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

The Hessian quadratic form of a real function can't be complex

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know that $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as ...
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5 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
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2answers
523 views

Simpler way to evaluate the Fourier transform of $\exp\left(i e^x\right)$?

I have the task to evaluate $|a(k)|^2$ with $$ a(k) = \int_{-\infty}^\infty \!dx\,\exp\left(i k x + i e^{x}\right).\tag{1}$$ The integral in (1) can be evaluated explicitly via the substitution ...
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1answer
72 views

Complex integration on NON-simple closed curve

Compute the following integral with the help of Cauchy's residue theorem. $$\int_C\cot z\,dz$$where , $C:z=4e^{4i\theta}$ , $-\pi\le \theta\le\pi$ Here , singularities of are given by $\sin ...
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1answer
16 views

Combining Moebius transformations

Moebius transformation in this case $\frac{az+b}{cz+d}$ for complex $z$. I have several transformations I want to apply to an initial $z$. For example first transform $f(a,b,z) = z + (a + bi) = ...
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1answer
33 views

On every simply connected domain, there exists a holomorphic function with no analytic continuation.

I am working on a question that requires me to prove that on every simply connected open subset of $\mathbb{C}$, there exists a holomorphic function that cannot be extended to a holomorphic function ...
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1answer
91 views

Is there a deep reason why replacing $\cos(x)$ with $e^{ix}$ and taking the real part often makes a contour integral work out?

I'm grading a complex analysis course right now and it turns out to involve a lot of contour integration. For instance, students are asked to find the integral $$\int_0^\infty \frac{\cos (ax)}{(x^2 ...
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2answers
30 views

Complex analysis proof about $|f(z)|$

I have to prove the following and have absolutely no idea where to start: If $f$ is holomorphic in $|z|>R$ and its limit at $\infty$ is $0$, then $\exists \; m \in \mathbb{N}$ such that $|f(z)| ...
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1answer
53 views

Find all entire function $f$ such that $\lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0$

If $f$ is an entire function such that $\lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0$ then find the function $f$. Replacing $z$ by $\frac{1}{z}$, we get $$\lim_{z\to 0}|zf(1/z)|=0$$This shows ...
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18 views

Explain about proof

Let $0 \leq R_1 \leq R_2 \leq \infty$ and let $f$ be holomorphic in the annulus $R_1 < |z - z_0| < R_2 $. Then, for any $r_1, r_2, z $ such that $R_1 < r_1 <|z-z_0| < r_2 < R_2$, we ...
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25 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
2
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0answers
30 views

Integral using Cauchy's integral formula and residue theorem

So, I'm having trouble getting the correct value for the integral $\int_0^{2\pi} \frac{\cos^2(3\theta)}{5-4\cos(2\theta)}\mathrm{d}\theta$. I substitute the exponential form of cosine into the ...
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1answer
118 views

why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarh's book on the Riemann Zeta Function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = 1 ...
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1answer
47 views

Is there a holomorphic function $f$ on the unit disc such that $|f(z)|\rightarrow\infty$ as $|z|\rightarrow 1$?

When I learnt that there exists a holomorphic function on the unit disc $D$ that cannot be continuously extended to a domain that is strictly larger $D$, I was taught the example ...
5
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1answer
25 views

Prove that the given condition implies analytic continuation

Here is an old qual problem I'm working on, I have some idea, but I'm not sure if I'm correct or not. I would be happy if anyone could possibly confirm or correct me: Let $U=\{z\in \mathbb{C} : ...
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1answer
241 views

Evaluate the following integrals/ Cauchy integral theorem

So I have two questions. 1) Evaluate $$ \oint\limits_{|z|=1} \dfrac{\cos(\pi z^2)}{(z-2)(z-4)^3} dz$$ and 2) Evaluate $$ \oint\limits_{|z|=6} \dfrac{\cos(\pi z^2)}{(z-2)(z-4)^3} dz.$$ Now I know the ...
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21 views

How to show $\Im\{z\cot(z)\}$ is not $0$ in the first quadrant?

I know that $\Im\{z\cos(z)/\sin(z)\}$ is non-zero in the open first quadrant of the complex plane, $\Im z > 0$, $\Re z > 0$, but somehow I cannot seem to show it directly. I think I must be ...
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1answer
42 views

Functions of complex numbers

Let $f(z) = \sqrt z$. Using the branch that is defined everywhere except where $z = x + iy$ with $y = 0$ and $x < 0$. What are the formulas for real valued functions $u(x, y)$ and $v(x, y)$ such ...
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42 views

Complex analysis, poles and singularities and boundedness

So I am on the following problem: Prove that an isolated singularity of $f(z)$ is removable as soon as either $\text{Re}f(z)$ or $\text{Im}f(z)$ is bounded above or below. The hint is to use a ...
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104 views

Confused about Pochhammer contour?

I know some theorems about complex analysis such as the argument principle. But I do not get the Pochhammer contour. I read about it on the wiki page of the beta function , but I do not understand a ...
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1answer
56 views

Simple example of the use of sheaves

What would be (one of) the simplest example of a mathematical result which is solved using the concept of sheaves?
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1answer
51 views

Describing the zero level set of a harmonic function

Let $p(k)$ be a complex polynomial of degree $n\in\mathbb{N}$. Let $A=\{k\in\mathbb{C}:\text{Re}\,p(k)=0\}$ The harmonic function $\text{Re}\,p(k)$ determines the behaviour of $A$. Fix $z\in A$. If ...
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1answer
55 views

What is the solution to this integral?

In some calculation, I encounter an integral of the form \begin{equation} \int_{-\infty}^\infty \text dz\ \frac{1}{z-i\varepsilon}e^{- a z^2+i b z}, \end{equation} where $a>0$ and $b$ are some ...
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1answer
39 views

Contour integral of $\frac{1}{\sqrt z}$ with branch cut

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals : ...
2
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1answer
42 views

Quartic equation or Sextic equation? And how to solve it?

In this arxiv paper (p. 11, eq. (3.2)) the authors claim that equation (3.2) is ... a quartic equation [...] which can be solved explicitly. The equation in question is \begin{equation} ...
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3answers
53 views

Complex integration by Cauchy's residue theorem

Evaluate the following integral by Cauchy's Residue Theorem $$\int_C\frac{2z^2-z+1}{(2z-1)(z+1)^2}\,dz$$where , $C:r=2\cos \theta$ , $0\le \theta \le \pi.$ I have problem about the contour ...
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2answers
495 views

Most general linear transformation which leaves the origin fixed and preserves all distances?

I'm working on the following problem from Ahlfors (complex analysis): Prove that the most general (linear) transformation which leaves the origin fixed and preserves all distances is either a ...
6
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1answer
37 views

Proving Plancherel's theorem using Cauchy integral formula

Plancherel's theorem says that $f(x) = \frac{1}{2\pi} \int^\infty_{-\infty} F(k) e^{ikx} dk$ where $F(k) = \int^\infty_{-\infty} f(x)e^{-ikx}dx$. I'm wondering if we can prove this using Cauchy's ...
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1answer
41 views

A question about current and Dirac measure

$0$ can be seen as a divisor of $\mathbb{C}$, and the current $[0]$ is defined as $[0](\varphi)=\varphi(0)$. Why is this reasonable?
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3answers
81 views

$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$

Prove that for all $\xi \in \mathbb{C}$, $$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$$ I don't really know how to compute this integral. Can you please help me?
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22 views

Radial limits of composition of functions

Is it true that if $f\in H(U)$ is a holomorphic function whose nontangential limits exist a.e and $g\in H^\infty(U)$ is a nonconstant function whose range is in $U$ and whose radial limits exist ...
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1answer
79 views

Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
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2answers
236 views

Why does $\int_0^{\infty}\frac{\ln (1+x)}{\ln^2 (x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?

I'd like to see the reason why $$\int_{0}^{\infty}\frac{\mathrm{ln}(1+x)}{\mathrm{ln}^2(x)+\pi^2}\frac{dx}{x^2}=\gamma$$ where $\gamma$ is the Euler-Mascheroni constant. I don't have any 'neat ...
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3answers
41 views

Proof for inequality with complex numbers

If $a,b,c$ and $d \in \mathbb{R}$ show that $$ac+bd \leq \sqrt{a^2+b^2}\sqrt{c^2 +d^2}$$ Let's use $z=a+bi$ and $g=c+di$ so $|z|=\sqrt{a^2+b^2}$ and $|g|=\sqrt{c^2 +d^2}$. So the equation is ...
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1answer
449 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
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42 views

(Theoretical) Complex Analysis Textbooks

Most books I've seen on complex analysis do not develop it theoretically, which can be somewhat infuriating for the budding pure mathematician. What I am looking for are some comprehensive, rigorous ...
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1answer
32 views

I might need some help on this Complex Fourier Series Problem

Here is the problem: Use the complex Fourier Series on $[-L,L] $ with complex coefficients to find a representation of $\frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} dx$ Here is my attempt: The ...
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2answers
64 views

geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
2
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3answers
89 views

Compute the integral

Compute the integral: $$\frac{1}{2\pi i}\int_{|z| = 1}\frac{(z-b)^m}{(z-a)^n}dz$$ where $|a| < 1 < |b|$; $m, n \in \mathbb{Z}$ My approach is using Cauchy integral formula, we have ...
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2answers
53 views

Extending the Riemann zeta function using Euler's Theorem.

Euler's theorem states that if the real part of a complex number $z$ is larger than 1, then $\zeta(z)=\displaystyle\prod_{n=1}^\infty \frac{1}{1-p_n^{-z}}$, where ...
6
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3answers
211 views

Infinite Series $\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}$

How can we prove the following formula? $$\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}=\frac{-2m+\sqrt{mn}\pi\cot\left(\sqrt{\frac{n}{m}}\pi\right)+n\pi^2\csc^2\left(\sqrt{\frac{n}{m}}\pi\right)}{4mn^2}$$ ...
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50 views

Inverse Gamma function for integers (Hankel)

So I want to prove that for all integers $n \in \mathbb{Z}$ it holds that $$F(n):= \frac{1}{2\pi i} \int\limits_{\gamma}^{} s^{-n}e^{s} ds = \frac{1}{\Gamma(n)},$$ with $\gamma$ the 'Hankel'-contour: ...
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2answers
43 views

compute the integral $\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$

Compute this integral: $$\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$$ my solution is I used derivative of Cauchy integral formula, which is $$f^{(n)}(z_0) = \frac{n!}{2\pi i}\int ...
2
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1answer
27 views

harmonic functions on the disk which agree on the real are identical?

The question is whether it is possible to find two distinct harmonic functions on the unit disk $\mathbb{D}=\{z: \ |z|<1 \}$ such that they agree on $\mathbb{D} \cap \mathbb{R}$. If yes, please ...
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1answer
34 views

the sequence of derivative cannot satisfy $|f^{(n)}(z_0)| > n!n^n$

Let $f: \Omega \to \mathbb{C}$. Prove that for any $z_0 \in \Omega$, the sequence of derivatives cannot satisfy $|f^{(n)}(z_0)| > n!n^n$ In this problem, I intend to prove by contradiction, and I ...
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26 views

Uniformly bounded family of harmonic functions

I am pretty sure other questions on this site can answer this problem, but I'm really interested in knowing if this particular solution is valid. Thanks. Question: Let $U$ consists of the set of ...
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33 views

Limits of complex line integrals as the Radius goes to infinity [on hold]

I have been stuck on this problem for a long time: Let $C$ be the circle $|z|=R$. Show $$\lim_{R \to \infty} \int_{C_R} \frac{(z^2 +2z-5)dz}{(z^2+4)(z^2+2z+2)} = 0$$ Use the result of 1. to deduce ...
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1answer
126 views

Show that there is no such entire function

This is an old qual problem I'm working on: Show that there is no entire function $f(z)$ satisfying $|f(z)-e^{\overline{z}}|\leq 3|z|$ for all $z\in \mathbb{C}$. I tried to use Liouville's theorem by ...
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35 views

Exact Differentials in Complex Variables [on hold]

I have been stuck on this problem for a long time :Let P and Q be continuous and have continuous partial derivatives in a region R .Let C be any simple closed curve in R and suppose that for any such ...
3
votes
1answer
89 views

When does $\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$

Considering the complex logarithm, when do we have $$\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$$ I originally wanted to try to prove it by showing $$\lim \prod^N ...