0,2\pi]$ is $\begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz \end{equation*}$. I approached this like a real integral in the hopes things ...