# Tagged Questions

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### Integral sign with circle (AND arrow on the circle) through it

I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path. So when I encountered in complex analysis the above integral sign but with ...
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### How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?

I need to solve the to following integral: $$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$ I tried this integral in Mathematica, but it was not able to solve it. ...
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### Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$

Can this integral be solved with contour integral or by some application of Residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$ It has two ...
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### Problem involving the computation of the following integral

I was solving the past exam papers and stuck on the following problem: Compute the integral $\displaystyle \oint_{C_1(0)} {e^{1/z}\over z} dz$,where $C_1(0)$ is the circle of radius $1$ around ...
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### Use the Residue Theorem to evaluate the following integral:

$$\int_{-∞}^{∞} \frac{x^4}{1+x^8} dx$$ I've found the zeros in the upper half plane to be $$e^{i \pi/8}, e^{i 3 \pi/8}, e^{i 5 \pi/8}, e^{i 7 \pi/8}$$ (right?) But then the calculation got really ...
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### Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.

Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$. The contour is ...
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### Evaluation of the contour integral $\int_\beta \frac{e^z}{e^z-\pi} dz$

Suppose $\beta$ is a loop in the annulus $\{z:10<\left|z\right|<12\}$ that winds $N$ times about the origin in the counterclockwise direction, where $N$ is an integer. Determine the value of ...
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### Is it possible to convert this circular contour into a different one?

It's fairly well known that we can eliminate all traces of a variable $x$ by contour integration. For instance, suppose we have a function: $$f(x) = c_0 x^0 + c_1 x^1 + c_2 x^2 + \dots +c_n x^n$$ ...
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### Is this Wick rotation correct? (self-intersecting closed contour)

I wonder if what is shown in figure 9.1 here is correct? Doesn't the contour self-intersect, i.e. it's not a simple closed curve hence the Residue theorem shouldn't apply to this closed contour, ...
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### theorems on analytic extension

I am a probability student and I have forgotten much of complex analysis I have learnt when I was an undergraduate. I have recently seen integrals evaluated using analytic extensions techniques when ...
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### Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$

How would I go about evaluating this integral? $$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
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### Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$

Evaluate: $\displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem. Given hint: consider $f(z) = \ln ( 1 +z)$. EDIT:: I know how to evaluate it, but I am ...
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### Evaluating the (complex) integral $\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$ using residues.

I am trying to evaluate the following integral. $$\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$$ where $\gamma$ is the path $\cos(t)+2i\sin(t)$ for $0\leq t <4\pi$. So, $\gamma$ is an ellipse ...
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### Complex analysis, evaluating a path integral

Evaluate the integral $\int_{\gamma}e^{z^2}+ \overline{z} \ \ dz,$ where $\gamma$ is the positively oriented unit circle.
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### Is always $\mathbb{D} \subset U$ in Poisson's integral formula?

Is always $\mathbb{D} \subset U$ in Poisson's integral formula? I mean for example for following example: Let $U=\{z|\operatorname{Im}(z)>0\}$ represent corresponding Poisson integral formula ...
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### Evaluate $\int\limits_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx$

I would like to show that $$\text{PV}\int_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx = \frac{\pi}{2}\mathrm{sech}(b)$$ using complex analysis. $a$ and $b$ are real numbers and $a \neq b$. ...
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### Calculating $\int_{C^+(2,2)} \frac {e^\sqrt z} {(z-2)^2}dz$ and $\int_{0}^{\infty} \frac 1 {1+x \sqrt x}dx$

I want to calculate $$\int_{C^+(2,2)} \frac {e^\sqrt z} {(z-2)^2}dz\quad\mbox{and}\quad\int_{0}^{\infty} \frac 1 {1+x \sqrt x}dx$$ using complex integration. In the first part $\sqrt z$ denotes the ...
### Explain how this reveals the number of roots in $f(z)$
If this takes place in a simple and smooth closed curve $\gamma$, that doesn't cross itself explain why for the polynomial, f(x), $\frac{1}{2\pi i}\int_\gamma\frac{f'(x)}{f(x)} dx$ tells you the ...