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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Equation of convolution of measures

Let $\mu_1,\mu_2$ be two locally finite complex regular Borel measures on $[0,+\infty)$ and $\delta_x$ be the Dirac measure at point $x\in[0,+\infty)$. Suppose that for all $x\in(0,+\infty)$ $$\...
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36 views

Degree of the Divisor of a Theta Function

Let $(\gamma_1, \gamma_2)$ be a base for a lattice $\Gamma$ in $\mathbb C$, and $\theta$ a theta function, ie an holomorfic function such that $\theta(z+ \gamma) = \theta(z)e^{2i\pi(a_\gamma z + b_{\...
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60 views

Modular Discriminant and Pentagonal Numbers

I am asked to show $$(2\pi)^{-12}\Delta(\tau) = q \cdot \Big (\sum_{n\in \mathbb{Z}} (-1)^n \cdot q^{(3n^2+n)/2} \Big)^{24}$$ where $\Delta:\mathbb{H} \to \mathbb{C}$ is the modular discriminant, $q=e^...
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37 views

Chain Rule of Complex Differentiable functions.

I want to formulate and prove chain rule for $\mathbb C-$ differentiable functions, I think it should be $(f\circ g)'(z)=f'(g(z))g'(z)$ But how to prove it ? Edit: I proved it using $\frac{f(g(z))...
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How to prove the Stoilov's theorem?

In a book of mine there was Stoilov's theorem which proof was omitted. How can I proof the following? Every continuous, open, discrete mapping $g$ of a plane can be represented as $g=f\circ h$ where $...
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37 views

Singularity at inifnity?

Is a necessary and sufficient condition for the function $f(z)$ to have a removable singularity at $\infty$ for $f(1/z)$ to have a removable singularity at $0$. Does the same hold for any other type ...
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53 views

Equivalent definitions of winding number

The concept of winding number of a closed curve relatively to a point not on the curve has several possible definitions. One can define the concept with the path integral $$Wn(\gamma,a) = \frac{1}{...
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74 views

Entire function approaching zero along upper half plane

Suppose $f$ is entire, i.e, $\;f: \Bbb C \to \Bbb C$ is analytic. Let $\Bbb H:= \{ z: Im(z)>0\}$ be the upper half plane. Suppose that $$\lim_{\substack{z \to \infty \\ z \in \Bbb H}} f(z)=0$$ ...
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55 views

Show that $f(z) = \ln r + i \varphi$ is differentiable in a neighborhood of $z_{0}$

I am faced with the following problem: Let $z_{0}\neq 0$ and let $f(z) = \ln r + i \varphi$, where $r = |z|$, $\varphi \in arg z$, and $\varphi$ is chosen so that $f$ is continuous in a neighborhood ...
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50 views

Complex Analysis with differential forms

I'm studying a little of Complex Anlysis and I have seen that I can thing the integrals of complex functions as integrals of differential forms in $\mathbb{R}^n$. For example I know that Cauchy ...
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49 views

Mistake in this definition from Conway's complex analysis book

I'm reading Conway's complex analysis book and on page 64 the author has enunciated the following definition: However, on page 81 the author has stated that: I think Conway was mistaken in the ...
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115 views

Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
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82 views

Fermat like equation for meromorphic functions.

I found this question in Conway, and really have no idea how to answer it. Can anyone provide any hints? For each integer $n\geq 1$ determine all meromorphic functions on $\mathbb{C}$ $f$ and $g$ ...
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31 views

“Multiple angle” addition formulae for Jacobi elliptic functions

The addition formulae for the Jacobi elliptic functions are given by $sn(u+v)=\frac{sn(u)cn(v)dn(v)+cn(u)sn(v)dn(v)}{1-k^2sn^2(u)sn^2(v)}$, $cn(u+v)=\frac{cn(u)cn(v)-sn(u)dn(u)sn(v)dn(v)}{1-k^2sn^2(...
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61 views

Is there only one way to prove the Riemann Mapping Theorem?

After looking through several complex analysis books and online resources (e.g. see here, here, and here), it seems that there is basically one well-known proof of the Riemann Mapping Theorem, which ...
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93 views

Analytic function with a non-essential singularity at $\infty$ reduces to a polynomial.

Show that a function $f(z)$, which is analytic in the whole plane and has a non-essential singularity at $\infty$ reduces to a polynomial. This question has been asked previously and I have gone ...
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71 views

Why does Conway define conformal maps in this way?

I'm reading Conway's complex analysis book and on page 46 the author defines conformal maps: This makes me think why he didn't define conformal maps as usual, i.e., as a function with angle ...
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62 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
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39 views

Three-gap problem, easy version.

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show ...
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48 views

invert a ''polynomial'' with poles (sounds ridiculous, but please refer to the details)

I recently encountered some polynomial inversion in some physics literature, the simplified version would be the following: $$f(x)=\sum_{n=0}^{\infty} f_nx^n=\frac{g(x)}{(x-x_1)(x-x_2)}$$ where $g(x)$...
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49 views

Which version of Taylor Theorem is this?

Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that $E(|X|^{r+2})<\...
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89 views

Good plan for studying Fourier Analysis?

I am planning to study Fourier Analysis from a mathematical point of view. I know that there are some pre-requisites, such as, elements of Functional Analysis and Complex Analysis. However, I would ...
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45 views

Cauchy Riemann Equations — Polar Coordinates

I want to show that the statement $f$ analytic and the Cauchy-Riemann equations in polar form are satisfied are equivalent statements. If we express $z$ in polar coordinates then $z = re^{i\theta}$ ...
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80 views

Integral of $e^{ix^2}$

How does one evaluate $$\int_{-\infty}^{\infty} e^{ix^2} dx$$ I know the trick how to evaluate $\int_{-\infty}^{\infty} e^{-x^2}dx$ but trying to apply it here I get a limit which does not converge: ...
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26 views

Complex line integration with assumptions

Let $f: \mathbb{C} \to \mathbb{C} $ be a holomorphic function with $$ \lim_{\lvert z \rvert\to\infty} \frac{f(z)}{z^{n-1}} = 0$$ for some $n\in\mathbb{N}$. How can I prove that $$ \lim_{r\to\infty} \...
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29 views

An example of a curve with all indexes

Give an example of a closed rectifiable curve $\gamma$ in $\mathbb{C}$ such that for every $k\in\mathbb{Z}$ there is some $a$ out of the curve such that $n(\gamma;a)=k$. Here, $n(\gamma;a)$ is the ...
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106 views

Contour Integral of Square root Function. Branch Cuts

I am doing a physics problem and have come across a contour integral that I just don't know how to solve. I do not have the complex analysis background and I am wondering if anyone can explain how to ...
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47 views

Can I split this integral to a sum over three contours?

I have the following integral $$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$ and this integral has ...
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Exact value of Gauss Sum

When I was studying quadratic reciprocity, my number theory professor used the following result without proof: $$S(n)=\sum^{n-1}_{x=0}\exp\left(\frac{2\pi ix^2}{n}\right)=\begin{cases} \sqrt{n}+\sqrt{...
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80 views

Fundamental theorem of algebra in different functional form

Consider the polynomial function: $f(x)=c_0+c_1x+c_2 x^2+\cdots+c_{n-1}x^{n-1}+x^n$, with $x$ and $c_0,c_1,c_2,\ldots,c_{n-1}$ are complex numbers. $|f(x)|$ is continuous and there exists closed and ...
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68 views

Example of Saddle-Point method

I am trying to solve using the saddle point method (large a>0): $$I(\alpha)= \int_{-i\pi/2}^{\pi/2}dz\, (1+z^2)e^{-a\cos(z)}$$ So I find that the point I want to expand about is z=0, because $\...
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64 views

Proving minimum modulus theorem using the maximum modulus theorem

I have been trying to prove minimum modulus theorem using the maximum modulus theorem. The problem statement is $f$ is continuous on a closed bounded region $R$ and it is analytic and not constant ...
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118 views

$f(z)=\sum a_n z^n$ ($a_n\geq0$) with radius of convergence $1$. Prove that $1$ is a singular point of $f$.

$f(z)=\sum a_n z^n$ ($a_n\geq0$) with radius of convergence $1$. Prove that $1$ is a singular point of $f$. I proved that $f$ converges on the unit circle. I tried to use uniqueness theorem to prove ...
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40 views

Order of zero for $\sin(z^2)$ about $z=\sqrt{\pi}$

What's the order of the zero $z=\sqrt{\pi}$ for the complex function $f(z)=\sin{(z^2)}$? Here my take... Letting $w = z^2$ such that $f(w)=\sin w$, we want the order of the zero $w=\pi$. Expanding ...
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72 views

Approximating $(1+\frac{1}{z})^z$ where $|z|$ is large

I know that $$\lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x=e$$ Is there an equivalent in complex analysis for $$\lim_{|z|\rightarrow \infty}\left(1+\frac{1}{z}\right)^z=?$$
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A question about the proof of $(z_1z_2)^a=z_1^az_2^a$

For $z_1,z_2\in \mathbb C$ if $\Im(z_1z_2)>0$ and $\Im(z_2)\ge 0$ prove that $(z_1z_2)^a=z_1^az_2^a$ , for $a$ is any real. I proved it like this: $z_1^az_2^a=\exp(a\log z_1)\exp(a\log z_2)=\exp(...
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63 views

Let $h(z) = g(f(z))$. If two of the three functions $f$, $g$, and $h$ are holomorphic and non-constant, must the third also be holomorphic?

If $h$ and $g$ are holomorphic it seems like the answer is no. Let $f(z) = f(re^{i\theta}) = \sqrt re^{i\theta/2}$ for $\theta \in [0,2\pi)$, and let $g(z)=z^2$. Then $f$ is discontinuous on the non-...
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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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152 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
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Is there a class of functions which is analagous to complex analytic functions?

As far as I am aware it is known that for any complex analytic function, the gradient of the real part of the function and the gradient of the imaginary part of the function are at right angles. For ...
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Is my proof of linear fractional transformations correct?

a) Prove that the most general $1-1$ conformal map of the upper half-plane onto itself is of the form $$z \to \frac{az+b}{cz+d}$$ where $a,b,c,d \in \mathbb{R}$ and $ad-bc =1$. b) Let $f$ be a $1-1$ ...
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Find $\int_0^{2\pi} \frac{d\theta}{2\pi\cos^{2n}(\theta)} \ n=1,2,3,\dots$ via Residue Theorem

So the question is as follows: Use the Residue Theorem to calculate $$\int_0^{2\pi} \frac{1}{2\pi\cos^{2n}(\theta)} d\theta \quad\quad n=1,2,3,\dots.$$ Now I believe the first step would be to use the ...
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40 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show that ...
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51 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
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66 views

A Question on Maximum Modulus Principle

Let $h : \mathbb C \to \mathbb C$ be an analytic function such that $h(0) = 0; h( \frac{1}{2 }) = 5$, and $|h(z)| < 10$ for $|z| < 1.$ Then, conclude that (a) the set $ \{z : |h(z)| = 5\...
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84 views

Using Rouche to find zeros of a polynomial

Given $P(z)=z^6+3z^4+z^2+z+9$, prove that all its zeros are contained in the annulus $1<|z|<2$, and find how many of them are in the first quadrant. I have been able to prove that they are ...
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154 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
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61 views

$f_n\to f$ iff for each closed rectifiable curve $f_n (z) \to f (z)$ uniformly for $z$ in the trace of the curve

I'd like to know if the following exercise is correct. I'm not completely sure about the last point but also I don't know what more I'd say. I really appreciate corrections or any suggestion you can ...
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115 views

Should a certain entire function be a polynomial?

Assume $f$ is an entire function such that $$\lim_{z\to\infty}\frac{|f'(z)|}{1+|f(z)|^2}=0,$$ then should $f$ be a polynomial? Picard's Theorem proves this instantly; which states: Let $f$ be a ...
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73 views

Does a Plancherel-style theorem for the Hardy space $\mathcal{H}^2(\mathbb{T})$ exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathcal{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...