# Tagged Questions

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### Any general methods to calculate integral of $P(x)/Q(x)$ from $0$ to $\infty$?

In complex analysis, we have general formula for $P(x)/Q(x)$ [$P$ and $Q$ are polynomials] from minus infinity to infinity, if $\deg Q - \deg P > 2$. Is it possible to have a general formula for ...
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### proving cauchy integral formula using integration by parts

I have found several proofs of Cauchy's generalized Integral formula, but I am looking to prove the first derivative case by using the fact that the first derivitave is analytic to state it as a ...
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### Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
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### Find $\int_\gamma \frac{dz}{z^2}$ wihtout explicit calculations

Evaluate the following integral without doing any explicit calculations: $\int_\gamma \frac{dz}{z^2}$ where $\gamma(t) = \cos(t) + 2i\sin(t)$ for $0 \le t \le 2\pi$. This exercise comes along with ...
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### complex sum-integration

Let $a_n \in \mathbb{R}$. Show that, $$0\le \sum_0^N\sum_0^N\frac{a_na_m}{n+m+\frac{1}{2}}\le \pi \sum_0^Na_n^2.$$ Hint: Integrate on the semicircle with radius R, the sum ...
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### Complex Integration of DTFT

Question A discrete-time signal $u \in \mathcal{l}^2(\mathcal{Z})$ has DTFT $$\hat{u}(\omega) = \frac{5+3\cos(\omega)}{17+8\cos(\omega)}$$ Use complex integration to find ...
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### Find all functions $f(z)$ that are analytic everywhere in the entire complex plane that satisfy $f(2-i)=4i$

This question is apparently related to Cauchy's Integral Formula and related theorems, but I honestly don't know how to start, other than potentially saying that $f(z)$ is every function such that ...
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### Evaluate $\oint_C\frac{dz}{z-2}$ around the circle $|z-2| = 4$

I don't completely understand how to approach these questions. I suppose the notation $\oint_C$ is something I'm not used to. So far, I have $\oint_C\frac{dz}{z-2} = \log(z-2)$. From here, I suppose ...
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### Evaluate $\oint_C |z|^2 dz$ around the square with vertices at $(0,0), (1,0), (1,1), (0,1)$

I don't think I quite understand how to go about this. My solution so far: $\oint_C |z|^2 dz = \oint_C (x^2 + y^2)dz = \oint_C (x^2 + y^2) d(x+iy) = \oint_C x^2 + y^2 dx + i\oint_Cx^2+y^2dy$. ...
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### Analytical Continuation

I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone ...
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### Complex double integral

I'm having trouble calculating following (complex) integral. $$\int_D z^n dA$$ where $D=\{ z \in \mathbb{C} \mid \lvert z \rvert \leq 1 \}$. I know how to calculate complex (line) integrals and real ...
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### Integral of the determinant of the Jacobian in complex variables?

Let $f$ be a function that maps a disc $D$ onto a unit square $A$ such that $f(D) = A$. $f$ is holomorphic and one-to-one. Why does $\int_D|f'(x+iy)|^2 \, dx \, dy = 1$?
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### Does $\int_a^b f'(\gamma(t))\gamma'(t)\,dt$ depend on the path for meromorphic functions?

Given a meromorphic function $f:\mathbb C\to\mathbb C$ and a smooth curve $\gamma:[a,b]\to\Gamma\subset\mathbb C$ with $\gamma(a)\neq\gamma(b)$, I am tempted to think the fundamental theorem of ...
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### Complex integral using residue theorem

I have $$\int_{|z|=1} z^m \sin\left(\frac{1}{z}\right)~dz,$$ for $m = 0,1,2,\dots$ I know that there is a singularity at $z=0$, and this singularity is within the curve, thus the residue theorem ...
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### How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help)

I want to carry out the following integration $$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$ which is trivial if calculated numerically with any value for b. But I really need to get an ...
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### Change of variable x=iy in improper integral

I'm trying to solve the following question: Let $I=\int_0^{\infty}\exp(-x^4)dx$. Take $x=i y$ to get $i\int_0^{\infty}\exp(-y^4)dy=i I$. Explain. This change of variable implies $I=iI$, with $I$ ...
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### Intuitive reason for why many complex integrals vanish when the path is “blown-up”?

It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along ...
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### Complex Analysis: Fundamental Theorem of Calculus

I was given the following question on a complex analysis exam but didn't answer it correctly: Evaluate $$\int \sqrt{z-1} dz$$ about the the unit disk $|z|=1$ using only the Fundamental Theorem of ...