Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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Pointwise limit of sequence of holomorphic functions given constraint on their derivatives at the origin

Consider a sequence of functions $g_n(z)$. $n$ takes values on the natural numbers, and $z$ is a complex variable. For all $n$, $g_n(z)$ is guaranteed to be an analytic function of $z$ within a disk ...
user196574's user avatar
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Improper Integral of a complex-valued function with a singularity on the real line

As part of an example in the Complex Analysis book by Bak and Newman, the authors use a nice trick which is: $$ \int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx = Im \int_{-\infty}^{\infty}\frac{e^{ix}}{x}...
giorgio's user avatar
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An inequality about the image of holomorphic functions on unit disk

Suppose $f:\mathbb{D} \to \Omega$ is a holomorphic surjective function on the unit disk $\mathbb{D}$,with $f'(z)\neq 0 $ for all $z \in \mathbb{D}$. Show that $\text{dist}(f(0),\partial\Omega)\leq f'(...
Isllier's user avatar
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Some questions on Gamma function expression as infinite product

Let´s consider the Gamma function expression due to Euler and Gauss: $$\Gamma(z) = \frac{1}{ze^{\gamma}\prod_{n=1}^{\infty} (1+\frac{z}{n})e^{\frac{-z}{n}}} $$ I am interested in showing that $\...
Mths's user avatar
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Multiplication operator is a closed

Let $\phi$ be a holomorphic function on the unit disk $\mathbb{D}$. We define the multiplication operator $M_{\phi}$ on the following domain $D = \{f \in H^{2}(\mathbb{D}): \phi f \in H^{2}(\mathbb{D})...
liamsi Meean's user avatar
1 vote
3 answers
38 views

Find the principal part at poles

I am asked to find the isolated singularities of the function and determine their typing, order, and finding the principal part at each pole. So for the first function, which is $\frac{e^z-e}{z^2-1}$, ...
robert lewison's user avatar
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Distinguishing poles and zeros in winding number calculation

I am trying to develop a numerical algorithm for finding the roots of a generic complex function $f(z)$ using winding numbers. This is done by dividing the desired complex domain into "boxes"...
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How to find coefficient $a_n$ for chebyshev approximation of an analytic function over a disc?

This is regarding Chebyshev approximation for an analytic function defined over a unit disc. For $e^z=\sum_{n=0}^{\infty}a_nT_n(z)$. How to find coefficient $a_n$? What could be the contour integral? ...
108_mk's user avatar
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1 answer
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Where in the proof is necessary the "simply connected" hypothesis in Caratheodory's Theorem?

The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265) Proposition 14.5. Suppose $D$ is a bounded simply connected open set in the plane, and let $\phi: D \...
MathLearner's user avatar
2 votes
2 answers
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Complex Integral ML Lemma

I must solve $$\int_{-\infty}^{\infty}\frac{x \text{sin}x}{x^2+4} dx$$ I simplified this to $$\int_{-\infty}^{\infty}\frac{x \text{sin}x}{x^2+4} dx = \frac{1}{i}\int_{-\infty}^{\infty}\frac{x e^{ix}}{...
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Using ML Lemma to Bound an Integral

I have $$\int\frac{ze^{iz}}{z^2+4} \text{dz}$$ which I am trying to bound using the ML Lemma over the arc in the complex plane of radius $R$ across points $(R,0),(0,R),(-R,0)$. I know $L= \pi R$ For $...
adisnjo's user avatar
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Hörmander's proof of Beurling's uncertainty principle and the application of Phragmén-Lindelöf

I am trying o understand the proof in the following paper. https://projecteuclid.org/journalArticle/Download?urlId=10.1007%2FBF02384339 where it is proven that if $$\int \int_{R^2}|f(x)\hat{f}(y)|e^{|...
Valsinator's user avatar
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1 answer
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Intuitive difference between complex differentiability and $R^2 \to R^2$ real differentiability

Let $g:R^2 \to R^2$ and $f:\mathbb C \to \mathbb C$ with $g(x,y) = (\operatorname{Re}(f(x+yi)), \operatorname{Im}(f(x+yi))$. It is a stricter condition to assume that $f$ is complex-differentiable (CD)...
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On Wikipedia's Proof of Cauchy's Integral Theorem from Green's Theorem

I struggle to understand two steps in Wikipedia's proof of Cauchy's Integral Theorem Assuming -for simplicity- that $\gamma$ is non-self-intersecting, what exactly is the definition of the region $D$ ...
Sam's user avatar
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2 votes
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Growth order of $e^{p(z)}$ for a complex polynomial, $p(z)$

Let $p(z):\mathbb{C}\to\mathbb{C}$ be a degree $m$ polynomial, and consider $e^{p(z)}$. I'm wondering whether it's true that the order of $e^{p(z)}$ is equal to the degree of $p(z)$. My working ...
Ty Perkins's user avatar
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Find the bounded harmonic function in the (closed) upper half-plane whose restriction to the real axis is sinx/x [closed]

Find the bounded harmonic function in the (closed) upper half-plane whose restriction to the real axis is sin(x)/x
maxim.rainin's user avatar
1 vote
2 answers
33 views

Find the $\int_C \frac{e^{-2z^2}}{(z-2i)(z-5)} dz$

I am trying to find $$\int_C \frac{e^{-2z^2}}{(z-2i)(z-5)}\,dz $$ where $C$ is the circle of radius three about the origin and I am not sure if my approach is correct. I noticed that out of the ...
Math Undergrad Student's user avatar
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1 answer
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Proving an infinite series is $o(1)$ using asymptotics

Prove that $\sum_{r=1}^{\infty}\left|f_N(r+1)-f_N(r)\right|=o(1)$, where $f_N(r)\sim\frac{(\log\log N)^r}{(r-1)!\log N}$ and these bounds are uniform in $r$. My naïve idea is that since these bounds ...
alidixon222's user avatar
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1 answer
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Method of Steepest Descent (deform contours where there are 2 saddles)

Question: use the method of steepest descent to obtain the first two non-zero terms in the asymptotic approximation $$\int_0^\infty \exp(ix(t^3/3+t))dt\sim i(1/x+2/x^3+...+a_n/x^n)$$ as $x\to\infty$ ...
vegetandy's user avatar
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4 votes
1 answer
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Cauchy product of zeta function

$\zeta(s)= \sum_{n=1}^{\infty} \frac{1}{n^s}; s \in \mathbb{C}$. For $Re(s) > 1 $, we have that the above series converges absolutely. And in this case, I wrote the Cauchy product of $\zeta(s)$ by ...
J P's user avatar
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2 answers
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Classifying Singularities of a Rational Function

I have $f(z) = \frac{cos(\frac{1}{z})(z+2)}{z-3}$ I know that $z=3$ is a singularity and I found that it is a simple pole because the limit as $z$ tends to $3$ of $f(z) \cdot (z-3) = 5\text{cos}(\frac{...
adisnjo's user avatar
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1 vote
1 answer
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Why is the Jacobi amplitude real when $k>1$?

I understand the Jacobi amplitude is defined as the inverse function of the incomplete elliptic integral of the first kind $$ \mathrm{am}(u,k) = F^{-1}(u,k), $$ where $$ F(u,k) = \int_0^u \frac{d\phi}{...
Khalid Wenchao Yjibo's user avatar
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Source of Proof of a theorem on Area of Pre-image under a complex polynomial

The following fascinating theorem ,attributed to Polya is mentioned in the introduction of the paper "The Areas of Polynomial Images and Pre-Images by Edward Crane" paper link.Could ...
AgnostMystic's user avatar
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1 answer
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Complex Integral and Residue involving multiple Branch Points inside the contour, without deforming it

I encountered the integral: $$\oint_{|z|=1} \frac{f(z) \ dz}{\sqrt{(z-a)(z-b)}} \ \ \ \ \text{with} \ \ \ |a|,|b| < 1$$ So that the branch points are inside the contour. I am not adding the ...
prikarsartam's user avatar
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Reading Ahlfors and Folland with some friend [closed]

I know the books aren't the easiest but I can't fight it I love to read hard and comprehensive books so what I want is some friend to read these books with and I want him to make me feel I am not good ...
Abed Urdnia's user avatar
0 votes
4 answers
66 views

Show that $f(\frac{1}{z})$ have a essential singularity at $0$.

We need to show that if $f(z)$ ,who is entire, periodic and non-constant , then $f(\frac{1}{z})$ have an essential singularity at 0. So we then need to show that: $f(\frac{1}{z})$ as not a pole at $z=...
Student_Number_249812341's user avatar
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0 answers
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Finding the imaginary part of $\ln\left(\ln\left(\frac{1+e^{2ix}}{2}\right)\right)$, where $x\in[0,\frac\pi2]$

I am looking for the imaginary part of the following expression $$\ln\left(\ln\left(\frac{1+e^{2ix}}{2}\right)\right)$$ where $x\in[0,\frac\pi2]$. The attempt I made was the following $$\begin{align} \...
Jessie Christian's user avatar
3 votes
0 answers
59 views

Question about uniform convergence in a proof

The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265) Proposition 14.5. Suppose $D$ is a bounded simply connected open set in the plane, and let $\phi: D \...
MathLearner's user avatar
5 votes
0 answers
124 views

Finding a closed form for $ \int_0^1 \frac1x \ln\left(\frac{\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $

I want a closed form for the following integral $$ \int_0^1 \frac1x\;\ln\left(\frac {\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $$ An integration by parts ...
Jessie Christian's user avatar
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0 answers
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Dirichlet's series

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of complex numbers. Furthermore, suppose that exists some $z_0 \in \mathbb{C}$ such that $\sum_{n=1}^{\infty}\frac{a_n}{n^{z_0}}$ converges. Now, my goal ...
J P's user avatar
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1 vote
0 answers
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Spivak "Calculus" understanding authors intend.

As a matter of fact, it is impossible to find a continuous f such that $(f(z))^2 =z$ for all $z$. In fact, it is even impossible for $f(z)$ to be defined for all $z$ with $|z|=1$. To prove this by ...
emil agazade's user avatar
-2 votes
0 answers
33 views

Show that $\phi\circ f$ is subharmonic [closed]

Let $f:G\rightarrow\Omega$ be a conformal equivalence and $\phi:\Omega\rightarrow\mathbb{R}$ be subharmonic. Show that $\phi\circ f$ is subharmonic. I tried to prove it but not succeed.
Tahmuras's user avatar
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0 answers
49 views

Can $\int_{-\infty}^\infty{\frac{\cosh(\xi t)}{\xi(\xi+\beta)\sinh(\xi g)}\mathrm e^{j\xi k}}\mathrm d\xi$ be calculated? [closed]

I used to think that this infinite integral problem was not difficult. But now I think I'm too naive. $$ \int_{-\infty}^\infty{\frac{\cosh(\xi t)}{\xi(\xi+\beta)\sinh(\xi g)}\mathrm e^{j\xi k}}\mathrm ...
adios518's user avatar
0 votes
1 answer
29 views

Computing Singularities of a function

We have $f(z) = \frac{e^{iaz}-e^{ibz}}{z(z^2-4z+5)}$, where $a$ and $b$ are not equal to each other and positive. I am trying to classify the singularities $z=2+i$ and $z=2-i$. I tried finding the ...
adisnjo's user avatar
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1 vote
0 answers
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Would it be possible to prove Liouville's theorem from the analyticity property of holomorphic functions alone?

Obviously a polynomial is either constant or unbounded, and from Liouville's theorem it seems to follow that the same is true for a power series with an infinite radius of convergence. (Such a ...
A A's user avatar
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2 votes
2 answers
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Examples of non-elliptic Doubly Periodic Functions

I'm reading something on elliptic functions and I do not understand why in various textbooks the authors do not give non-trivial examples of doubly periodic and elliptic functions. Indeed, they only ...
TheWanderer's user avatar
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-1 votes
2 answers
54 views

Prove that, except for the identity function, a holomorphic map of the open unit disk into itself has at most one fixed point in the disk. [duplicate]

Prove that, except for the identity function, a holomorphic map of the open unit disk into itself has at most one fixed point in the disk. By Schwarz Lemma, if $f$ is the holomorphic map, and $f(0)=0,$...
Tapi's user avatar
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1 vote
1 answer
37 views

Applying the Identity Theorem to Analytic Functions Agreeing on 1D Curves

I'm working through a complex analysis problem and have encountered a problem that I'm struggling to understand. The context involves two meromorphic functions, g and h, which are given on the unit ...
zich's user avatar
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1 vote
0 answers
49 views

Entire function $f$ such that $f(z_0 +z) =f(z_1 +z) =f(z)$ is constant [duplicate]

Take $z_0,z_1$ $\in \mathbb{C}$ to be $\mathbb{R}$ linearly independent. The exercise is to prove that if $f:\mathbb{C} \rightarrow \mathbb{C}$ is entire such that $f(z_0 +z) = f(z)$ and $f(z_1 +z) = ...
Lucas G's user avatar
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2 votes
2 answers
77 views

$\sum_{n=R}^{\infty}{\binom{n}{R}\frac{a^n}{(1+a)^{n+1}}} = a^R$

for any $R \in \mathbb{N}$ I want to show the identity $$\sum_{n=R}^{\infty}{\binom{n}{R}\frac{a^n}{(1+a)^{n+1}}} = a^R$$ I had an argument that I really liked but unfortunately it has some mistakes, ...
Paul's user avatar
  • 1,344
5 votes
1 answer
70 views

Rudin theorem $7.8$

There is the definition of $(D\mu)(x)$: Accordingly, let us fix a dimension $k$, denote the open ball with center $x\in\mathbb{R}^k$ and radius $r>0$ by $$B(x,r)=\{y\in\mathbb{R}^k:\lvert y-x\...
JohnNash's user avatar
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1 vote
1 answer
37 views

exponential map property

Consider $f:[0,1)\rightarrow \mathbb{S}^1$ given by $f(t)=\exp(it)$ Note, $f'(t)=i\exp(it)$ so $|f'(t)|=1$. $|f(t)-f(s)|=\int_s ^t f'(t) dt \leq |t-s|$ for all $s,t\in [0,1)$ We know that $f$ has an ...
monoidaltransform's user avatar
0 votes
1 answer
117 views

How is $|z|^2$ not analytic?

I am aware that the C.R equations are not satisfied for $|z|^2$. However, I tried to prove using limits. My approach was using the formula for derivative of a complex function: $f'(0) = \lim_{z\to 0}\...
Praneel65's user avatar
0 votes
2 answers
67 views

Sum of reciprocals of norms of Gaussian primes

A Gaussian integer is of the form $m+ni$ for $m,n\in\mathbb{Z}$. $m+ni$ is a subset of the Gaussian primes (denoted $\mathbf{P}$) if $m^2+n^2$ is a square of a prime congruent to $3\pmod{4}$ in $\...
alidixon222's user avatar
2 votes
0 answers
37 views

Entire functions which satisfy a coefficient property

Let $f$ be an entire function. Then we can write $$f(z) = \displaystyle \sum_{n=0}^{\infty}c_n z^n$$ for some $c_0, c_1, \ldots$. For a positive real $r$, let $$M_r = \sup \{ |f(z)| : |z| = r \}$$. ...
idk31909310's user avatar
2 votes
1 answer
60 views

Meromorphic continuation of Euler product

Short version: What can be said about the meromorphic continuation of the Euler product $$\prod _{p}\left (1+\frac {p^{-s}}{p-2}\right )?$$ Longer version: I realise I have some misconceptions about ...
tomos's user avatar
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41 views

Show that $\left\lvert e^{ix}-\sum\limits_{k=0}^{n}\frac{(ix)^k}{k!}\right\rvert\leq\min\left \{\frac {2|x|^n}{n!},\frac {|x|^{n+1}}{(n+1)!}\right\}.$

Show that $$\left \lvert e^{ix} - \sum\limits_{k=0}^{n} \frac {(ix)^k} {k!} \right \rvert \leq \min \left \{\frac {2 \left \lvert x \right \rvert^{n}} {n!}, \frac {\left \lvert x \right \rvert^{n + 1}}...
Akiro Kurosawa's user avatar
1 vote
1 answer
35 views

Confusion in proof of $H^1(X,\mathcal{O})=0$ where $X$ is an open disk

In Otto Forster's Lecture on Riemann Surfaces, I have faced a small confusion in which I am certainly overlooking something in the proof of Theorem 13.4 that for $X:= \{z\in \mathbb{C}: |z|<R\}, 0&...
Hushus46's user avatar
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0 votes
2 answers
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Calculate $\int_{-\infty}^{\infty}\frac{e^{itz}}{(z+i)^2}\,\mathrm{d}z$

Trying to calculate $$ \int_{-\infty}^{\infty}\frac{e^{itz}}{(z+i)^2}\,\mathrm{d}z $$ using contour integration. for $t > 0$, I use the lower semicircle that encapsulates the singularity at $z = -i$...
giorgio's user avatar
  • 463
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0 answers
18 views

Conformal equivalence from $D = \{z : 0 < \arg z < 3\pi / 2\}$ to $S = \{z: 0 < \Im z < 1\}$

I would like to verify if the conformal equivalence from $D = \{z : 0 < \arg z < 3\pi / 2\}$ to $S = \{z: 0 < \Im z < 1\}$ I have found is correct. First, consider the map $f: D\to \mathbb{...
idk31909310's user avatar

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