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

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
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1answer
75 views

Prove or disprove the existence of polynomials.

Prove or disprove that there exist non-constant polynomials $p$ and $q$ such that $p(z)e^{p(z)}+q(z)e^{q(z)}=1$ for all $z\in \mathbb{C}$.
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2answers
70 views

Show that $e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}$ for all $y \in \mathbb{R}$ with $|\mu_1| \leq 1$

How to show the expansions \begin{gather*} e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}\\ e^{iy} = 1 + iy - \frac{1}{2}y^2 + \frac{\mu_2 |y|^3}{3!} \end{gather*} where $y \in \mathbb{R}$ and $|\mu_1| \leq 1$ ...
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1answer
30 views

proof verification analytic function

Exists an analytic function $g(z)$ such that $g^{'}(z)= \frac{1}{z^{2}-1}$ in the annulus $1<|z|<2$. My answer is yes i calculate the integral of $g$ which is $\frac{1}{2}(\log(z+1) + ...
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1answer
39 views

Question on radius of convergence

Can anyone help me with the following problem: I have a solid geometric picture of what is going on in my head, but I can't seem to turn that into a proof.
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2answers
94 views

not following two steps in proof that $\int_{0}^{\infty} cos(x^2) = \frac{\sqrt{2 \pi}}{4}$

Hi: I'm reading some notes I found on complex analysis on the internet. In the example, they prove that $$\int_{0}^{\infty} \cos(x^2) = \int_{0}^{\infty} \sin(x^2) = \frac{\sqrt{2\pi}}{4}.$$ I ...
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2answers
98 views

Do there exist nonconstant holomorphic functions $f$ and $g$ on the open unit disk such that $e^{f(z)}+e^{g(z)}=1$?

This is a question from an old qualifying exam that I was trying to solve for practice: Prove or disprove that there exist nonconstant holomorphic functions $f$ and $g$ on the open unit disk ...
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1answer
25 views

Computing the radius of convergence of a given series

Could anyone help me with the following problem? I'm getting $1$ as the answer, but I found the solution (without justification) to this problem online and it says the answer is $1/2$. Determine ...
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1answer
21 views

Showing uniform continuity of function giving radius of convergence

Let $f$ be an analytic function on an open disk $D$ and let $R(z)$ denote the radius of convergence of the power series of $f$ about a point $z$. Is there an easy way to show that $|R(z_1) - R(z_2)| ...
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1answer
59 views

If $f$ is the limit of polynomials with only real zeros, then all zeros of $f$ are real

Problem Let $f$ be a non-constant entire function. Suppose that there is a sequence of polynomials ${P_n(z)}$, $n=1,2,...$ such that $P_n(z)$ converges uniformly to $f$ on every bounded set ...
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50 views

Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...
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1answer
61 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
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0answers
94 views

Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$ \int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0 $$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
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1answer
45 views

A question about complex power series.

My book says the following: Let $\sum_n a_nz^n$ be a convergent complex power series with radius of convergence $r$. Then there exist $C,A\in\mathbb{R}$ such that $|a_n|<CA^n$, where ...
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2answers
46 views

Is it possible to realize a general compact Riemann surface in $\mathbb CP^2$?

Let $X$ be a compact Riemann surface with smooth boundary $\partial X$. Is it always possible to realize $X$ as a complex submanifold of $\mathbb CP^2$? In other words, is it true that there exists a ...
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0answers
26 views

Analytic cohomology on Zariski site vs analytic cohomology on analytic site

If I have an affine algebraic complex manifold (in fact it is Stein), what is known relating the cohomology of analytic sheaves using only Zariski opens vs the cohomology of analytic sheaves using the ...
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1answer
32 views

Radius of convergence of series given radius of convergence of another series

I'm hoping someone might be able to verify my solution to the following problem: Suppose that the series $\sum c_n z^n$ has radius of convergence $R$. Find the radius of convergence of the $\sum ...
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1answer
38 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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1answer
29 views

Cauchy-Riemann and Analytic Functions

Using the Cauchy-Riemann conditions, tell if $f(z) = z^{*}$ is analytic I have tried this: $Z = x + iy$ $f(x + iy) = Z^{*} = x - iy$ $U(x,y) = x$ $V(x,y) = -y$ $U_x = 1$ Deriving respect to $x$ ...
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1answer
40 views

An example of a complex power series. [closed]

I am looking for a complex power series which is convergent for some $z\in\Bbb{C}$ but not absolutely convergent. In other words, $a_0+a_1z+a_2z+\dots$ is convergent but ...
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5answers
330 views

Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
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2answers
309 views

Find a conformal map from semi-disc onto unit disc

This comes straight from Conway's Complex Analysis, VII.4, exercise 4. Find an analytic function $f$ which maps $G:=$ {${z: |z| < 1, Re(z) > 0}$} onto $B(0; 1)$ in a one-one fashion. ...
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1answer
107 views

Conformal mapping of the domain bounded by a line segment and a circular arc

I am trying to construct a conformal map from the region $R$ which is the set of points in the complex plane bounded by the segment connecting $i$ and $1$ and the part of the unit circle in the first ...
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0answers
71 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
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0answers
75 views

Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want ...
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2answers
42 views

Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
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1answer
41 views

Usage of the term $\arg(z)$

Consider the complex number $z = -1 - i$. Is it mathematically correct to say that $\arg(z) = 5\pi/4$? Sure, $5\pi/4$ is not the principle argument of $z$, but it is an element of the set $\arg(z)$. ...
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57 views

At most one connected component of unbounded portion of entire function.

Suppose $f$ is an entire complex analytic function and $R$ a positive real number. Define the set $E:= \{z\in\mathbb{C};|f(z)| < R\}$ to be the set of $z$ whose image is bounded by $R$. I want to ...
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1answer
118 views

Why does convolution not maintain asymptotic equality of coefficients?

Assume I have four (generating) functions $f$, $f'$, $g$ and $g'$. If that is interesting, we can assume that they all share the same radius of convergence $\rho > 0$. In addition, we know that ...
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1answer
41 views

Quotient of 2 holomorphic functions which may be holomorphic

Let f and g be two holomorphic functions on a domain $\Omega$. Suppose that $\frac{f}{g}$ is always finite (while g can be zero at some points). Is it true that then $\frac{f}{g}$ is holomorphic? ...
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1answer
192 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
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54 views

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}$

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}; f(z)=\frac{1}{(z-i)^2}$ and $f(z)=e^{(z-1)^-1}$ Good evening, I have been trying to solve the above exercises. However, I'm not sure if my procedure ...
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1answer
72 views

Conformal mapping between regions symmetric across the real line

In Conway's Functions of One Complex Variable, the section on the Riemann Mapping Theorem has the following exercise: Let $G$ be a simply connected region which is not the whole plane, and suppose ...
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1answer
88 views

Sketching regions is complex plane

When sektching the region $\left|\frac{2z-1}{z+i}\right|$$\geq$1 on the argrand diagram, how should we go about identifying the region, should we take $\left|2z-1\right|\geq\left|z+i\right|$ or ...
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1answer
27 views

Describe the family of analytic functions with the following properties:

Find the family of all functions $f$ analytic in $\mathbb{D}$ (the open unit disk) and continuous on $\overline{\mathbb{D}}$ such that $|f(z)|=e^{\text{Re}(z)}$ for all $z\in\mathbb{D}$. My intuition ...
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0answers
42 views

Entire function with $L^2$ modulus is identically zero [duplicate]

I want to show that if $f$ is entire and $\int_{\mathbb{R}^2}\left| \:f\: \right|^2 < \infty$, then $f \equiv 0$. I was thinking of assuming $f$ is not identically zero; then, since a bounded ...
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0answers
96 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
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1answer
49 views

Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
2
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0answers
187 views

Contour integration with merged pole/branch-cut type behavior?

I have the expression $$f(z)=\frac{-i}{\sqrt{z^2-a^2}},$$ where $a$ is a purely real number and $z$ is a complex variable. Numerical plotting gives the following. This leads me to the following ...
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1answer
42 views

winding number in several complex variables

Is there any analogue of the concept of winding numbers in the theory of several complex variables? If so, can anyone provide me references for studying it?
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87 views

Solving the equation $(z-2)^{4}+(z+1)^{4}=0$

$(z-2)^{4}+(z+1)^{4}=0$ I tried starting by solving $z^{4}=1$ with the solutions being , $1cis (\frac{n\pi }{2})$, where $n = -1, 0, 1, 2$ I am unsure about how to proceed from here, I tried to ...
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1answer
21 views

Extrema of the set of values $|f(-1/2)|$ for analytic functions $f \colon \mathbb{D} \to \mathbb{D}$

I have a past qual question here: consider the set $S = \{ |f(-1/2)| \colon \textrm{$f \colon \mathbb{D} \to \mathbb{D}$ is analytic and has a triple zero at the origin} \}$, where here $\mathbb{D}$ ...
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1answer
64 views

given analytic $f(z)$ in $f(z)/(1-z)$ , derivative $f '(z)$ seems to have singularity at $z=1$

Quick version: I want $f'(1)$, where $$F(z)=\frac{f(z)}{1-z}$$ with $f$ analytic at $z=1$. But when I follow a seemingly valid line of reasoning, I reach the conclusion that $f'(z)$ is not analytic ...
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1answer
37 views

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$ Prove that if $f(\mathbb{D})⊂\mathbb{D}$ or ...
3
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1answer
78 views

Evaluation of Sum of $ \sum_{n=1}^{\infty}\frac{\sin (n)}{n}$.

If $\displaystyle S = \sum_{n=1}^{\infty}\frac{\sin (n)}{n}.$ Then value of $2S+1 = $ Using Fourier Series Transformation I am Getting $2S+1=\pi$ But I want to solve it Using Euler Method and Then ...
2
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1answer
64 views

Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
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4answers
88 views

Help with Complex integration

I have to calculate the following integral $$\int_{-\infty}^{\infty} \frac{\cos(x)}{e^x+e^{-x}} dx$$ Anyone can give me an idea about what complex function or what path I should choose to calculate ...
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1answer
79 views

Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
8
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2answers
235 views

General method to “naturally interpolate” to a complex map?

Given a region of the complex plane and a map $z \to f(z)$, is there a general way to "naturally interpolate" the point $z$ to $f(z)$ in such a way that the movement follows a "natural" smooth path ...
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1answer
43 views

Uniformly analytic functions

Consider the following definition: Let $\Omega$ be an open set of $\mathbb{R}_x^n$, $x = (x_1, ..., x_n)$. A $\mathcal{C}^{\infty}$-function $\varphi(x)$ on $\Omega$ is said to be uniformly analytic ...