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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33 views

To prove ring properties of analytic functions

Let $R$ be the ring of entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ that are analytic at every point of $\mathbb{C}$ with respect to point-wise addition and multiplication. Then show that ...
2
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1answer
46 views

Entire function $f$ and a function $g = \overline{f(\bar z)}$

Let $f$ be an entire function $f$ and a function $g = \overline{f(\bar z)}$. Then which of the followings are true. a) if $f(z) \in \mathbb R$ for all $z\in \mathbb R$, then $f =g$. b) if $f(z) \...
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0answers
48 views

To solve multivariate polynomial equations

For a system of multivariate polynomial equations like this: $$ \left( {\begin{array}{*{20}c} {\frac{{124}} {3}} & { - 24} & {\frac{{ - 68}} {3}} & {\frac{{68}} {3}} \\ {32} & {...
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2answers
81 views

How to solve $\int_{-\infty}^{\infty} \frac {sin(t)}{t^2+1} dt$?

I'm considering here the fact that $$\lim\limits_{R\to\infty} \int_{\Gamma_R} \frac {e^{iz}}{z^2+1} dz=0$$ , where $\Gamma$ is a contour defined as a semicircle centred about the origin, of radius $...
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0answers
26 views

Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
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0answers
15 views

Upper bounds for the modulus of $f(s)=\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}s\right)$

Let the complex variable $s=x+iy$, and $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisors function (is a known multiplicative function in number theory, for example $\sigma(1)=1$ and $\sigma(6)=1+2+3+6=...
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0answers
39 views

What can be said of the Convergence radius of a P-Series? [closed]

What can be said about the convergenceradius r of a p-series $P(z)=\sum_{n\ge0}a_n z^n$ if the coeffizient sequence $(a_n)_n$ satisfies one of the following conditions: a) there exists C>0 and N$\in\...
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1answer
19 views

Edge of convergence radius behaviour

what do i have to do if the excercise is "examine the behaviour at the edge of the convergence radius". I don't even know if that's the correct translation, please fix if not. For my actual ...
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127 views
+50

How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
2
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1answer
79 views

Is $|z^2|$ entire function?

I want to know whether $|z^2|$ is entire function or not. If I am not wrong then $z^2$ is entire but $|z|$ is not entire (Am I wrong here?) So, now how to say anything whether $|z^2|$ is entire or ...
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1answer
163 views

Is there a way to prove that i²=-1? [duplicate]

I have 4 questions regarding the imaginary and complex numbers. (And some ideas) My questions are about the way that I’m trying to come up with a proof to the equation i²=-1 (and from there maybe ...
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0answers
25 views

Basics on Schwarz -Christoffel Integral

I've just began to study the Schwarz-Christoffel integral, but I'm having trouble to understand some very basic points. For example, take $S:\mathbb{H}\to \mathbb{C}$ (where $\mathbb{H}:=\{z\in \...
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1answer
26 views

Is there a meaningful measure on analytic functions?

Let $\mathcal{B}$ be the functions analytic on the unit disk and continuous on its boundary. With the supremum norm this becomes a Banach space. Is there any way to define a meaningful measure on ...
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1answer
47 views

Flip integral boundaries of Delta function to get contradiction

Look at this equation: $\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}dy\delta\left(x-y\right)f(y)=\int_{-\infty}^{+\infty}dxf(x)$ If I flip integration boundaries of both integrals, minus ...
1
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1answer
33 views

Seeming Contradiction With Complex Exponents

I was fooling around while waiting for a page to load and came across the following "contradiction". Let $x=(-1)^{i}$. Then $x^{i}=(-1)^{i\cdot i}=(-1)^{-1}=-1$. Thus, $x=\left(x^{i}\right)^{-i}=(-1)^...
3
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1answer
48 views

Improper integral complex analysis $\int_{-\infty}^\infty \frac{e^{ax} \, dx}{\cosh(x)}$

I tried the following problem but I don't think I got the right answer. I checked it by substituting $a=\frac{1}{2}$ into the integral and putting that through Wolfram Alpha but it didn't match the ...
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0answers
26 views

Holomorphic function restricted to curves

Supppose $F:B_{r}(z_0) \to \mathbb{C}$ holomorphic with $F(z_0) = F'(z_0) = 0$ and $F''(z_0) \neq 0$. Show that there exist two curves $\gamma_1, \gamma_2: [0, 1] \to B_{r}(z_0)$ such that: i) $\...
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2answers
27 views

Complex function $f$ is either constant or unbounded, but maximum value still does exist even if $f$ is not constant?

In Complex Variables and Applications, Brown & Churchill (9th edition), I stumbled upon a chapter which got me somewhat confused. On page 175 of the book, there is the theorem, which states the ...
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2answers
72 views

How to read that sum?

My exercise is to show that $\sum_{n=0}^{\infty}{N+n \choose n}z^{n} =1/(1-z)^{N+1}$ where $N\in\{0,1,2,\ldots\}$ and $z\in\mathbb{C}$ with $\lvert z \rvert <1$. Now, this doesn't even work with $...
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1answer
26 views

Modulus of function values on circle is necessarily less than or equal to modulus at centre?

Consider a complex function $f(z)$, a circle $C_\rho$ with radius $\rho$, centred at $z_0$, which can be defined as $z=z_0+\rho e^{i\theta}$ $(\theta \in [0, 2\pi]$). In the book on complex variables ...
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1answer
44 views

Power Series Extensions into the Complex Plane

I'm working through Complex Analysis by Serge Lang, and I came across a part that I can't figure out on my own, and I was hoping that I can get help here. This is the theorem in question: Theorem ...
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2answers
65 views

Proof that $\overline{P(z)} = P(\overline{z})$ for polynomial $P$ with real coefficients

Let $$ a_0, a_1, a_2, a_3, \ldots , a_n \quad (n \ge 1)$$ denote real numbers, and let $z$ be any complex number. With the aid of $$ \overline {z_1 +z_2+ \ldots +z_n} = \overline z_1 +\overline z_2+ \...
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2answers
72 views

Prove $\sum \frac{\cos nz}{n!}$ converges on compact sets. [closed]

Prove that $$\sum \displaystyle\frac{\cos nz}{n!}$$ converges on compact subsets of complex plane.
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0answers
44 views

Inequality in proof of thm. 10.30 Rudin's Real and Complex Analysis.

I'm trying to understand the proof of theorem 10.30 and I'm missing something at the very beginning. Theorem 10.30 Suppose $\varphi\in H(\Omega)$, $z_0\in \Omega$, and $\varphi'(z_0)\neq 0$. ...
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0answers
32 views

Specific fucnction has 11 different zeros

Let $f : \mathbb{C} \to \mathbb{C}$ be given by $$ f(z) = z^{11} + 4 e^{z + 1} - 2 $$ Show that $f$ has 11 different zeros in the annulus $\{z \in \mathbb{C} : 1 < |z| < 3\}$. This is an old ...
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2answers
42 views

Show complex equation of closed curve integral

I need to show this equation: $$\frac{1}{2ia} \cdot \oint _{\gamma } \frac{e^{iz}}{z-ia}dz = \frac{e^{-a}}{2ia} \cdot \oint _{\gamma } \frac{1}{z-ia}dz$$ I have an hint to using Taylor. I have no ...
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1answer
34 views

An entire function of irrational order

It's easy to construct an entire function of order $\frac{p}{q}$ for any positive integers $p,q$. But is there an example of an entire function of irrational order ($\sqrt2$ for example)?
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27 views

Complex Analysis Problem (Argument principle or Rouché's Theorem ?)

My problem: Let f be analytic in $\overline{B(0;R)}$ with $f(0) = 0$, $f'(0)\ne0$ and $f(z)\ne0$ for $0 < |z| \le R$. Put $\rho = \min\limits_{|z|=R} |f(z)| > 0$. Define $N: B(0; \rho) \...
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0answers
18 views

Fourier transform spherically symmetric function with complex constant

In Gradshteyn's section 17.24 on Fourier transform pairs for spherically symmetric functions, the third entry relates $\frac{e^{-ar}}{r}$ and $\sqrt{\frac{2}{\pi}}\frac{1}{(a^2 + k^2)^2}$. I think ...
0
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1answer
26 views

biholomorphic on unit disk

Let $D$ be the unit disk and $f: D\rightarrow G$, $\; p_1$ the maximum value of $dist(f(z),f(0))\;$ and $p_2$ the minimum value of $dist(f(z),f(0))$ for $z\in \partial \bar G$ Prove that : $|f(z)-...
2
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1answer
51 views

Pullback of euclidean metric on the disc.

$\newcommand{\Im}{\operatorname{Im}}\newcommand{Re}{\operatorname{Re}}$Consider the biolomorphism $$f : D \to H$$ where $H$ is the complex upper hyperplane $\{\Im(z) > 0\}$ and $D$ is the unic disc,...
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1answer
39 views

Cauchy rieman equation. What is u?

Using cauchy rieman equation, i want to show the function is analytic. So i want to decompose from f(z) to two term (real part and imaginary part) With rectangular form or polar form. But it is so ...
6
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1answer
46 views

Fourier Transform of $\frac{1}{\sqrt{|x|}}$

I want to find the fourier transform of $\frac{1}{\sqrt{|x|}}$. I checked the table of common fourier transforms in Wikipedia, and I know the answer should be $$\sqrt{\frac{2\pi}{|\omega|}}$$ What I ...
0
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2answers
32 views

Complex plane mapping of circle

Could anyone help me with this question, I get a different answer to the textbook. Question The unit circle $\left|z\right|=1$ in the z-plane is transformed to the w-plane by the transformation $w=\...
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0answers
41 views

Laurent expansion of $f(z)=\frac{z}{z^2+1}$

The exercise: Find the laurent expansion of $f(z)=\frac{z}{z^2+1}$ in $K_{1,2}(-i)$. My thoughts: $K_{1,2}(-i)$ denotes the annulus. 1 and 2 the radiuses. First thing I did is decompose in partial ...
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0answers
23 views

meromorphic function on torus

Consider the familly of meromorphic function on the square torus (endowed with the corresponding complex structure) with $p$ simple poles and $p$ simple zeros and $L^1$-norm equal to $1$ : $\mathcal ...
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0answers
30 views

Possible root of a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+…a_1x-1$ [duplicate]

Let $p$ be a real polynomial of the real variable $x$ of the form $p(x)=x^n+a_{n-1}x^{n-1}+....a_1x-1$. If $p$ has no roots in the open unit disc and $p(-1)=0$, then can we predict the other possible ...
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1answer
18 views

Locally discrete and globally discrete set in $\mathbb{C}$

My aim is about constructing an entire function with prescribed set of zeros in $\mathbb{C}$. The question is what set should we take? Certainly we can't take $\{1,1/2,1/3,\cdots\}$. Can we take $\{1, ...
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0answers
61 views

If $f(z) = sinz$ has infinitely many solutions , then f is constant. [closed]

Let $f(z)$ be complex polynomial. Prove that If $f(z) =\sin z $ has infintely many solutions , then $f$ is constant. I think it may be proved by Rouche theorem.
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2answers
69 views

Construct a harmonic function that appears to be discontinuous on the unit circle.

Construct a harmonic function $u$ in $D(0,1)$ that satisfies $$ lim_{r \to 1^-}u(re^{i\theta}) = \begin{cases} 1 & 0 < \theta < \pi \\ 0 & \pi < \theta < 2\pi \...
6
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0answers
74 views

Rigorous justification for “complex” change of variable in integration

Suppose that I have $X_1,\ldots,X_n$ i.i.d. $\sim $ $X$ and $Y_1,\ldots,Y_n$ i.i.d. $\sim$ $Y$ for some continuous $X$ and $Y$. Consider the r.v.'s $\bar{X}=\frac{1}{n}\sum_jX_j$ and $\bar{Y}=\frac{1}{...
3
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3answers
49 views

$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx $

$$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx$$ where $a>b>0$ What shall I do? I have diffucty when I meet multi value function.
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1answer
44 views

Evaluating an integral in terms of the gamma function

I need to evaluate the following integral $$\int_0^\infty x^{\mu}\exp(-{\lambda}x^\kappa)\,dx$$ (where $\mu$, $\lambda$, and $\kappa$ are are all real) in terms of the Gamma function $\Gamma(t)=\int_0^...
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2answers
39 views

Existence (?) of an analytic $f$ such that $|f(z)|=e^{|z|},~|z|<1.$

Prove or disprove: there is an analytic function $f$ on $D=\{z:~|z|<1\}$ such that $|f(z)|=e^{|z|},~z\in D.$ Attempt. I believe that such a function does not exist (besides, a candidate ...
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1answer
33 views

Pointwise limit of $\sum{\frac{1}{n} \left(\frac{2z-i}{2+iz}\right)^n}$?

I am trying to find the set $\Omega$ where the series $$ \sum_{n \geq 1} \frac{1}{n} \left(\frac{2z-i}{2+iz}\right)^n $$ exhibits pointwise convergence. I have thought of several approaches: ...
2
votes
5answers
94 views

Evaluate $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}$ using contour integration

This question is Exercise 10 of Chapter 3 of Stein and Shakarchi's Complex Analysis. Show that if $a>0$, then $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}=\frac{\pi \log(a)}{2a}.$ The hint is ...
0
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0answers
49 views

Imaginary, holomorphic and bounded function, zero everywhere at $(i+n^\beta)$. Blaschke condition?

Is there a function $h\ne 0$, holomorphic and bounded on $\mathbb{H_+=\{z \in \mathbb{C}: Im\,z > 0}\}$ zero exactly at at all points $(i+n^\beta)$, $0 \lt \beta \lt +\infty$, $n\gt 0$ ? I have ...