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

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?
1
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1answer
29 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 ...
2
votes
2answers
24 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 ...
11
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1answer
57 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 ...
2
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1answer
30 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} ...
2
votes
0answers
33 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 ...
5
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2answers
55 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 ...
4
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0answers
17 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 ...
2
votes
3answers
84 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 ...
3
votes
2answers
38 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
votes
1answer
26 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 ...
3
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1answer
28 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 ...
6
votes
1answer
124 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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0answers
21 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 ...
3
votes
1answer
34 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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0answers
31 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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0answers
29 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 ...
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0answers
20 views

Three and a half basic questions on the Weil restriction of scalars

I am currently trying to get familiar with the Weil Restriction functor. For a finite field extension $L|K$ it associates a variety over $K$ to every variety $X$ over $L$ as the representing object ...
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0answers
17 views

Counting sectors in the complex plane.

Let $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(k)=\alpha\left(k+\frac{\beta}{n\alpha}\right)^{n}$, $n\in\mathbb{N}$, $\alpha,\beta\in\mathbb{C}$, $\alpha\neq0$, and $\text{Re}\, f(k)\geq0$ when ...
0
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2answers
25 views

How to write exp(2iz) in the form u(x,y)+iv(x,y)?

I took an exam on Complex Analysis recently, and questions involving the complex logarithm and exponential were a sticking point for me. Questions such as: Q. The function $f$ is defined by $f(z) = ...
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2answers
68 views

Simplify integral's to a real number

$$ I=\int_{-\infty }^{\infty}\frac{x^{2}dx}{1+x^{6}} $$ Simplify answer until you get an expression involving real numbers only. I have racking my brain on this and still can't get anywhere. Firstly ...
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1answer
183 views

Physical interpretation of residues

What is physical interpretation of residues of poles (of any order) of a complex function? Poles represents the points where a complex function cease to be analytic and residues are calculated to ...
6
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1answer
133 views

Conjecture on zeros of analytic function

I have a conjecture that I can´t prove nor disprove, any help on doing so will be very grateful. Let $f: \{z: |z|<2\} \to \mathbb C$ be a non constant analytic function such that if $|z|=1$ ...
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votes
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 ...
2
votes
1answer
32 views

Attempt at understanding Weierstrass points

I'm reading through Springer - Riemann surfaces and Farkas and Kra - Riemann surfaces and theta functions. I'm attempting to get an understanding of Weierstrass points. I've come up with a (hopefully) ...
1
vote
1answer
51 views

Evaluation of Real Integral

Given the following definition:$$I=\int\limits_{0}^{2\pi}e^{-i\theta n}\left(\frac{1}{n}\right)^{\rho e^{i\theta}}d\theta$$ Is there an analytic method for evaluating this integral? Best Regards
1
vote
1answer
36 views

Cauchy's Integral Formula, Evaluate

Evaluate, $$ \oint_{C}\frac{\cos2z}{z^{2}(z^{2}-z+1)}dz $$ where $C$ is the circle of radius $2$ centred at the origin. Answer should be in the format $J=A\cos(2z+)+B\cos(2z-)+C$. Really ...
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1answer
30 views

Stein & Shakarchi, Complex Analysis, Ch.3 Ex.7

Suppose $f : \mathbb{D} \to \mathbb{C}$ is holomorphic, and $d = \sup_{z,w \in \mathbb{D}} |f(z) - f(w)|$. Show that $$ 2 |f'(0)| \leq d$$ This entire exercise is a complete mystery to me and I am ...
4
votes
2answers
48 views

$\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues

I'm trying to find $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}dx$ where $\alpha>0$ is real. My approach was to take an integral along the real line from $1/R$ to $R$, around the circle ...
2
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0answers
15 views

A stochastic variant of the heat equation modulo $2\pi$ has weird unstable particle-antiparticle solutions. Does this equation have a name?

I implemented a discretization of a weird 2D heat equation "mod $2\pi$", $$\dot{f}(\mathbf{x},t)=\Delta^*f(\mathbf{x},t)$$ where (WARNING: handwavy, I'm not sure I understand it) ...
2
votes
1answer
40 views

Analytic function on the open unit disc

Let $\Bbb D$ be the open unit disc and $f:\Bbb D\to\Bbb C$ be an analytic function such that $|f(z)|\le |f(z^2)|$, for all $z\in\Bbb D$. Prove that $f$ is constant. Here is my proof: For any ...
2
votes
3answers
63 views

A question about complex integration of $\frac{1}{p(z)}$

Let $p(z)$ be a polynomial of degree $n\ge 2$. Is it true that, there is a $R>0$ such that $$\int\limits_{|z|=R}{\frac{1}{p(z)}dz}=0?$$ My attempt is: there is a $R>0$ such that $|p(z)|\ge ...
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0answers
25 views

central limit theorem for complex-valued martingale

Does martingale strong law of large numbers and martingale central limit theorem extends to complex-valued martingale?
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votes
0answers
32 views

what is the first derivative and second partial derivative of complex function?

what is the first derivative and second partial derivative of complex function to be used in Taylor's series expansion? Suppose this function is second order differentiable. Is it ...
3
votes
1answer
49 views

Is there a proof for the maximum principle without the Cauchy integral theorem?

All the theorems about holomorphic functions seem to rely on the Cauchy integral theorem: Liouvilles theorem about bounded whole functions, the maximum principle, the open mapping theorem for ...
3
votes
0answers
43 views

Which permutations of $\mathbb{C}$ commute with the Riemann zeta function?

I'm trying to figure out whether the permutations of $\mathbb{C}$ which commute with the Riemann $\zeta$ function are necessarily continuous or not. Obviously both the identity and the complex ...
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votes
1answer
38 views

Complex Roots Made Visible [on hold]

Can anyone point me to a pdf of the article, "Complex Roots Made Visible", by Norton and Lotto. Thanks, Ron
2
votes
1answer
30 views

Conformal mapping and its application in finding roots of polynomial

So for a polynomial, if we want to find the roots in a complex plane. Rouche's theorem is the first tool in my head. However, I saw several problems of finding the roots in the first quadrant or upper ...
1
vote
3answers
53 views

Using complex variables to find sums of Fourier series

Use complex variables to find the sum of the Fourier Series: $$\sin(\theta) + \frac{\sin(2\theta)}{2^{2}} + \frac{\sin(3\theta)}{2^{3}}+\cdots$$ where $\theta$ is a real variable.
2
votes
1answer
47 views

Green's function for Helmholtz equation for the plane with a hole

That is find $G$ which satisfies \begin{align} (\nabla^2+k^2)G(\mathbf{x}, \mathbf{y},\omega) = \delta(\mathbf{x}- \mathbf{y}) \end{align} subject to $$\frac{\partial G}{\partial y_n} = 0 ...
5
votes
1answer
73 views

Identity Principle type question: Prove that $f=g$

While reading a complex analysis textbook the following assertion came up Since $f,g:D\equiv D(a,r) \to \mathbb{C}$ are analytic and injective functions such that $f(D)=g(D)$, $f(a)=g(a)$ and ...
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0answers
35 views

Analytic $N$th roots: can my statements here be generalized?

I'm generally having trouble seeing when and why analytic functions have analytic $N$th roots. I know the following statements to be true, but I know that they can also be generalized in various ...
5
votes
1answer
52 views

Let $f(z)$ be a function analytic in a domain containing the segment $[0,1]$ and satisfying $f(z+1)=azf(z)+p(z)$.

Let $f(z)$ be a function analytic in a domain containing the segment $[0,1]$ and satisfying $$ f(z+1)=azf(z)+p(z) $$ in that domain, where $a\in\mathbb{R}$ and $p$ is a polynomial. Show that ...
5
votes
1answer
239 views

Riemann hypothesis reformulation - again

Yesterday I started to write a paper about the reformulation of the Riemann Hypothesis. My idea was to map the function such that all of the trivial zeros are outside of the unit disk, and the ...
1
vote
1answer
37 views

Show that Mergelyan's theorem cannot extend to the case in which $S^2-K$ has infinitely many components.

This is an exercise in W. Rudin's Real and Complex Analysis. For $n=1,2,\ldots$, let $D_n=D(\alpha_n;r_n)$ be disjoint open discs in (the unit open disk) $U$ whose union $V$ is dense in $U$, ...
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1answer
23 views

When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$ Note: I have tried using trigonometric formulas but it didn't work. Maybe I ...
0
votes
1answer
56 views

Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$ [on hold]

$Z_1,Z_2$ and $Z_3$ are affixes of points of equilateral triangle $ M_1 ,M_2$ and $M_3$. Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$.
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vote
1answer
82 views

Equivalence of holomorphic functions

Given that $$\left(1-\frac{z}{\zeta_j}\right)^{-z}=\sum\limits_{k=1}^\chi\frac{z^k}{k\zeta_j^k},$$ where $\chi$ is the largest nonnegative integer $k$ for which ...
2
votes
1answer
53 views

Do analytic functions on open subsets of $\mathbb{C}$ with an analytic square root form a sheaf? [duplicate]

I'm trying to learn algebraic geometry and am trying to think about what kinds of things are presheafs but not sheafs. One exercise I had was to show that bounded holomorphic functions on open ...