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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18 views

Orthogonal complex matrices: polar decomposition

Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)? thanks ...
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48 views

Integration of hyperbolic functions. [closed]

Kindly solve this integral. I shall be very grateful. $$ \int_0^{\infty}\frac{\mathrm{e}^{-x}}{\mathrm{e}^{ax}-1}dx $$ Thanks.
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1answer
49 views

Computing $\int_{\partial S} \frac{1}{1+z^n} dz$

Let $S=\{re^{it} : 0<r<R, 0< \varphi < 2\pi/n\}$ for some $R>1$ and $n\geq 2$. How can we compute $$\int_{\partial S} \frac{1}{1+z^n} dz?$$ I can't compute it directly, so I assume I ...
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0answers
27 views

Beyond Schwarz Lemma

Let $f(z)=a_1z+a_2z^2+a_3z^3... $ be a Schwarz function then by lemma $|a_1|\leq 1 $.But what is known about the higher coefficients? For example ; what can be said about $ max [a_1+a_2]$ ? Is there ...
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1answer
17 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
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3answers
42 views

Radius of convergence of power series of complex $\log$

Let $f(z) = \log(z)$ for $z\in \Bbb{C}\setminus (-\infty,0]$. Since $f$ is holomorphic on its domain, we know it has a power series development about each point $z_0\in \Bbb{C}\setminus (-\infty,0]$. ...
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0answers
37 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
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1answer
33 views

Non-constant entire function-bounded or not? [duplicate]

Show that if $f$ is a non-constant entire function,it cannot satisfy the condition: $$f(z)=f(z+1)=f(z+i)$$ My line of argument so far is based on Liouville's theorem that states that every bounded ...
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1answer
50 views

A consequence of Schwarz lemma

Suppose that for some $\epsilon>0$ the function $f$ is holomorphic on $B(0,1+\epsilon)$ such that $f(a) = 0$ and $|f(z)|\leq1$ if $|z| \leq 1$. Prove for $|z| \leq 1$: $$|f(z)|\leq ...
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0answers
23 views

$f_n\to f$ iff for each closed rectifiable curve $f_n (z) \to f (z)$ uniformly for $z$ in the trace of the curve

I'd like to know if the following exercise is correct. I'm not completely sure about the last point but also I don't know what more I'd say. I really appreciate corrections or any suggestion you can ...
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1answer
18 views

If $f$ is holomorphic and $f(a) \neq 0$, then $\exists B(a,r)$ such that $f(z) \neq 0$ $\forall z \in B$

Let $G$ be a region and $f$ holomorphic in $G$. If there exists an $a$ such that $f(a) \neq 0$, then because $f$ is holomorphic, it is continuous, so there exists a $B(a,r) \subseteq G$ such that ...
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1answer
20 views

Don't understand proof of minimum modulus principle

Minimum modulus principle: If $f$ is a non-constant holomorphic function a bounded region $G$ and continuous on $\bar{G}$, then either $f$ has a zero in $G$ or $|f|$ assumes its minimum value on ...
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0answers
22 views

Irreducibility of analytic varieties

Let $V$ be an analytic variety and $V^{*}$ denote the locus of its smooth points. From Griffiths & Harris, page 21, we have that an analytic variety $V$ is irreducible iff $V^{*}$ is connected. ...
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22 views

Hölder continuity of measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function and $\mu$ the (via Stieltjes inversion formula) associated measure, which is a finite Borel measure on $\mathbb R$ and let $C(\lambda)$ be the function ($\alpha \in ...
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0answers
18 views

Singular locus of analytic subvarieties

In Griffiths and Harris page 21, it is proven that the singular locus, denoted $V_{s}$ is contained in an analytic subvariety of the complex manifold $M$ not equal to $V$ which is the analytic ...
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1answer
16 views

Index of essential singularity

Let $f$ be a holomorphic function on a punctured disk $\Delta^*$ with essential singularity at puncture. Furthermore suppose that it has no zeroes on $\Delta^*$. Question: Does this integral have to ...
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1answer
21 views

Application of Residue theorem

Let f(z,w) be holomorphic in $\mathbb{C}^{n}$ and not identically zero on the w-axis. Let {$b_{j}$} be the set of singularities of f(z,w) in some disk of radius $|w| < r$. Why does the residue ...
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1answer
26 views

Schwarz-Pick type inequality for $f:\mathbb{D}\to D(0,R)$ holomorphic

Let $f:\mathbb{D}\to D(0,R)$ be a holomorphic function and let $a_i\in \mathbb{D}, 1\leq i\leq n$ such that $f(a_i)=0$ for every $i$. Show that $$|f(z)|\leq R\prod_{i=1}^n ...
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1answer
26 views

Analyticity of log f(z)

In a solution to a problem, I read that, if $f(z)$ is entire, $f(z)\neq0$ and the domain of definition of $f(z)$ is simply connected, then it is possible to choose a branch of log $f(z)$ that is ...
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4answers
24 views

Constructing a Fractional Linear Map

I am working on a practice prelim question: "Construct a nonlinear fractional map $\phi(z) = \frac{az+b}{cz+d}$ with $c \ne 0$ such that $\phi(\phi(\phi(z))) = z$. I feel like I just need to take ...
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1answer
19 views

How is the integral of $\frac{f(\zeta)-f(z)}{\zeta - z}$ over $C_{\epsilon}$ $0$?

I am trying to understand a proof of this theorem: Suppose $f$ is holomorphic in open set that contains the closure of a disk D. If C denotes the boundary circle of this disk with positive ...
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3answers
46 views

Can I use the Dirichlet's test to prove the convergence of $\sum_{n=1}^N \frac{e^{in}}{n}$?

I am trying to state that $$\sum_{n=1}^\infty \frac{e^{in}}{n}$$ converges. Is it correct that $|\sum_{n=1}^N e^{in}|\leq M$ for every positive integer $N$? I.e use $e^{in}$ as the $b_n$ term in ...
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1answer
23 views

Why can a function in $L^1(\partial \mathbb{D})$ be represented by a Fourier series?

I am looking for a reference to the claim that for any $f\in L^1(\partial \mathbb{D})$, where $\partial \mathbb{D}$ is the unit circle in $\mathbb{C}$, ...
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1answer
38 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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1answer
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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0answers
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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1answer
35 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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0answers
37 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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1answer
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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1answer
24 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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1answer
40 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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0answers
20 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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1answer
18 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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0answers
15 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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0answers
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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0answers
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
65 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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1answer
129 views

Find the largest number for which a Laurent Series converges [on hold]

Determine the largest number $R$ such that the Laurent series of $$f(z)=\frac{2}{z^2-1}+\frac{3}{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$. Not really sure where to start with this. Any ...
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0answers
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
83 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
23 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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1answer
42 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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25 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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1answer
39 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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1answer
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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2answers
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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0answers
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 ...
3
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2answers
78 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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0answers
29 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 $A$ ...