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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308 views

Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this? I would like some sort of ...
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165 views

History Question - Branch Cut

My professor began discussing branch cuts in class today and mentioned that he did not know the origin of the term. Does anyone know the origin of the term and perhaps a source that talks about it?
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923 views

Choosing the branch of a logarithm

The problem: I am integrating complex logarithms over an angle $\phi$ over $[0,2\pi]$. It is quite complex (pun not intended) and I called Mathematica in to aid me. I am calculating an energy of a ...
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336 views

The Milnor Conjecture on the Unknotting Number of a Torus Knot

Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = ...
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76 views

I need your suggestions: Does this series converge or diverge?

I have this series given by $$\sum\limits_{n=1}^\infty \frac{1}{n^3 \sin^2n}.$$ I know that the terms are all nonnegative. Can I get a subsequence of $\frac{1}{n^3 \sin^2n}$ such that its sum diverges ...
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36 views

query about the cosine of an irrational multiple of an angle?

de Moivre's identity $$ (\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta $$ only applies as written when $n \in \mathbb{Z}$. if the exponent is a fraction $\frac{m}{n}$ then there will ...
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35 views

True or False: If f is analytic and maps the deleted unit disk to the unit disk, then 0 is not a pole for f.

I am studying for my final exam in complex variables and I ran across this true or false question. True or False: If $f:D(0;1)\setminus\{0\} \rightarrow D(0;1)$ is analytic, then $0$ is not a pole ...
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25 views

A metric such that every Blaschke factor is an isometry

Let $\mathbb{D} := \{z \in \mathbb{C} : |z| < 1\}$. Define $d : \mathbb{D}\times \mathbb{D} \rightarrow \mathbb{R}$ to be $$ d(z,w) := \left| \frac{z-w}{1-\overline{w}z} \right| $$ I am supposed to ...
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104 views

Analytic variety is a countable union of complex manifolds

In an article on real analytic manifolds I came across the following remark: Let $W$ be a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$ and let $S$ be its singular locus. ...
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36 views

Show $\frac{\partial^2}{\partial ^2 x}+ \frac{\partial^2}{\partial ^2 y}= 4 \frac{\partial^2}{\partial z \partial{ \overline{z}}} $

I want to solve the following exercise: Show that: $$\frac{\partial^{2}}{\partial ^{2}x}+ \frac{\partial^{2}}{\partial ^{2}y}= 4 \frac{\partial^{2}}{\partial z \partial{ \overline{z}}} $$ My ...
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32 views

Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...
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45 views

Obtaining a single-valued branch of $\ln \left( \frac{z-a}{z-b} \right)$ with a branch cut

It is rather easy to see that the function $$f(z) = \ln \left( \frac{z-a}{z-b} \right)$$ has branch points at $z=a$ and $z=b$, My question is why considering a branch cut "connecting" $a$ and $b$ ...
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69 views

Prove there is no branch of arg $z$ on $0 < z < 1$.

If $G$ is an open connected subset of $\mathbb{C}$ that does not contain the origin, we call a continuous function $\alpha$ satisfying $\alpha(z) = \text{arg} z$ for all $z \in G$ a branch of arg $z$. ...
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47 views

A periodic entire function which must have a fixed point

I would like to check my work on the following problem: Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point. So my attempt is: ...
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76 views

Integrating $\int_{-\infty}^\infty \frac{1}{1 + x^4}dx$ with the residue theorem

Calculate integral $$\int\limits_{-\infty}^{\infty}\frac{1}{x^4+1} dx$$ with residue theorem. Can I evaluate $\frac 12\int_C \dfrac{1}{z^4+1} dz$ where $C$ is simple closed contour of the upper ...
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173 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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36 views

$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$ This is a problem from an old ...
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64 views

Does anyone have a good reference on calculating contour integrals around the unit circle (numerically or otherwise)?

I am looking for a reference that will help me calculate contour integrals around the unit circle or other curve. I have a particularly ugly function which isn't likely to have a nice closed form so I ...
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92 views

Holomorphically simply connected implies simply connected

In my book on complex analysis a "Holomorphically simply connected" set is defined as a set where for any holomorphic function $f $ and any closed path $\gamma _1 $ we have that $\int_{\gamma _1 } ...
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142 views

Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
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38 views

Can anyone prove this identity without passing through the complexified tangent space?

Let $\rho: \mathbb{C} \to \mathbb{R}$ be a smooth function, $\Omega = \{ z : \rho(z) <0 \}$, and suppose $|\nabla \rho| = 1$ on $b\Omega$. It is true that $$\int_{b\Omega} f(z) d\bar{z} = -2i ...
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97 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
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112 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} ...
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107 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
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144 views

Interpretation of the Argument Principle

Recall that the argument principle states that given a meromorphic function $f$ and a compact region $K \subseteq \mathbb{C}$ whose boundary determines a simple contour and on which $f$ has no ...
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91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
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354 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
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82 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
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57 views

Integrating $\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz$ using residue calculus.

I'm trying to use the residue calculus to evaluate $$\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz,$$ where $s>0$, and where $\text{Arg}$ is the principal argument, ...
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306 views

Joukowski Aerofoil Plot

I've just had a go at plotting flow around aerofoils and I've come across a problem where I can't spot where I've gone wrong. I've previously worked out that the complex potential flow around a disk ...
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83 views

Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
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136 views

solution set in $\mathbb{C}$ of $ z^{\frac1{z}}=\left(\frac1{z}\right)^z$

If $z \in \mathbb{C}$ what can be said about the solution set of: $$ z^{\frac1{z}}=\left(\frac1{z}\right)^z $$ aside from the fact that it contains the fourth roots of unity? I will add as a footnote ...
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85 views

Verma modules and delta function

What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory (http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf): The ...
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266 views

A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261) *3. Using Ex. 2, show that $p + q$ maps $\Omega$ in a one-to-one manner onto a region bounded by convex contours. Comments: ...
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87 views

Applications of Microfunctions

Can anyone suggest good (a) uses/applications or (b) construction of micro-functions (introduced by Mikio Sato in 1971) in analysis? I am trying to understand the subject better. Suggestions of ...
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84 views

Is there a way to calculate $\int \limits_0^1\frac{x^3}{\sqrt{x^2-1}}\frac{1}{1-a^2x^2}\frac{1}{1-b^2x^2}\frac{1}{c-x}\mathrm dx$

I want to calculate $\displaystyle \int \limits_0^1\dfrac{x^3}{\sqrt{x^2-1}}\dfrac{1}{1-a^2x^2}\dfrac{1}{1-b^2x^2}\dfrac{1}{c-x}\mathrm dx$ $a$ and $b$ are real parameters, c could be complex and is ...
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423 views

Determine the number of zero points of $z^8-5z^3+z-2$ within the open unit circle (Rouché?)

How many zero points does the polynomial $z^8-5z^3+z-2$ have within the open unit circle? Hello, consider $$ \gamma\colon [0,2\pi]\to\mathbb{C}, \varphi\longmapsto\exp(i\varphi) $$ and ...
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96 views

Simplifying a sum?

Define polynomials $P_{j,s}^{(r)}$ via the generating series $$\left(\frac{d^s}{dz^s}f(z)\right)^r=\sum_{j=0}^{\infty} P_{j,s}^{(r)}z^j,$$ where $r\geq 1$. Here, $f(z)=z+a_2z^2+a_3z^3+\cdots.$ I was ...
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71 views

A Hankel Transform Integral: $\int_0^\infty\frac{1}{k^2-k_p^2}J_0\left(k\rho\right)\;k\,dk$

$$ \int_0^\infty\frac{1}{k^2-k_p^2}J_0\left(k\rho\right)\;k\,dk $$ Suppose that $k_p$ is in the first quadrant in the complex plane, and that $\rho$ is purely real. $J_0$ is the Bessel function of ...
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255 views

An entire function of strict order 2

Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way. Suppose $f(z)$ is an entire ...
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98 views

Extracting Taylor coefficients of a quotient

I was wondering if anybody has come across functions of the form $$\Phi_n(z):=\frac{f(z)^{n+1}}{zf'(z)-f(z)}\quad (n\geq 1).$$ Here, $$f(z)=\sum_{k=0}^{\infty} a_kz^k$$ is holomorphic on the open unit ...
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172 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
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168 views

How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
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397 views

extension of Cauchy's Integral formula

This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6. Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
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244 views

Möbius transformations form a simple group

How to show the group $M$ of Möbius transformations is a simple group? I know: $SL_2(\mathbb C)/\{+I,-I\}\cong M$ then if $A \lhd M \implies \phi^{-1}(A) \lhd SL_2(\mathbb C)/\{+I,-I\}$. So if ...
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112 views

An entire function with finite covering group is a polynomial.

Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
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90 views

Invertibility of a Toeplitz operator

Let $\phi$ be a real-valued function. I am trying to show that the Toeplitz operator $T_\phi$ is invertible if the function 1 is in the range of $T_\phi$. Here is what I got so far: There exists a ...
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284 views

Jacobi theta function

This is a question from Stein & Shakarchi's complex analysis book. Show that if $\rho$ is fixed with $Im(\rho)>0$, then the Jacobi theta function $$\theta(z|\rho)=\sum_{n=-\infty}^\infty ...
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74 views

Is the inverse of any elementary function asymptotic to some elementary function?

Is the functional inverse of any elementary function asymptotic to some elementary function ? For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See ...
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428 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...