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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60 views

Can you help me with this problem?

How can I understand this? $$-q\sum_{n=0}^{\infty}\frac{t^{p+nq}}{p+nq}=\sum_{n=0}^{q-1}\omega^{-np}\ln(1-\omega^{n}t)$$ Here $p,q\in\mathbb{N}$. Let $\omega=e^{2\pi i/q}$ be a primitive ...
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1answer
251 views

Radius of convergence of $\displaystyle\sum\limits_{n=0}^\infty2^{-n^2}z^n$

I was reading examples to find the radius of convergence for power series. The power series is defined as $\displaystyle\sum\limits_{n=0}^\infty c_n(z-z_0)^n$. And to find the radius of convergence ...
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3answers
830 views

Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)

An arithmetic sequence of step $d$ is a set of the form: {$a, a+d, a+2d, a+3d, ...$} where $a, d$ are positive integers. Show that the positive integers cannot be partitioned into a finite number ...
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73 views

How to plot $\{z\in\mathbb{C}:|z|^2\ge z +\bar z\}$?

How would I plot plot $\{z\in\mathbb{C}:|z|^2\ge z +\bar z\}$? So far I did: \begin{align*} |z|^2 & \ge z +\bar z \\ |a+ib|^2 & \ge a+ib+a-ib \\ a^2 + b^2 & \ge 2a\\ a(a-2)+b^2 & \ge ...
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0answers
84 views

Calculating the Fourier transform of $\frac{e^{-k(x-k)^{2}}}{x}$

I would like to calculate the Fourier transform of the function: $\frac{e^{-k(x-k)^2}}{x}$ but the integral $$\int_{-\infty}^{+\infty} \frac{e^{-k(x-k)^{2}}}{x} \ e^{i\omega x}\,dx$$ it is very ...
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175 views

Composition of multi complex gaussian normal distribution

assume $w_0$, $w_1$, $w_2$, $w_3$ are circular symmetric complex Gaussian distributions, and the composite of $$ h = e^{j\theta_0}w_0 + e^{j\theta_3}w_3 - e^{j\theta_1}w_1 -e^{j\theta_2}w_2 $$ so ...
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1answer
2k views

Goursat's Theorem

A very first theorem that is proved in the first course of Complex Analysis would be the Gousart Theorem. Here it is: Theorem (Goursat). Let $f:U\rightarrow\mathbb{C}$ be an analytic function. Then ...
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332 views

Using Summation By Parts on $\sum_{i=1}^{\infty} z^{i}\frac{1}{i}$

I'm trying to show that the power series $\sum_{k=1}^{\infty} z^{k}\frac{1}{k}$ converges for $z$ s.t. $|z| = 1$ (convergence on the unit circle), except for $z=1$ . My exercise book says to do it ...
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71 views

Liftings of curves $u\cdot v$ and $v\cdot u$ with respect to the sine covering map.

I'm trying to work through the exercises in Otto Forster's book on Riemann Surfaces. While most of them seemed not that hard, this one gives me a headache: Let $X=\mathbb{C}\setminus\{\pm1\}$ and $Y ...
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2answers
481 views

An entire function with periodic bounds

This is a bit of a repost from an old question, but it doesn't seem like it was fully answered before and this is a bit of an abstraction from that post. I'm trying to show the following: Let $f(z)$ ...
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119 views

An integral transform.

Let's consider a complex function that can be represented in the following form: $$ K(z)=\int_{-\infty}^{\infty}A(\alpha)z^\alpha d\alpha $$ Writing $z=re^{i\theta}$, we get: $$ ...
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5answers
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Why: A holomorphic function with constant magnitude must be constant.

How can I prove the following assertion? Let f be a holomorphic function such that |f| is a constant. Then f is constant. Edit: The more elementary the proof, the better. I'm working my way ...
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2answers
104 views

$\lim_{z \to 0} z \exp(f(z))$ for holomorphic $f$?

How would one show that if $\displaystyle \lim_{z \to 0} z \exp(f(z))$ exists, where $f: \mathbb{C}^\ast \longrightarrow \mathbb{C}$ is holomorphic, then it must be zero? Is this even true?
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1answer
298 views

Towards the solution of the Problem : Field Extension problem beyond $\mathbb C$ (Question 1)

I am posting this problem in order to break the problem in my previous post Field Extension problem beyond $\mathbb C$. Notation: $M(\mathbb C):=$ Field of all meromorphic functions on $\mathbb C$, ...
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2answers
75 views

How performs the function $ z^{1/2}$ in the complex plane?

How performs the function $ z^{1/2}$ in the complex plane? Thanks for your help I know it's a multi-valued function and that we must be careful with the branch on which it is defined, one of my ...
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4answers
906 views

Sheaves and complex analysis

A complex analysis professor once told me that "sheaves are all over the place" in complex analysis. Of course one can define the sheaf of holomorphic functions: if $U\subset \mathbf{C}$ (or ...
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3answers
326 views

Other functional equations for $\zeta(s)$?

For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ ...
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1answer
171 views

Can a Herglotz-Nevanlinna function attain real values?

Let $\mathbb{H}^+=\{z \in \mathbb{C}\mid \Im(z)>0\}$. We say that an analytic $F\colon \mathbb{H}^+\to\overline{\mathbb{H}^+}$ is a Herglotz-Nevanlinna's function. Question Can it be that ...
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2answers
279 views

Weierstrass Factorization Theorem

Are there any generalisations of the Weierstrass Factorization Theorem, and if so where can I find information on them? I'm trying to investigate infinite products of the form $$\prod_{k=1}^\infty ...
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1answer
627 views

The Laurent Series of $\sqrt{(z-1)(z-2)}$ at the point $z=\infty$?

How can we find the Laurent Series of $\sqrt{(z-1)(z-2)}$ near point $z=\infty$? By definition, it's equivalent to asking the Laurent Seires of $\sqrt{\frac{(t-1)(t-2)}{t^2}}$ near the point $t=0$. ...
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1k views

An entire function

I have been struggling on the following problem. Suppose $f$ is an entire analytic function such that $|f(z)|>1$ if $|z|>1$. Show that $f$ is a polynomial. My idea is as followed: all zeros of ...
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2answers
627 views

Show either $f$ is constant or $g(z)=0$ for all $z$ in the region

let $G$ be a region, and $f$ and $g$ be holomorphic function on $G$. if $\bar{f}\cdot g$ is holomorphic, show that either $f$ is a constant or $g(z)=0$ for all $z$ in $G$.
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1answer
982 views

Finding roots of unity?

The $n$th roots of unity are the complex numbers: $1, w,w^2,...,w^{n-1}$, where $w=e^{\frac{2\pi i}{n}}$. Why is this true? I understand why $w$ is 1 root of unity, but why are $w^0,..., w^{n-1}$ ...
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0answers
242 views

Question on harmonic functions

By the maximum principle, every harmonic function on a bounded domain is uniquely determined by its boundary values. However, for unbounded domains, we can have infinitely many harmonic functions with ...
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3answers
182 views

compute integral $\int_0^{2\pi} \frac{1}{z-\cos(\phi)} d\phi$

Can anybody help me to compute the integral $$\int_0^{2\pi} \frac{1}{z-\cos(\phi)} d\phi$$ where $z \in \mathbb{C}$ denotes a complex number? Thank you!!
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280 views

Using Rouche's theorem

Let $p>1$. Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$. Function $\phi(p)$ is analytic on its domain. It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
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1answer
75 views

Holomorphic map, preimages

Is there a neat way of determining the domain that maps to the upper half plane by the map $f(z)={z^2+4\over z}$? i can see that the map is holomorphic everywhere except at $0$. And therefore the ...
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1answer
182 views

Laplace inverse of the sine function

I was wondering if there is a closed-form Laplace inverse of the sine function. I have tried the following: $$ \sin(as)=\sum_{n=0}^{\infty}\frac{(-1)^{n}(as)^{2n+1}}{(2n+1)!} $$ an $n$-th power of ...
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186 views

Uniform convergence of a power series when avoiding a point of divergence

Here's the exercise: Let $\delta\in(0,1)$ and let $(a_n)_{n\in\mathbb{N}}$ be a real, monotonic decreasing sequence that converges to $0$. Show that $\sum a_nz^n$ converges uniformly on ...
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0answers
489 views

Convergence of complex power series

I'll post the full problem before I'll show my (rather limited) progress: i) Find all $z \in \mathbb{C}$ so that the following power series converge around $0$: a) $\sum_{k=0}^\infty z^k$, b) ...
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1answer
95 views

Proving $|w\overline{z}+\overline{w}z|\leq 2|wz|$

I want to prove $|w\overline{z}+\overline{w}z|\leq 2|wz|$. My attempt: $$\begin{array}{c c}|w\overline{z}+\overline{w}z| & =|(c+id)(a-ib)+(c-id)(a+ib)| \\ & =|2(ac+bd)| \\ & ...
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2answers
568 views

Roots of unity?

The $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even: The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = ...
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1answer
659 views

Complex Analysis: Finding the level curves of a function?

Consider the function $f(z)=z^2$. Prove that level curves of $Re(f(z))$ and $Im(f(z))$ at $z=1+2i$ are orthogonal to each other. I am not sure how to apply level curves or contour lines for complex ...
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1answer
263 views

Field Extension problem beyond $\mathbb C$

There are lots of fields between $\mathbb C$ and Meromorphic Functions on $\mathbb C$. For example set of "All Even Meromorphic Functions on $\mathbb C$'' is a subfield between $\mathbb C$ and ...
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1answer
167 views

Estimates involving a holomorphic function on the unit disc

Assume that $f$ is an analytic function on the unit disc $\mathbb{D}$ and continuous up to the closure. Therefore $f(z)=\sum\limits_{n=0}^\infty c_nz^n$ for all $z \in \mathbb{D}$. If $f$ have $m$ ...
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1answer
539 views

Residue at $z=\infty$

I'm a bit confused at when to use the calculation of a residue at $z=\infty$ to calculate an integral of a function. Here is the example my book uses: In the positively oriented circle $|z-2|=1$, ...
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401 views

Characterization of Cauchy-Riemann operator

Let $U \subset \mathbf C$ be an open subset of the complex plane and suppose we have a differential operator of order 1, $L: \mathcal C^{\infty}(U) \to C^{\infty}(U)$ such that $Lu = 0$ if and only ...
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1answer
399 views

How to show set of all bounded, analytic function forms a Banach space?

I am trying to prove that set of bounded, analytic functions $A(\mho)$, $u:\mho\to\mathbb{C}$ forms a Banach space. It seems quite clear using Morera's theorem that if we have a cauchy sequence of ...
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1answer
403 views

Holomorphic Automorphism Group

By a domain I mean an open connected subset of ${\mathbb C}$. If $D$ is a domain, let $\operatorname{Aut}(D)$ denote the collection of holomorphic bijections $f:D\to D$. It is well-known that if $f$ ...
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138 views

Translation of entire functions along the real axis

Given an entire function $f(z)$, and $0\neq a\in \mathbb R$. We define the translation operator: $$T_{a}f(z)=f(z-a).$$ What properties the new function $f(z-a)$ could have? It is entire function! ...
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195 views

Limit conditions of a subharmonic function imply that it is constant

Let $u$ be a subharmonic function on $\mathbb{C}$. Suppose that $$\limsup_{z\to \infty} \frac{u(z)}{\log|z|}=0$$ I'm trying to prove that this implies $u(z)$ is constant. I have a feeling that it may ...
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2answers
5k views

Multiplying complex numbers in polar form?

Could someone explain why you multiply the lengths and add the angles when multiplying polar coordinates? I tried multiplying the polar forms ($r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot ...
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0answers
305 views

Jensen's Inequality for complex functions

Jensen's inequality states that if $\mu$ is a probability measure on $X$, $\phi$ is convex, and $f$ is a real-valued function, then $$ \int \phi(f) \, d\mu \geq \phi\left(\int f \, d\mu\right).$$ Is ...
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3answers
628 views

Rectangular form of a complex number?

Why does rectangular form serve as an accurate description of a complex number? Why not $a * bi$(multiplication) or another operation? Why does addition describe the relationship between the real part ...
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1answer
297 views

Branch Points of Riemann Surfaces

Can a Riemann surface of a complex-valued function have three branch points? I've been learning about Riemann surfaces from Brown's complex analysis book and the exposition isn't too general, so if ...
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129 views

Bounds on unit disc imply boundedness at origin

Suppose $f$ is holomorphic on $D_{1}(0)$ the open unit disc. Let $\Gamma_{1} = \{z : |z| = 1, x>0, y>0\}$ where $z = x+iy$ and define $\Gamma_{2}, \Gamma_{3}, \Gamma_{4}$ similarly. On ...
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233 views

Construction of conformal mapping

Let $\epsilon>0$. I was asked to find a conformal mapping from $(\mathbb{R}\times(0,2))-((-\infty,i-\epsilon ] \cup[i+\epsilon,i+\infty))$ (An infinite horizontal strip but chopped a fine strip ...
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1answer
89 views

How to integrate $\int_{\gamma_1} \frac{dz}{z(z-i)}$ with $\gamma_1 = Re^{it}$, $R>1$?

I am stuck calculating the integral $$\int_{\gamma_1} \frac{dz}{z(z-i)}$$ over $\gamma_1 = Re^{it}, R>1$. If I had to integrate over $\gamma_2 = re^{it}, r < 1$, I could just expand the ...
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1answer
178 views

Finding the number of analytic functions which vanish only on a given set.

Let $S = \{0\}\cup \{\frac{1}{4n+7} : n =1,2\ldots\}$. How to find the number of analytic functions which vanish only on $S$? Options are a: $\infty$ b: $0$ c: $1$ d: $2$
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1answer
113 views

Difference of two subharmonic functions

Is it true that for a smooth real-valued function $h(z)$ on some neighborhood of the closure of a bounded domain, that $h$ can be expressed as the difference of two smooth subharmonic functions? If ...