# Tagged Questions

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### Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
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### Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0$ $1)$ show that $2+i$ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
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### Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
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### How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
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### Calculating $\int_0^\infty \frac{\ln x}{(x^2+9)^2} dx$

I try to calculate $$\int_0^\infty \frac{\ln x}{(x^2+9)^2} dx$$ I use a book that tells me to replace $\ln x \$ by $\ \ln(|x|) + i\phi_z$ where $\phi_z$ denotes the argument of $z$, chosen between ...
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### How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$\int_C {e^{3z}-z\over (z+1)^2z^2}$$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
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### Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...
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### How to find $\frac{f(z)}{z-a}$

I hope that you can help me to find some residues. I know two ways to find the residue in a value $a \in \mathbb{C}$: Straight forward calculation: $\int_{C(a,\epsilon)^+} f(z) dz$ Rewriting a ...
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### Singularities of complex functions.

How do I determine the singularities of a function? What is a singularity? In the functions below which are the singularities? a)$$f(z)=\frac{1}{(z^4+2z)}$$ b)$$f(z)={e^{1/z}}$$
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### Determining Laurent Series expansion and residues

Determining Laurent Series expansion and residues of $f(z)=\frac{z}{(z+1)(z+2)}$ around $z = -2$. What is the validity of the expanded region? What is $res(f, -2)$??
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### Which way will produce the following integral?

Which way $\gamma$ will produce the following integral? $$\int\limits_{\gamma}\frac{3+i}{z^5 - z}dz = 0$$
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### Calculate the integral using the Cauchy integral formula

$f(z) = \frac{z}{z^2 -1}$ Calculate $\int_\gamma f(z) dz$ where $\gamma(t) = i + e^{it}$ for $t\in[0,2\pi]$ using the Cauchy integral formula.
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### Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i$ are $$z_1,z_2,z_3,z_4,z_5,z_6$$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
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### Complex solutions of polynomial question

$2z^3-6z^2+mz+n = 0$ $m, n$ are real and $1+\sqrt{ 2} i$ is a solution. Find $m$ and $n$. Attempt to solve : Giving the known theorem $1-\sqrt{2}i$ is also a solution, so we can substitute each time ...
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### A question on normal family

Let $f$ be an entire function. Suppose that the sequence $F_0=\{f(0),f'(0),f''(0),\cdots,f^{(k)}(0),\cdots \}$ is bounded. Show that $F_z=\{f(z),f'(z),f''(z),\cdots,f^{(k)}(z),\cdots \}$ is a normal ...
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### Rouche's theorem for unit disk

I have a function $F(z)=2z^4+5.41z^3+10.24z^2+4.83z+1.414$. And I need to prove how many roots does it have in unit circle, on unit circle and out of unit circle. By Rouche's theorem I was able to ...
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### How to integral : (1-i) / cosh z on C

Let C be the triangular contour connecting 3i, -1-i, 1-i. How to interal (1-i) / cosh z on C ? Don't use the residue theorem..
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### Show that $\sum_{k=1}^N\frac{1}{(k+a)(k+b)}=\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k}-\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k+N}$

I am quite stuck on this problem and I don't know how to proceed. The question states: Let $a,b,N\in\mathbb{N}$, $b>a$, $N\geq b-a$. Show that ...
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### Complex integration Question - Contour Method [duplicate]

I'm asked to find: $$\int_{-\infty}^\infty \frac{\ln(x^2+1)}{1+x^2} dx$$ Attempt Considering $$\oint \frac{\ln(z^2+1)}{(z+i)(z-i)} dz$$ So first I find the branch points of the function. This ...
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### Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
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### Find all value of $z$ for which each equation holds.

$$(a) \sin z=\cosh 4$$ $$(b) \cos z=2$$ $$(c) \sin z=i\sinh 1$$ $$(d) \cosh z=1$$ my answer (a) $\sin z= \sin x \cosh y+\cos x \sinh y=\cosh 4$ so, $x=2n\pi+{\pi\over2}$ and $y=4$ (b) since ...
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### Evaluating the definite integral of a trig function via complex analysis methods.

Show that $$\int_0^{\pi} \frac{1}{1 + \sin^2\theta} {\delta}\theta= \frac{\pi}{\sqrt{2}}$$ Normally if the limits of integration were from $0$ to $2\pi$ what I would, and have been taught ...
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### show analytic function such $f(z)={\operatorname{Log}(z+5)\over z^2+3z+2}$

Show that $f(z)=\dfrac{\operatorname{Log}(z+5)}{z^2+3z+2}$ is analytic everywhere except at the point $-1,-2$ and on the ray $\{(x,y):x\le -5,y=0\}$. i think that separate denominator and ...
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### Limit of a complex function (resulting in divide by zero situations)

I have the following limit which I'm attempting (and failing) to evaluate: $\lim_{x \to i} (z-i)\frac{e^{imz}}{(z^2+1)^2}$ Evaluating directly, we get $\frac{0}{0}$, so can rewrite using L'Hopital's ...
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### show that there exists $n_0$ such that for $n\geq n_0$ $f_n(z)$ has $k$ roots.

The situations is as follows. We have sequence of holomorphic functions $\{f(n)\}$ which is defined on a open $U$ in $\overline{B(0,1)}$. Suppose now that this sequence converges uniformly to a ...
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### Show that $f(z)\leq \left|\frac{z-a}{1-\overline{a}z}\right|$

Given is that $|a|<1$ and the transformation $$T\colon z\mapsto\frac{z-a}{1-\overline{a}z}$$ This maps $B(0,1)$ onto $B(0,1)$. Now suppose for some $\epsilon >0$ the function $f$ is holomorphic ...
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### zeros of a polynomial

Given $P(z)=z^6+6z+10$, find how many roots are in each quadrant I have already seen that $P(z)$ has six different roots, and that none of them are real or of the form $ki$, $k\in \Bbb R$. Since ...
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### Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function

So I'm working a problem that states: A function $f$ is analytic in an open set $U$. Define $g$ by $g(z)=\overline{f(\overline{z})}$ (just because the notation can be hard to read, this is the the ...
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### Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
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### Integration of complex function with respect to complex variable

I was given as homework to calculate the complex integral limit $$\lim_{T\rightarrow \infty} \frac {1}{2\pi i}\int_{c-iT}^{c+iT}\frac {x^s}{s^{k+1}}ds$$ where $c>0$ and $k\geq1$ is an integer. ...
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### Existence of non-constant bounded analytic function on $\mathbb{C}\setminus \mathbb{Z}$

Show that there is no non-constant bounded analytic function on $\mathbb{C}\setminus \mathbb{Z}$. In this homework problem i tried to proceed in the following way: If possible, let $f$ be a ...
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### Find all holomorphic functions on $\mathbb{C}$, except for some singularities, such that $|f(z)|\leq C(|z|^{3/2}+|z-1|^{-3/2}), z\in\mathbb{C}-\{1\}$

First I wrote the Laurent series of $f(z)$ around $z=1$: $$f(z)=\sum_{n=-\infty}^{-1}c_n(z-1)^n+\sum_{n=0}^{\infty}c_n(z-1)^n.$$ Now if $|z|$ becomes very large, the first sum with the negatives ...
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### Integrating $e^{e^{ix}}$

Evaluate $\int_0^{2\pi}e^{e^{ix}}dx$. Attempt: $e^{ix}=\cos{x}+i\sin{x}$, so we can write $$e^{e^{ix}}=e^{\cos{x}}e^{i\sin{x}}$$ and then use the same identity to get ...
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### A problem in complex analysis from Stein's book [duplicate]

I'm trying to solve an exercise from Stein's Complex Analysis: Let $t>0$ be given and fixed, and define $F(z)$ by $F(z)=\prod _{n=1}^\infty (1-e^{-2\pi nt}e^{2\pi iz})$ Show that $F(z)$ is of ...
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### Vanishing Partial Derivatives of a Harmonic Function

If $u$ is a harmonic function such that all of its partial derivatives vanish at some point $z$, show that $u$ is constant.
I have a question about Complex Analytical functions. I have some homework that asks: let $f(z) = u(x,y) + iv(x,y)$. Indicate the following functions for which u(x,y) may be analytic: $6(x^2-y^2)$ | ...