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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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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13 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 ...
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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
62 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
162 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 ...
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1answer
126 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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1answer
42 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 ...
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1answer
28 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) ...
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1answer
47 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
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1answer
33 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
29 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 ...
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2answers
46 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 ...
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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) ...
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1answer
39 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 ...
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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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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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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 ...
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1answer
48 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 ...
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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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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
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1answer
26 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 ...
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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.
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1answer
46 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 ...
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1answer
71 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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34 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 ...
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1answer
50 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 ...
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1answer
237 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 ...
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1answer
36 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 ...
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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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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 ...
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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 ...
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2answers
35 views

Complex Analysis ( Open/Closed Set).

let $z = re^{i\theta}$ , How do we prove that , $0\leq \operatorname{arg}(z)\leq\dfrac{\pi}{4}$ ($z \neq 0$), is neither a open set nor a closed set. $\operatorname{arg}(z)$ is nothing but $\theta$ ...
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3answers
39 views

Is the boundedness necessary to extend harmonically?

"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question ...
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38 views

Complex dot product

I know that the complex dot product is defined as $\boldsymbol{a}\cdot\boldsymbol{b}=\sum_{i}a_ib_i^*$. Is there a standard name for the operator ...
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1answer
66 views

How to differntiate $\int_{0}^{2\pi} u(re^{i\theta}) d\theta$?

Suppose $u$ is a twice continuously differentiable function on $a< |z|<b, \ z\in \mathbb C,$ which is harmonic that is, it satisfies $u_{rr}+\frac{1}{r}u_r + u_{\theta \theta}=0.$ (If we put ...
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1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
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2answers
37 views

A question regarding a proof in Ahlfors

Ahlfors says the following: if $f (z) $ is analytic on a disc, then its integral along any closed path contained in the disc is $0$. The proof for this is the following: Let $F (z)=\int_\sigma {f ...
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1answer
27 views

If $\phi$ is entire and satisfies $|\phi(z)| \leq e^{|z|}$, then $|\phi'(z)| \leq c e^{|z|}$ for some $c > 0$.

If $\phi$ is entire and satisfies $|\phi(z)| \leq e^{|z|}$, then $|\phi'(z)| \leq c e^{|z|}$ for some $c > 0$. I saw this problem on a practice qual but I had no idea what to do. It looks ...
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1answer
49 views

Functions with real domain but complex range, do they have any use?

For example if we define the square root function like this: $$\text{Sqrt}({x})= \begin{cases} \sqrt{x} & x\geq 0 \\ i\sqrt{-x} & x<0 \end{cases}$$ Or we could have an exponential ...
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2answers
53 views

Show that $\lim\limits_{z \to a}\frac{\log|f(z) - f(a)|}{\log |z - a|}$ is an integer.

Let $f$ be analytic in a neighborhood of $a$. Show that $\lim\limits_{z \to a}\frac{\log|f(z) - f(a)|}{\log |z - a|}$ exists and is an integer. We have $$\frac{\log|f(z) - f(a)|}{\log |z - a|} = ...
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26 views

how to solve complex differential equation [on hold]

how to solve this differential equation $$ a_1\cdot \phi'(x)^+ + a_2 \cdot \phi'(x)^-=c_2\cdot g(x) $$ where $\phi(x)$ is a complex analytic function thanks
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53 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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35 views

Holomorphic proper function on $\mathbb{C}$ is a polynomial

I want to show that every holomorphic proper map $f:\mathbb{C}\to \mathbb{C}$ is a polynomial. Since $f$ is continous and proper, it can be extended to a continous map $f:S^2\to S^2$, where $S^2$ is ...
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1answer
34 views

Prove that a harmonic function is an open map.

I'm trying to solve the following exercise of the book Functions of one complex variable, John B. Conway on page 255: 4. Prove that a harmonic function is an open map. (Hint: Use the fact that the ...
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1answer
39 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
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1answer
34 views

Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
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34 views

A doubt regarding a property of a complex function

I just wanted to confirm that when we define a complex function $f (z) $, is it just a function in terms of $z $, or also that in terms of $\overline {z} $? This is because if the latter is true, ...
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1answer
43 views

Variant of Riemann mapping theorem

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk) and let $\Omega$ be non empty open simply connected in $\mathbb C$ and $\Omega \neq \mathbb C.$ Then Riemann mapping theorem tells us that there exists ...
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17 views

Complex Normal Gaussian noise

I would like to create complex normal Gaussian noise with dimensions $(M,N)$ The noise should have zero mean and $var=1$. How can I do so?