# Tagged Questions

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### Integration Error

Sorry if this doesn't make any sense or if I did something obviously wrong, I was just playing around with taylor series' and then I got stuck. I know from the geometric series that: ...
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### An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
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### Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
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### Complex integral and parametrization of a circle

I am trying to compute the following integral of $$\int \frac{1}{z^3+3} dz$$ over a circle of radius $2$, centerd at $(2,0)$. Thus I am trying to compute the residue and have found that the function ...
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### Calculating a complex integral by rewriting as a contour integral on |z|=1.

I need to show that $\int_0^{2\pi}\frac{d\theta}{2+i\:sin\theta}=\frac{2\pi}{\sqrt{5}}$ I used $sin\theta=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})$ and substituted $z=e^{i\theta}$. I ended up with ...
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### Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
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### Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
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### How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
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### Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ?

I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$ But apparently the answer is $$i\cosh(x)+C$$ I was just wondering, is this ...
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### Integration $1/x$ - complex number

Why there is no integral $$\int_{-e}^{e}\frac{1}{x}$$ And why integral $$\int_{-e}^{-1}\frac{1}{x}= -1$$ and not $$\int_{-e}^{-1}\frac{1}{x}=(-1 + i\cdot\pi)$$ E.g. ...
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### Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
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### Integral over a complex plane

I am wondering if closed-form solutions exist for the following integral: $$\int_\mathbb{C}e^{-\frac{|u-v|^2}{c}-|v|^2}|v|^{2n}d^2v$$ where $u$ is a complex number, $c>1$ a real constant and $n$ ...
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### Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n}$?

Where are the solutions of the equations $$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$ Since the ...
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### Integration With i

Why does this approach to integration not work? If there is an integral $1/\sqrt{a^2-x^2}$, the answer is $\arcsin(x/a)$. But if the integral is $1/\sqrt{x^2-a^2}$ then it is $\log(x+\sqrt{x^2-a^2})$. ...
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### Contour integration of complex number confuses me, still.

Given $f(z) = (x^2+y)+i(xy)$ and we integrate it using the Parabola Contour. For a parabola, $\gamma(t) = t + it^2$. So, $f(\gamma(t)) = 2t^2 + it^3$. What was ...
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### Contour integration with square root in the denominator [closed]

$$\oint \dfrac{1}{\sqrt{(z+u)(z+1/u)}(z-a)^2}dz$$ where the contour parametrized by $z$ ,includes $-u$ and $-1/u$ but not $a$. Also note that $u \in \mathbb{R}$ and $a \in \mathbb{C}$.
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### Find the integral of $\overline{z}$

Question: Find $\int\overline{z}$, when the contour is a parabola. Interval is from 0 to 1. My Attempt: $z = x + iy \Rightarrow \overline{z} = x - iy$ $f(z) = x - iy$ Since the contour is a ...
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### Difficulty in understanding integrals of complex numbers

I understand what integration of real numbers is. I know how the definition of it is made. I have trouble in understanding how it works for complex numbers. I am referring to the notes here: ...
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### Contour integration that is reduced to integration over unit circle

I want to evaluate $\displaystyle \frac{1}{2\pi}\int_{0}^{2\pi} \frac{1}{1-2rcos\theta + r^2} d\theta$ for $0 < r< 1$. I was thinking or replacing $2cos\theta = (e^{i\theta} + e^{-i\theta})$ ...
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### Can the series $1/(i-0) + 1/(i-1) + 1/(i-2) + \cdots$ be reduced to $\log(i)$

Can the series $\dfrac1{i-0} + \dfrac1{i-1} + \dfrac1{i-2} + \cdots$ be reduced to $\log(i)$? It looked similar to the harmonic series, so I checked wikipedia for Harmonic series, and found the ...
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### About problem in complex integrals

I solved this problem in complex integrals. Is my answer a correct ? Here $z$ is a complex value: $$C:|z-1|=1 \ \ \ \ \ \mbox{integral path}$$ $$\int_C\ \frac{2z^2-5z+1}{z-1}\ dz$$ My answer ...
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### How do we find the integral of this conjugate? [closed]

I'm not sure how to find $$\int_C \overline{z}^2\ dz$$ where $C$ is the circle $|z|=2$ traveled counterclockwise once.
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### Proof $\int\Re(f(x))\,\mathrm{d}x=\Re(\int f(x)\,\mathrm{d}x)$

I have a function $f: \mathbb{R}\to\mathbb{C}$. How can I proof/argue that $$\int\Re(f(x))\,\mathrm{d}x=\Re\left(\int f(x)\,\mathrm{d}x\right)$$ (and the same for the imaginary part)? I'm afraid I ...
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### Cauchy Integral Formula Confusion

According to Cauchy's Integral Formula, we have: Let $U$ be an open subset of the complex plane. Let $f: U \rightarrow \mathbf{C}$ be a holomorphic function. Let $\gamma$ be the boundary of some ...
### Evaluate $\int_0^{2\pi} |x \cos(\theta)+y \sin(\theta)|\, d\theta$
I am required to prove that $\displaystyle \int_0^{2\pi} |x \cos(\theta)+y \sin(\theta)|\, d\theta= 4\sqrt{x^2+y^2}$, $\ x$ and $y$ are real. I let $\sin\theta = \frac yz$, $\cos\theta=\frac xz$, ...
I was reading Rudin and I stumbled upon a proof that I do not seem to understand. It is on page 325 of Baby Rudin $3^{rd}$ edition. In case you do not have a copy I shall write some background ...