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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1answer
15 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
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0answers
12 views

Existence of analytic function on a unit dsic (Converse of Schwaraz Pick Lemma )

(Schwarz - Pick Lemma) Suppose that $f$ is analytic on the Unit Disk $\triangle$ and satisfies the following two conditions : (1) $|f(z) \leq 1$ for all z $\in \triangle$ (2) $f (a) = b $ for some ...
1
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1answer
11 views

Contour integral of non analytic exponential function

The value of the integration of the function $f(z)$ over the circle of radius 3 centered at $z=1$, where $f(z) = e^{\frac{-1}{(z-1)^2}}$ this function has a pole at $z=1$ of $2$nd order. I don't ...
3
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2answers
63 views

Is entire function a polynomial? [duplicate]

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function, and suppose that for every $z\in \mathbb{C}$ there exists $n_z\in \mathbb{N}$ such that $f^{(n_z)}(z)=0$. Is $f$ necessarily a polynomial?
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1answer
10 views

Classification of conformally equivalent annuli via periods

How does one define the periods that appear in this question and show they are conformally equivalent? Are the details worked out in a textbook somewhere? Presumably we do something like take the ...
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1answer
16 views

Normal Family complex

I search about a family of holomorphic complex function that not normal but their derivative is normal. the definition of normality similar to Montel's Thrm
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1answer
31 views

Given $z$ show that $\left | z\right | = 2\sin\theta$ and $\arg z = \theta$

I've been attempting this complex-related question but couldn't quite crack the challenge. (b) Given that $$z={1-\cos 4\theta+i\sin 4\theta\over\sin 2\theta+2i\cos^2\theta},$$ show that ...
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1answer
16 views

Complex Analysis Triangular Inequality

I recently started learning Complex Analysis and as of now don't have much command over it, I am stuck up with this assignment Question of mine which is as follows: If $|Z_i| < 1$ and $V_i ≥ 0$ ...
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0answers
24 views

Complex Analysis Modulus Property

My teacher has asked an extra question apart from the one mentioned below in the picture: What will happen if |z| = |w| = 1 ? With respect to following answer: I don't see any point in that ...
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1answer
13 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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1answer
44 views

Integration of hyperbolic functions. [on hold]

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
43 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
23 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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votes
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 ...
3
votes
3answers
27 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
28 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 ...
2
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1answer
32 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 ...
3
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1answer
41 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 ...
2
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0answers
22 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
18 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
19 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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21 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
17 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 ...
3
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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
23 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
17 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
42 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
20 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 $\mathbb{C}$, ...
1
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1answer
17 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 ...
3
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1answer
32 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
26 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
21 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
21 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
29 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
15 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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0answers
9 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
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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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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31 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
40 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
43 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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0answers
29 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 ...
3
votes
2answers
75 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" ...