# Tagged Questions

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### How to describe two integration contours as set? [on hold]

Friends I need support to understand how one can describe two integration contours as set? can anyone please explain it with the help of a example?
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### What is integration contour and how to discribe it? [on hold]

We knew that an integration contour can be described as a set of points. How one can describe the two integration contours as sets.? Can anyone help me with examples.
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### Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta$ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
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### Inverse of Mellin transform

I would like to invert the following Mellin transform $M(s)$ of a function $f(x)$ defined on $[0,a]$ with $a>0$ (or get the $x\rightarrow 0$ asymptotics): $$M(s) = \frac{2a^s}{s-2(1-a^s)}$$ We ...
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### Switching $\int$ and $\sum$ proof

Been reading through this proof which seems incorrect: Let $f_n$ be continuous on the curve $C$ and $\sum f_n$ converge uniformly on $C$. Then $\sum\int_Cf_n(z)dz=\int_C\sum f_n(z)dz$ PROOF: ...
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### Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
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### Integral equation solution

I have an integral equations of the form $\int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
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### Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
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### Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx$$ Here's what I have tried. We have to make a ...
### Find a Harmonic conjugate $v(x,y)$ to $u(x,y)$.
Show that $u(x,y) = \frac{y^2}{x^3+y^3}$ in some domain and find the harmonic conjugate $v(x,y)$ to $u(x,y)$.