# Tagged Questions

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

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### Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
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### Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
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### Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
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### Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral ...
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### Exchange series and integral in a complex context.

Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1}$$ if $k>1/2$ ...
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### Complex line integral over a square

Evaluate following complex line integral.Let $c=\{z|\max\{|\text{Re}(z)|,|\text{Im}(z)|\}=1\}$ be the square with orientation $+1$. Calculate $$\int_c \frac{z\ dz}{\cos(z)-1}$$ with ...
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### Find the complex integral over a path

I have to integrate the complex function $$z+1/z$$ which is parameterized by $\gamma(t), 0 \le t\le 1$ and satisfies $Im\gamma(t) > 0$, $\gamma(0) = -4+i$ and $\gamma(1) = 6+2i$. Can I assume the ...
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### complex analysis2 [closed]

let $f=u+iv$ be an entire function, and suppose that its imaginary part $v$ is non negative in the upper half plane but is equal to zero at all points of the real axis. (a) prove that ...
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### Solution to nonlinear ODE with square root

How do I solve the following equation? $\dot{x}=\sqrt{x^{2}-\frac{2}{3}x^{3}}$ with $x(0)=0$? I'm guessing I have to work with $dt=\frac{dx}{\sqrt{x^{2}-\frac{2}{3}x^{3}}}$ and integrate in [0,t'] ...
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### How do you compute a complex exterior derivative?

The context is deriving cauchy riemann equations using green's/stoke's theorem. The function is the complex function $f(x,y)=u(x,y)+iv(x,y)$ with associated one form $u(x,y)dx+iv(x,y)dy$. Here is my ...
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### Solving an integral by Cauchy Formula

I want to solve the integral $$\oint_{|z|=\frac{1}{2}}{\frac{e^{1-z}}{z^3(1-z)}dz}$$ Its a long time ago that I solved such integrals. Is it just by definition of the line integral? Maybe someone can ...
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### Find all possible values of the integral

Find all possible values of $\displaystyle I= \int_C \frac{dz}{1+z^2}$, where $C$ is a curve with initial point $0$ and final point $1$ that does not meet the poles of $\dfrac{1} {1+z^2}$. It looks ...
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### Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
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### Is integrating $e^{iz^{2}}$, along the real axis in the complex plane the same as integrating the riemann integral of $e^{x^2}$?

In the title, $z\in \mathbb{C}$ and $x\in\mathbb{R}$. More specific to my problem, I am hoping that $\int_{0}^{R}e^{iz^{2}}dz=\int_{0}^{R}e^{x^{2}}dx$. Maybe this is obvious but I want to make sure. ...
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### Lower bound on the distance fom a point to the border of a region.

I'm trying to prove the following result in complex analysis: If $f$ is an analytic and bijective function from the unit disc to an open connected region $A$ then the distance from $f(0)$ to the ...
I am trying to evaluate the integral $I=\int_0^\infty e^{-ix^2}\,dx$ as one component of evaluating a contour integral but I am dropping a factor of $1/2$ and after checking my work many times, I ...