The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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26 views

Function poles and divergence of series

Yesterday I tried to calculate the residues of a function the way below, but soon I realized it won't work. Now I have a question about the poles of a function, and a series representing it. $$z\in ...
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30 views

Show that $\sum_{n=1}^\infty \dfrac{1}{4^n}\sin (nz)$ comveges pointwise

Condiser $$\sum_{n=1}^\infty \dfrac{1}{4^n}\sin (nz)$$ in the region $|z|\leq1$. Show that $f_n$ converges uniformly and decide whether it converges uniformly. Well, I think that I should start with ...
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17 views

Estimate for measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function (https://en.wikipedia.org/wiki/Nevanlinna_function) and let $\mu$ be the measure associated to $F$ via the Stieltjes inversion formula: ...
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2answers
25 views

Show that $\sum_{n=0}^\infty\frac{3 n+6 i}{(1+2 i)^n}$ converges.

Show that $\sum_{n=0}^\infty\frac{3 n+6 i}{(1+2 i)^n}$ converges. So I am applying the root test: $$\lim_{n\to\infty} |\sqrt[n]{\frac{3 n+6 i}{(1+2 i)^n}}|$$ and I have some difficulties with this ...
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34 views

Conformal maps onto open right half plane

On the Big Rudin there is the conformal map $$\varphi(z) = \frac {1+z}{1-z}$$ which sends $\{-1, 0, 1\}$ to $\{0, 1, \infty\}$. The book says: The segment $(-1, 1)$ maps onto the positive real ...
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27 views

Cauchy's Integral Formula Question- Calculating an integral with z^4 + 16 on the denominator

I think the first part of this question is okay. For the second part, I have found the roots and then calculated the absolute difference between these roots and i and, as they are all greater than ...
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23 views

Maximum /Minimum Modulus theorem for Harmonic Function ( Corollary 6.16 )

Suppose thatt $u(x,y)$ is a real valued non constant harmonic function on a bounded domain D. Then $u(x,y)$ can not attain its maximum or minimum value in $D$. I am studing complex ...
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23 views

Radius of convergence of power series development of $\frac{1}{\cos(z)}$

What is the radius of convergence of the power series development of $f(z) = \frac{1}{\cos(z)}$ at $z_0=i$? The function $f$ is defined on $D=\{z\in \Bbb{C} : \cos(z)\neq 0\}$. The largest open disk ...
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31 views

Square root of an even polynomial is holomorphic

Given an even degree polynomial $p(x)$, all of whose roots satisfy $|z| < R$. Explain why there is a holomorphic (i.e. analytic) function $h(z)$ defined on the region $R < |z| < ∞$ which ...
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17 views

How to write the sum of series as Laurent series?

How to write the sum of series as Laurent series ? $-\frac12\sum\limits_{n=0}^{\infty}(\frac z2)^n+-\frac4z\sum\limits_{n=0}^{\infty}(-z^{-2})^n$ I have somehow a blackout, how can I combine $2$ ...
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17 views

Sequence of holomorphic functions and approximation by polynomials.

Let $\Omega=\{ z\in \mathbb{C}:$ $Im$ $z>0,$ $|z|>1\}\cup\{z \in \mathbb{C}:$ $Im$ $z<0$ $|z|>1\}$ I know that since $\hat{\mathbb{C}}\setminus \Omega$ is connected there's a sequence of ...
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14 views

Troubles understanding task for complex logarithm.

I have troubles understanding this question and what to do, the goal is to show that there is no complex determination of the logarithm and square root and those two are just some parts of the whole ...
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15 views

Struggling understanding isolated Singularities

I'm struggling to understand this. I must find out the type of isolated singularity at $z_0 = 0$. for $f_1 = \frac{z}{e^z - e^{-z}}$ , $f_2 = z^2 + 1$, $f_3 = \frac{z+1}{sin(z)}$ on the Annulus(0,1) ...
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32 views

how to use complex integration to calculate $\int_0^{\pi}(1/a+\cos(x))dx$?

I have so far replaced $dx$ by $1/zi \ dz$, but I don't know how to deal with $\cos(x)$
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1answer
41 views

Let $f$ be an odd meromorphic function , what can I deduce about $res (f,0)$

Let $f$ be an odd meromorphic function. What can I deduce about $res(f,0)$?
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46 views

Find the largest number for which a Laurent Series converges

Not really sure where to start with this. Any help greatly appreciated
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30 views

Factorization of the sine

I am working on the Basel problem for a project for my Mathematics study. I need to proof that one could write the sine as a factorization of its linear roots. I know the proofs is in general done bye ...
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2answers
77 views

Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...
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1answer
21 views

Can Runge's approximating rat. fns. be required to take certain prescribed values?

Suppose $f$ is analytic on an open set $U$ containing the compact set $K$, and $\{r_n\}$ is a sequence of rational functions provided by Runge's theorem (having poles in some prescribed set $A$). For ...
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41 views

Prove $f(z)$ is a polynomial if $|f(z)| \leq (1 + |z|)^n$

Prove $f(z)$ is a polynomial if $f(z)$ is entire and $|f(z)| \leq (1 + |z|)^n$ $\forall z \in C$. Here is what I wrote for my proof: $f(z)$ can be represented as a power series ...
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23 views

Proof of Weierstrass Preparation Theorem

In Griffiths and Harris, Principles of Algebraic Geometry, on page 8, near the end of the proof of the Weierstrass Preparation Theorem, he states that $h(z,w)$ has only removable singularities in the ...
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38 views

Laurent Series Expansion of $\frac{-3z^2+8z+1}{(z-2)(z^2+1)}$

Laurent Series Expansion of $\frac{-3z^2+8z+1}{(z-2)(z^2+1)}$ on the annulus $A(1,2)$ I think $A(1,2)$ denotes the set $\{z:1<|z-0|<2\}$, so it excludes the poles. using partial fraction ...
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38 views

Easy application of the Riemann Mapping Theorem

Riemann Mapping theorem Every simply connected region $\Omega \subset \mathbb C$ is conformally equivalent to the open unit disk (except $\Omega = \mathbb C$) What are application of this ...
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39 views

If $\lvert f(z)\rvert \leq e^{Re(z)}$, then $f(z) = \lambda e^z$

Prove that if $f$ is entire and $\lvert f(z)\rvert \leq e^{Re(z)}$ $\forall z \in \Bbb C$, then $f(z) = \lambda e^z$ where $\lambda$ is a constant. I know $e^{Re(z)} = \lvert e^z \rvert$, so ...
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18 views

Singularities, Removable, Essential and Poles.

I'm taking a complex analysis course this term and I'm having trouble understanding the theory behind Laurent series, orders and singularities. Can anyone give me a hand in understanding these ideas ...
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2answers
77 views

Find $\int_{\gamma}\frac{dz}z$

If $\gamma$ is a path from $-i$ to $i$, whose image is contained in $\mathbb C\setminus\mathbb R^-$, find $\int_{\gamma}\frac{dz}z$ Does the integral converge ?, because the path $-i+2it, 0\le ...
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27 views

Properities of complex sine function

please help me solve the following: Suppose we have an integer $k$. We define $A = \left\lbrace x+iy: \ (2k-1)\frac{\pi}{2} < x < (2k+1)\frac{\pi}{2} \right\rbrace$. Proof that sine maps $A$ ...
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30 views

Branch points of functions defined as convolution integrals

I am studying sets of equations containing convolution integrals of the following type: $$ u\mapsto \int_D dz g(z) f(z-u), $$ where $g$ is analytic, but $f$ has a pole at the origin (so colloquially ...
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20 views

existence of roots for a general function over the complex plane

For some more general functions other than polynomials, are there any fixed conditions for the existence of roots in a general sense? For instance, function like $z\mathrm{sin}z-1$
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24 views

Conformal map from disk with smaller disk removed to upper half plane

I'm working on a problem that was a previous complex qualifying exam at my university. I believe I have a solution, but I'm not entirely confident in it. The problem is as follows: Find a ...
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1answer
34 views

Find a Linear Fractional Transformations (LFT) $w(z)$

I have absulotly no idea how to approach this question, Can anyone please provide with a hint or any kinda information so I can solve this question. Thank you very much for you help
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2answers
34 views

Complex Differentiation

Can anyone give a hint to how to approach this question?
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1answer
37 views

Complex integral in all $ \mathbb(R) $

I have to evaluate the next integral $$ \int_{-\infty}^{\infty}\frac{1}{(\lambda -ip)^n}e^{ipx}dp \quad p \in \mathbb{R} \ \ \ n\in \mathbb{N} $$ I don't know what strategy to follow, because I ...
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1answer
26 views

General Solution of ODE (complex eigenversion)

I am trying to figure out the general solution to the following matrix: $ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} -3 & -5 \\ 3 & 1 \end{pmatrix}\mathbf{Y}$ I got a solution, but it is so ...
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2answers
28 views

Singularities and their Residues

i have the function $$f(z)= \frac{\sin(z)}{(z-1)(\sinh(z))}$$ and i need to find the residue for the singularities. I have found the two singularities to be $z=1$, and $z=0$ I found the residue for ...
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18 views

Proving Riemann-Hurwitz formula for riemann sphere

Given a rational map $f:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$, where $\hat{\mathbb{C}}$ is the Riemann sphere, I need to prove that $2\deg(f) - 2 = \sum (v_f(p)-1)$, i.e. prove the Riemann-Hurwitz ...
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3answers
144 views

Find$\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 2x + 4}\,dx$ and $\int_{-\infty}^{\infty} \frac{\sin(x)}{x^2 + 2x + 4}\,dx$

Find $$\int_{-\infty}^{\infty} \dfrac{\cos(x)}{x^2 + 2x + 4}\,dx$$ and $$\int_{-\infty}^{\infty} \dfrac{\sin(x)}{x^2 + 2x + 4}\,dx$$ I find it really difficult. Much appreciate it if anyone can ...
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1answer
44 views

Cauchy-Riemann question

I have managed to do parts a, b and c(i). However, I am stuck on the remainder of the question. I was wondering if I could get any hints?
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11 views

Show that a bilinear form is complex skew-Hermitian.

A complex skew-Hermitian form (also called an antisymmetric sesquilinear form), is a complex sesquilinear form $s : V × V → \mathbb C$ such that $$s(w,z) = -\overline{s(z, w)}.$$ Prove that the ...
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31 views

Partial fractions for $\frac{1}{(z^2+1)^2}$

I am trying to decompose the following: $$\frac{1}{(z^2+1)^2}$$ for the calculation of the integral $$\int_{|z|=2} \frac{e^{iz}}{(z^2+1)^2}dz$$ by using Cauchy's formula. The only thing I am stuck ...
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1answer
46 views

Complex limit? $\lim_{x \rightarrow \infty} e^{(a+bi)x}=0$

What condition have to satisfy a and b to get the next result:$$\lim_{x \rightarrow \infty} e^{(a+bi)x}=0$$? Thank you :)
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how can I show that $\cot\pi z$ and $\csc \pi z$ have simple poles for every integer $n$? so then I can calculate residues at those poles?

how can I show that $\cot\pi$z and $\csc\pi$z have simple poles for every integer $n$? so then I can calculate residues at those poles?
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34 views

Laurent Series expansion of $f(z)=(z-1)sin{1\over z}$

I need to find the Laurent series expansion of the function: $$f(z)=(z-1)sin{1\over z}$$ about $$A= z ∈ \Bbb C : 0<|z|<∞ $$ Any help would be appreciated!
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31 views

Partial fractions for $\left (\frac{z^2+1}{z^2-1} \right )^2$

I am trying to decompose this function: $$\left (\frac{z^2+1}{z^2-1} \right )^2$$ into partial fractions. What I've done up to now is this: $$\frac{z^2+1}{(z-1)(z+1)}=\frac{A}{z-1}+\frac{B}{z+1}$$ ...
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2answers
53 views

Evaluating the complex integral $\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx$

I stumbled upon this particular integral a few minutes ago, and I have no idea how to go about it : $$\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx$$ I looked up on the internet and I managed to ...
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12 views

contour integral of complex function

How would one compute the contour integral of along the wedge shape contour for the function $f = z^{-3/2} = \dfrac{1}{r^{3/2}}\dfrac{1}{\cos(\dfrac{3\theta}{2})+i\sin(\dfrac{3\theta}{2})}$ or ...
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26 views

Inversion map is a Conformal map

I'm studying PDE by Evans book and I need to show that the inversion map $f:\mathbb{R}^n-\{0\}\to \mathbb{R}^n$, defined by $$f(x)=\frac{x}{\|x\|^2}$$ is conformal. So I have a hint, show that ...
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1answer
32 views

Analytic function in upper half plane.

Let $f$ be an analytic function in the upper half-plane with $|f(z)| <1$. Now if $f( \iota )=0$ then find the maximum possible value of $|f(2 \iota)|$. Clearly Reflection principle is not working ...
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2answers
34 views

Proving that if $u,v$ are harmonic on $D$ and $uv = 0$ on an open subset of $D$, then $u$ or $v$ is 0 on $D$

Let $D$ be an open connected set. Suppose $u$ and $v$ are harmonic functions on $D$ and that $u(z) v(z) = 0$ on an open subset of $D$. Obviously on this open subset at least one of $u$ or $v$ is 0, ...
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1answer
67 views

Normal families of holomorphic functions

If a set $F=\{f: f$ is holomorphic on $G, G $ is open in $C \}$ is normal. we want to show that $F'=\{f': f\in F\}$ the set of the derivatives of function that contained in F is normal. And what about ...