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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Alternate proof for the Casorati-Weierstrass theroem

I have to use the following theorem to prove the Casorati-Weierstrass theorem: If $f$ is analytic in a closed disk $D$ of radius $R$, centered at $z_0$, with $f(z_0)=0$, and if $|f(z)|\leq M$ on the ...
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1answer
13 views

Residue of $1/(\sin(1/z))$ defined at $z=0$? Trying to derive Laurent Series of $\csc (1/z)$ to find it.

This question is related to this one. I was able to figure out on my own that the residue of $\displaystyle \sin \left(\frac{1}{z} \right)$ is defined at $z=0$ by finding the Laurent Series of ...
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2answers
47 views

How to integrate the following? $\int_{0}^{+\infty}\frac{1-\cos x}{x^{\alpha+1}}\,dx$

$$2\alpha\int_{0}^{\infty}\frac{1-\cos{x}}{x^{\alpha+1}}dx=?$$ I know that it should be solved by integrating on a contour of two semicircles with radius $\epsilon$ and $T$, and the real line. ...
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3answers
113 views

Equation of a Riemann surface?

Intuitively in complex analysis I know what a Riemann surface is. It is a surface such that at every point on it the value of a function $f(z)$ is single-valued. However, how would I go about finding ...
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1answer
20 views

Classifying singularities and determing orders of complex functions

Here are a few functions for reference purposes: $f(z) = \frac{sin(2z)}{z^3}$, $ \space g(z) = \frac{sin(z)}{tan(z)}$, $ \space h(z) = z^2 sin(\frac{1}{z})$ Suppose I was calculating the ...
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3answers
64 views

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$?

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$? So far I have seen the following results: If $f\colon U\to\mathbb C$ has an antiderivative $F$ on $U$ then ...
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1answer
27 views

Essential singularity of the resolvent operator of an unbounded operator

Is there an unbounded operator with isolated points in the spectrum, not all of which are eigenvalues? For unbounded operators it is known that isolated spectral points are either poles or essential ...
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668 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...
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1answer
33 views

calculating contour integral

It would like to find an expression for $ \dfrac{1}{2 \pi i}\int \limits_{c-i \infty}^{c+i \infty} n^s (s-1)^{-e^{iu}}/s^2 ds$ where $0<|u| \leq \pi/6$, so $\Re(e^{iu}) \geq 1/2$ and $\log(s-1) ...
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19 views

Gaussian integral $\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)\,\mathrm d x$ along $[-R,R]+\mathrm i[0,Y]$

Use integration along $\partial Q$ of $Q=[-R,R]+\mathrm i[0,Y]$ to show that for all $Y\geq 0$ it holds that $$\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)~\mathrm dx = \int_{-\infty}^\infty ...
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37 views

Intergration $\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]$

I need to calculate the integral: $$\int^{\infty}_{0}\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]dx$$ I guess complex analysis is suitable for this integral, but I still have no ideas which kinds of ...
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1answer
13 views

Holomorphic function on a domain has a primitive

I am wondering if this is true. i did a proof but i'm not sure about it. take $D \subseteq \mathbb C$ to be a domain and let $f: D \to \mathbb C$ be a holomorphic on $D$. then $f$ is analytic on $D$ ...
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1answer
295 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
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18 views

Contour integrals along $\gamma(t)=t+\mathrm it^\alpha$ and a rectangle

Compute the following contour integrals with $\gamma(t)=t+\mathrm it^\alpha,t\in[0,1],\alpha > 0$ and $R=[a,b]+\mathrm i[c,d]=\{x+\mathrm iy\mid x\in[a,b],y\in[c,d]\}$. ...
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3answers
103 views

Lagrange inversion theorem application

Can someone give me an example of where the Lagrange inversion theorem is applied in such a way it inverts a formal series? For example, say I have $$\sum_{i>-1} a_it^i = u.$$ Can someone show ...
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43 views

(solved) Holomorphy on open unit disk and continuity to the closure implies absolutely convergence of coefficients?

I am having trouble proving that the space of holomorphic functions continuous till the closure in the unit open disk coincides with the power series whose coefficients form an absolute convergence of ...
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1answer
25 views

Evaluating a complex function integration on a circle

Evaluate $$ \int_c \frac {1}{z \sin z} dz$$ where: $$ C: |z| = 3 $$ I know that z = 0 is a non-removable singularity. It also might be an essential singularity since $sin z$ will cause a singular ...
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0answers
15 views

Dirichlet series with abscissa of absolute convergence $= \frac{1}{2}$

I'm trying to figure out a Dirichlet series which has its abscissa of absolute convergence $=\frac{1}{2}$. I've been trying to think about using the formula for this abscissa: ...
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1answer
26 views

Proving a function is entire

Let $f(z)$ be an entire function. Prove that: $$ g(z) = \left\{\begin{align}\frac {f(z)-f(0)}{z},\;z\ne0\\ f^\prime(0),\;z =0\end{align}\right. $$ is also entire. I know that for $ g(z) $ to be ...
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25 views

Given two entire functions $f_1,f_2$ without common zeros, prove that one can find some entire functions $g_1,g_2$ such that $f_1g_1+f_2g_2=1$ [duplicate]

The question Let $f_1,f_2$ be some entire functions without zeros in common, so for every $z∈ℂ$ we have $|f_1(z)|^2+|f_2(z)|^2≠0$. Prove that there exist two entire functions $g_1,g_2$ such that: $$ ...
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22 views

Show that such a function exists

Let $F_1,F_2:\Bbb R^2\to \Bbb R$ be the function $F_1(x_1,x_2)=\dfrac{-x_2}{x_1^2+x_2^2}$,$F_2(x_1,x_2)=\dfrac{x_1}{x_1^2+x_2^2}$, Show that $\exists$ a function $f:D\to D$ where $D$ is the open ...
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14 views

Existence/non-existence of Holomorphic Function

Let $D=\{z\in \Bbb C:|z|<1\}$. Show that there exists a holomorphic function $f:D\to D$ such that $f(\frac{3}{4})=-\frac{3}{4}$ and $f^{'}{(\frac{3}{4})}=-\frac{3}{4}$ Show that there exists ...
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1answer
63 views

Finding a conformal map to the upper half-plane

Find a conformal map from the set $$\{z \in \mathbb{C}: |\operatorname{Im}z| < \pi \}\setminus \left[-\pi i; 0 \right]$$ to the upper half-plane. I have used a composition of the following maps: ...
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1answer
11 views

Lower bound on the distance fom a point to the border of a region.

I'm trying to prove the following result in complex analysis: If $f $ is an analytic and bijective function from the unit disc to an open connected region $A$ then the distance from $f(0)$ to the ...
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0answers
17 views

To prove a complex function f(z) is analytic in a defined region [on hold]

What is the minimum set of conditions that should be fulfilled, in order to prove that the complex function f(z) is analytic ?
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2answers
34 views

Analytic continuation of Riemann zeta-function

I showed that $$\zeta(s)=\frac1{1-2^{1-s}}\sum_{n=1}^\infty (-1)^{n-1}\frac1{n^s}.$$ Right hand side should be analytic when Re$(s)>0$ and $s\neq 1$, but there are problems at points ...
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1answer
46 views

Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
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1answer
16 views

Bound for complex roots of polynomial

I am trying to prove that if $p(z)=z^n+a_{n-1}z^{n-1}+\dots+a_0$ then all the zeros lie in a circle of radius $R= \max\{1,|a_0|+|a_1|+|a_2|+\dots+|a_{n-1}|\}$ I'm trying to use induction and perhaps ...
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1answer
29 views

Analytic function on upper half plane

Here is the problem: Let $f$ be an analytic function defined on $\mathbb{H}$:={$z \in \mathbb{C}:Imz > 0 $}. Suppose that $|f(z)|<1$ for all $z \in \mathbb{H}$. Prove that for every $z \in ...
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0answers
23 views

Types of singularities of a complex function

Suppose $z \in \mathbb{C}$ is a pole of $f$. Prove that $z$ is an essential singularity of $g:=e^{f}$. I thought we need to porve that $z$ is neither a removable singular nor a pole of $g$. But when ...
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36 views

Riemann mapping theorem - Personal project

I would like to work on the Riemann mapping theorem this summer. Does anyone could give me some good references linked to this objective. For your information, I now currently finishing a degree in ...
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1answer
54 views

Derive zeta values of even integers from the Euler-Maclaurin formula.

Euler showed: \begin{equation} B_{2 k} = (-1)^{k+1} \frac{2 \, (2 \, k)!}{ (2 \, \pi)^{2 k}} \zeta(2 k) \end{equation} for $k=1,2, \cdots$. We could from here find $\zeta(2k)$ in terms of the ...
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21 views

What is the explicit expression for the Laurent series?

Is the following an explicit expression for the Laurent series: $$f(z)=\sum_{n=0}^{\infty}(z-z_0)^n\frac{f^{(0)}(z_0)}{n!}?$$ The reason I ask is because this is what I have seen being derived when ...
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1answer
192 views

Choosing between a semicircular contour and a rectangular contour

In a paper I came across (page 10, section 7), the authors state that $\int_{-\infty}^{\infty} \frac{dx}{(b^{2}+x^{2})\cosh ax} $ can be evaluated by "closing the real axis with a semi-circle centered ...
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1answer
23 views

If $f(z)=v(x)+iu(y)$ is entire then $f$ is linear

Suppose that $f$ is entire and can be written as $f(z)=v(x)+iu(y)$, that is, the real part of $f$ depends only on $x=\Re(z)$ and $y=\Im(z)$. Prove that $f(z)=az+b$ for some $a\in\mathbb R$ and ...
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1answer
24 views

how can I show that $e^{i\varepsilon e^{i\theta}}\underset{\varepsilon\to 0}{\longrightarrow }1$ uniformly?

How can I show that $e^{i\varepsilon e^{i\theta}}\underset{\varepsilon\to 0}{\longrightarrow }1$ uniformly ? Nothing I did work.
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0answers
15 views

Using Jacobi-Anger Expansion to prove Bessel function property

The Jacobi-Anger expansion is represented as $$ e^{ix \cos\theta} = \sum_{n=-\infty}^{\infty} i^{n}J_{n}e^{in \theta} $$ With this known value, I'm attempting to show that $$ \int_{- \pi}^{\pi} ...
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17 views

Showing a set is open/closed/both/neither

Consider the set $U=\{z\in\mathbb{C}\,\colon |z|>1, \Im{(z)}\ge0\}$. Is this set open, closed, both or neither. I stated it was neither since if we take a point $z\in U$ with $\Im{(z)}=0$, we ...
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2answers
27 views

Finding the order of zeros for complex functions

I kinda know the answer, but what I am looking for is a proof. Let $ f(z) $ be analytic on $ D $ with a zero of order $ m $ at $ z = a $ Let $ g(z) $ be analytic on $ D $ with a zero of order $ n $ ...
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25 views

Evaluate complex integral

Evaluate $$ \int_c \frac {\cos z}{(1-z)^2} dz$$ where: $$ C: |z+2| = \frac {1}{2} $$ Can I simple say that the integral is equal to zero by Cauchy-Goursat theorem because the contour is a closed ...
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1answer
20 views

Find the orders of zeros for a complex function

Find the orders of zeros for the following functions at z = 0: 1.$$ z^2 (e^{z^2} - 1) $$ 2. $$ 6 \sin(z^3) + z^3 (z^6 - 6) $$ The question means that I should set both functions to zero and find ...
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1answer
45 views

Runge's Approximation Theorem

This is a homework problem, so my apologies for a seeming lack of motivation. I want to understand what's going on in this problem more than I want someone to just write down the solution, so if ...
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21 views

Bilinear Transformation from upper half plane to upper half plane

Define $H^+=$ {$z\in \mathbb C: y>0 $} $H^-$={$z\in \mathbb C: y<0$} $L^+$={$z\in \mathbb C: x>0$} $L^-$={$z\in \mathbb C: x<0$} The function $f(z)=\frac {z}{3z +1}$ maps $H^+$ on to ...
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5answers
94 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
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1answer
14 views

Lauret series around point $z_0 $=$1$

The function is: $f(z) = \frac{1}{(z-1)(z+1)}$. I compute that tis is equal to $\frac{1}{2}(\frac{1}{z-1} - \frac{1}{z+1})$. Now I need to show the radius of convergence and I started with $|z-1| ...
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Proof of an inequality in $\mathbb{C}$

Let $z\in \mathbb{C}, n \geq 2$. Show this complex inequality $$|z^n-1|^2\le |z-1|^2\left(1+|z|^2+\dfrac{2}{n-1}\Re{(z)}\right)^{n-1}$$ For $n=2$ the inequality is easy to prove: $$|z^2-1|^2\le ...
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1answer
21 views

Closed complex integral in an annulus

I have the function $$f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)},$$ and I want to find $$\oint_{|z|=1}f(z)dz.$$ What I know:  Let $A=\{z\in\mathbb{C}|r<|z|<R\}$ be the annulus with ...
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0answers
22 views

Evaluating a complex integral with a pole

I am asked to evaluate the integral $$\int_\gamma \frac{e^{2z^2}}{z^{77}}\,{\rm d}z$$ where $\gamma$ is a circle centre $0$ traversed once anti-clockwise. Clearly the integrand has a pole of ...
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0answers
22 views

Development of the Taylor series of $f(z)= \frac{1}{1+e^{-z}}$ around the point $z_0 = o$

Can I just write $\frac{1}{1+e^{-z}}$ = $\frac{e^z}{e^z + 1}$ and because $e^z$ = $1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + ... = \sum_{n=0}^{∞}\frac{z^n}{n!}$ is $\frac{e^z}{e^z + 1}$ = ...
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1answer
18 views

Development of the Taylor series of $f(z)= \frac{1}{1+z^2}$ around the point $z_0 = 1$

I tried with: $\frac{1}{1+z^2}$ = $\frac{1}{(1+zi)(1-zi)}$ Then $w = z-1$ and $z=w+1$ ... $\frac{1}{(1-(w+1)i)(1+(w+1)i)}$ = $\frac{A}{1-wi-i} + \frac{B}{1+wi+i}$ -> $A=1/2$ and $B=1/2$ I put A and B ...