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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2
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1answer
335 views

Complex integral over circle using Cauchy's formula

I have to integrate the complex function $$ \frac{e^z-1}{z^5} $$ over the curve $\gamma(t)=1+re^{-5it}$ where $t \in [0,2\pi]$. The curve has winding number -5 with respect to a point inside the disc ...
7
votes
2answers
462 views

Is Wikipedia incorrect on the Cauchy - Riemann equations (sufficient condition for differentiability)?

According to Wikipedia, "Moreover, the equations are necessary and sufficient conditions for complex differentiation once we assume that its real and imaginary parts are differentiable real functions ...
1
vote
1answer
172 views

analytic function $f$ such that $f(z)^2 = z$

Delete from the complex plane the non-positive part of the imaginary axis. How do we explicitly define an analytic function $f$ on our "modified complex plane" satisfying $f(z)^2 =z$? This was an ...
4
votes
4answers
714 views

Fibonacci( Binet's Formula Derivation)-Revised with work shown

Okay so here is the revised question with my current work. Links to previous post(s)(Just for Gerry): Fibonacci Numbers - Complex Analysis Here's my attempt on the problem set thus far: (Note ...
2
votes
1answer
262 views

Inverse Laplace transform and Jordan's Lemma

I'm trying find the inverse Laplace transform $f(t)$ where I have $F(p)=\dfrac{9}{p(p+3)^2}$. I know $f(t)$ already to be $1-3te^{-3t}-e^{-3t}$. I have the integral $$f(t)=\dfrac{9}{2 \pi ...
-2
votes
0answers
177 views

Fibonacci Numbers - Complex Analysis [duplicate]

Possible Duplicate: Complex Analysis - Integral over a circle of radius R Hey guys~ Does anyone know where to find the solutions to this problem set on page 106 involving the fibonacci ...
4
votes
1answer
360 views

A qualifying exam problem involving Schwarz lemma

This is a problem in the book "Berkeley Problems in Mathematics", which I think the solution given is wrong, can someone help? The following problem appeared in Spring 1991. Let the function $f$ ...
4
votes
2answers
392 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
1
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1answer
983 views

Radius of convergence for the exponential function

I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
1
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1answer
43 views

$f$ is a meromorphic function, Suprimum of the number of solutions to $f(z)=w$ is finite. Then $f$ is rational.

$f$ is a meromorphic function in $\mathbb{C}$. For each $w \in \mathbb{C}$ we denote by $N_{f}(w)$ the number of solutions to the equation $f(z)=w$ in $\mathbb{C}$. Prove that if $\sup_{w\in ...
3
votes
1answer
318 views

Complex Exponential as a limit

I need some help with a homework problem. This is Ahlfors exercise 1 p. 178: Using Taylor's Theorem applied to a branch of $\log (1 + \frac{z}{n})$ prove that $\lim (1 + \frac{z}{n})^n=e^z$ ...
1
vote
1answer
49 views

contour integration ambiguity

A paper provides the following derivation: say you have to solve: $ \displaystyle R \sim - \frac{1}{2}\int_{-\infty}^{\infty} \mathrm{d}w \frac{dp/dw~e^{2iw}}{p(w)}$ Then if we have poles at ...
2
votes
2answers
218 views

Are limit superior and limit inferior defined for $z_n$ being a complex sequence?

All the definitions of limit superior and limit inferior I have seen (even in the books about complex analysis) define them for a real sequence only. What could stop us from defining it as follow for ...
6
votes
1answer
137 views

What is $\lim\limits_{z \to \pi/2} \tan^2(z) $ for $z \in \mathbb C$?

I am trying to evaluate the following limit ($z \in \mathbb C$): $$\lim\limits_{z \to \pi/2} \tan^2(z) $$ I get the following solution: $$\lim\limits_{x \to \pi/2} \tan^2(x) = \lim\limits_{x \to ...
3
votes
2answers
184 views

The equation $2 \cosh(3.1786803659501505 z) = z$?

Let $a$ be a positive real number and $z$ a complex number. I was wondering about the equation $2 \cosh(a z) = z$ where we solve for $z$. Clearly if $z$ is a solution than so is its conjugate. It ...
25
votes
1answer
891 views

What is wrong with this fake proof $e^i = 1$?

$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$ Obviously, one of my algebraic manipulations is not valid.
2
votes
2answers
293 views

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$?

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$.
0
votes
1answer
85 views

There are no $f,g:\mathbb{C}\to\mathbb{C}$, entire end $e^{f(z)}=e^{g(z)}+c$.

I am trying to prove Picard's little theorem. Here are my steps: If $a,b\notin Im(\phi(\mathbb{C}))$ then $\phi_1(z)=\phi(z)-a$ and $\phi_2(z)=\phi(z)-b$ don't vanishes any where. then there are ...
4
votes
1answer
65 views

Holomorphic 1-forms in $y^2-(z-a_1)\ldots(z-a_n)$

I know that the surface $y^2-(z-a_1)\ldots(z-a_n)$ is a Riemann Surface (that is the Riemann surface of $\sqrt{P(z)}$ with $P(z)=(z-a_1)\ldots(z-a_n)$) of genus $g$ and that ...
1
vote
2answers
156 views

Estimating the integrated Tchebychev function and calculating its error

I would like to understand how to derive (2) from (1) below. Problem: If $\psi_1$ is the integrated Tchebychev function below $$\psi_1(x)=\frac{1}{2\pi i} ...
6
votes
2answers
167 views

About the determination of complex logarithm

Although it must be a silly question, I am really confused. For complex logarithm, in general, $$\log(z_1z_2)\neq\log(z_1)+\log(z_2)$$ even if the logarithm is already determined. I think this is ...
3
votes
3answers
273 views

Find the residues of singularities of the following function:

Let $$f(z)=\frac{e^{\pi iz}}{z^2-2z+2}$$ and $\gamma_R$ is the closed contour made up by the semi-circular contour $\sigma_1$ given by, $\sigma_1(t)=Re^{it}$, and the straight line $\gamma_2$ from ...
2
votes
3answers
197 views

Where does the iteration of the exponential map switch from one fixpoint to the 3-periodic fixpoint cycle?

In the answering of another question in MSE I've dealt with the iteration of $x=b^x$ where the base $b=i$. If I reversed that iteration $x=log(x)/log(i)$ then I run into a cycle of 3 periodic ...
2
votes
1answer
50 views

Prove that $\lim_{R \rightarrow \infty} \int_{\sigma_1}f(z) dz=0$.

Let $$f(z)=\frac{e^{\pi iz}}{z^2-2z+2}$$ and $\gamma_R$ is the closed contour made up by the semi-circular contour $\sigma_1$ given by, $\sigma_1(t)=Re^{it}$, and the straight line $\gamma_2$ from ...
1
vote
0answers
150 views

Complete Elliptic Integral of the 3rd Kind - Residual Computation

Let us consider the following function $f(a,k)$ in the interval $a,k\in (0,1]$ : $$f(a,k)=\frac{2 \sqrt{1-a^2} \sqrt{a^2-k^2}}{\sqrt{a^2}}\Pi\left(a^2,k^2\right)$$ where $\Pi\left(a^2,k^2\right)$ is ...
2
votes
2answers
168 views

Evaluate the following contour integral:

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t\leq 2\pi$. Evaluate $$ \int_{\gamma(0,1)}\dfrac{z^2+1}{z(z^2+4)}dz $$ I've tried to use the binomial expansion with ...
2
votes
1answer
89 views

If $f$ a holomorphic function is defined on a connnected open set $U$ and $f$ is locally zero near some $z\in U$

Does this imply $f$ is zero on the whole of $U$? If $U$ is a disc and $f$ is zero near the center then from the Taylor series I see that $f$ is constant. If $U$ is any connected opens set, then since ...
1
vote
2answers
253 views

branch of logarithm

What's the shortest way to show that there is no analytic function $f$ on $\mathbb{C} \backslash \lbrace 0 \rbrace$ such that $$\exp(f (z)) = z$$ for all nonzero complex numbers $z$? I came across ...
-1
votes
1answer
167 views

Buddhabrot Sewing machine [closed]

The Buddhabrot fractal traces the orbits of the points outside the Mandelbrot set. What design considerations need to be taken into account to create a computerised sewing machine that traces out ...
1
vote
1answer
226 views

Fibonacci Generating Function of a Complex Variable

So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function for the recursive fibonacci numbers. I have two questions: 1. Why is it useful to use a ...
0
votes
1answer
80 views

Justification of taking limit under an integral

In the solution of a complex analysis problem I'm working through the following comment was made: $\displaystyle-\lim_{\epsilon \rightarrow 0} \int_{\pi}^{0} i e^{i\epsilon \displaystyle e^{i ...
4
votes
1answer
412 views

Möbius transformations on the upper half plane

Let $\phi$ be a holomorphic function from the unit disk onto the upper half-plane such that $\phi(0)=\alpha$. Give a method to find an upper bound for $\lvert\phi ′(0)\rvert$? To apply ...
2
votes
4answers
122 views

geometrical multiplication of complex numbers

It is taught that $\mid z \mid ^{2} =x^{2}+y^{2}$ and $\bar{z}z=x^{2}+y^{2}$. Algebrically its fine to understand it. But what is the geometrical meaning of it? I tried multiplying couple of numbers ...
1
vote
1answer
136 views

Analytic function in open connected set that is bounded by another analytic function

Let $G$ be an open connected set and $f, g$ analytic functions on $G$. If $|f|\le |g|$ then there exists an analytic function $h$ such that $f(z)=h(z)g(z)$. We know $|f/g|\le 1$ everywhere in ...
1
vote
3answers
890 views

Complex Analysis- Research

I was interested in doing some research in complex analysis. I already have a basic understanding of the subject. i.e. I read Saff and Snider's book "Fundamentals of Complex Analysis". But now I would ...
9
votes
3answers
905 views

Applications of Residue Theorem in complex analysis?

Does anyone know the applications of Residue Theorem in complex analysis? I would like to do a quick paper on the matter, but am not sure where to start. The residue theorem The residue ...
7
votes
2answers
223 views

Interpolation of analytic function on unit disk

Been thinking about this problem for a long time without any progress, can someone help? Consider a bounded function $f: \mathbb{D} \rightarrow \mathbb{D}$ with the following property : for every ...
4
votes
3answers
93 views

When does the next complex split occur?

So I was thinking about complex numbers and how they came about and someting interesting occured to me: the formation of complex numbers occurs because there exists a function (namely $f(x)=x^2$) ...
6
votes
1answer
130 views

$f:\mathbb{C}\rightarrow\mathbb{C}$ is entire function such that $g(z)=f(1/z)$ has a pole at $z=0$, then is $f$ surjective?

$f:\mathbb{C}\rightarrow\mathbb{C}$ is entire function such that $g(z)=f(1/z)$ has a pole at $z=0$, then is $f$ surjective? I can prove that $f$ will be a polynomial. and hence $f$ is surjective. am ...
3
votes
2answers
85 views

$z_{0}$ is a zero of order $m$. Prove that $|z_{0}|^m\geq|f(0)|$ where $f$ is analytic in the unit disc

$f$ is an analytic function in the unit disc, so that $|f(z)|\leq1$. Let $z_{0}$ be a zero of order $m$. Prove that $|z_{0}|^m\geq|f(0)|$ My approach: We can write: $$(1) \ \ \ ...
5
votes
1answer
481 views

Properties of the Mandelbrot set

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology? Considering the fact that the set is based on a quadratic function, are there any ...
5
votes
1answer
197 views

Prove that the only root of the equation $z-\sin(z)$ in the unit disk is zero.

Prove that the only root of the equation $z-\sin(z)$ in the unit disk is $z=0$. My first thought is Rouche's Theorem, but I don't know any bounds on $|\sin(z)|$. Suggestions?
2
votes
1answer
80 views

we need to pick out the cases where $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic but not neccessarily constant.

we need to pick out the cases where $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic but not neccessarily constant. $1.$$ \Im(f'(z))>0$ for all $z$ $2.$ $f(n)=3\forall n\in\mathbb{Z}$ $3$ ...
1
vote
1answer
248 views

any two simply connected open set in the plane R^2 are diffeomorphic

Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ...
1
vote
1answer
127 views

Holomorphic function with uncountable set of zeros?

I am aware that on a region, this is only possible if the function is identical to zero. If the domain is not a region, is it possible to have a non-trivial holomorphic function with uncountable zero ...
2
votes
0answers
127 views

Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$ The function is ...
5
votes
2answers
120 views

Computing $\lim_{s \to 1} \Gamma \left(\frac{1-s}{2}\right) (s-1)$

I want to evaluate the following limit: $$\lim_{s \to 1}\; \Gamma \left( \frac{1-s}{2} \right) (s-1).$$ I know that the gamma function has simple poles at $-n$ for $n \in \mathbb{N}_0$ with residue ...
2
votes
2answers
179 views

$f^2+2f+1$ is a polynomial implies that $f$ is a polynomial

This is a complex analysis problem. Let $f$ be an entire function and $f^2+2f+1$ be a polynomial. Prove that $f$ is a polynomial.
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0answers
89 views

Modular form weight 0

Why is an entire modular form of weight 0 must be a constant? In particular, does a function defined on the upper half plane that is analytic everywhere, including i/infty, imply boundedness? Can we ...
2
votes
3answers
162 views

Prove inequality in complex numbers in an unit circle

Given $|\omega| < 1$, $\omega \neq 0$ and $|z| < 1$. Prove inequality: $$\frac{|\frac{|\omega|}{\omega}z+1|}{|1-z \bar \omega|} \le \frac{2}{1-|z|}$$ It is simple but i have problems with it. ...