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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7 views

Supremum of $\cot(\pi z)$

I try to estimate the supremum of $|\cot(\pi z)|$ where $z=(n+1/2) e^{i t}$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and expanding it $|\cot(\pi ...
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1answer
19 views

Commuting $\operatorname{Re}$ with integral

Is the following always true? $$ f:\mathbb{C}\to\mathbb{C},\ \operatorname{Re}\left(\int f(z)d z\right) = \int\operatorname{Re}(z)dz $$ $$ \frac{d\operatorname{Re}(f)}{dz} = ...
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0answers
14 views

Lang's proof of the Weierstrass preparation theorem

Relevant Google Books link. I'm having problems with the final step in the proof of Theorem 9.1. It's not clear to me why the function $I + \tau \circ \frac{\alpha(f)}{\tau(f)}$ should be ...
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1answer
24 views

Meaning of $\partial f /\partial x$

I have an exercise in complex analysis that begins, If $U\subset \mathbb C$ is an open set and $f:U\to \mathbb C$ is real differentiable.... Later on, it allows me to assume $f$ is holomorphic. ...
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12 views

Runge's Theorem Application

Below is a question out of Gamelin's Complex Analysis which I cannot quite figure out. Any tips would help appreciated! "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that accumulates ...
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19 views

Use differential form to prove meromorphic function on compact riemann surface has same zeros and poles

I am reading mine's modular form note, proposition 1.12 states that the sum of residues of a differential form on compact Riemman surface is 0. Then he states that applies this to $df/f$, then we can ...
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2answers
39 views

Why are the zeros of $f$ isolated?

I'm reading Conway's complex analysis book and on page 79 he proved the following theorem: I think the theorem he is mentioning is a corollary which says that each zero of $f$ has a finite ...
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1answer
32 views

Proving that a complex-valued function has limit infinity at a finite point

Question: Show using the $\epsilon -\delta$ definition that $$\lim \limits_{z \to i} \frac{z-1}{z^2+1} = \infty$$
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0answers
37 views

Find a Mobius Transformation that carries the points $ -1, i, 1+i$ to the following:

My goal is to find a Mobius transformation that transforms $-1, i, 1+i$ onto the points a) $0, 2i, 1-i$ b) $i, \infty, 1$ For part a, I know that the Mobius transformation $M$ will be such that ...
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4answers
67 views

How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$?

$\lim_{s \rightarrow \infty} \zeta(s) = 1$ I have seen a proof using the fact $1 \leq \zeta(s) \leq \frac{1}{1-2^{1-s}}$ but this relies on proving the inequality first which is quite cumbersome. I ...
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1answer
24 views

Calculate the vertical asymptote of the absolute of a complex rational function

I have a function with the following shape: $$ f(x) = \left|\frac{a_0 + a_1x+a_2x^2 + ... + a_nx^n}{b_0 + b_1x+b_2x^2 + ... + b_nx^n} \right| $$ The constant $b_0=1$ (I don know if it matters) When ...
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2answers
32 views

given $-\pi < \theta \leq \pi$ prove $f(z) = z^{1/3}$ is not entire.

I don't want the solution at all, but I'm incredibly stuck, and I really need some (hopefully not much) help. What I've considered: Liouville's Theorem Not applicable because f is not bounded. ...
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1answer
16 views

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$ [on hold]

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$. This outcome is the result of this link. So I think I have to use the previous result and the ...
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0answers
42 views

Why is that if $z^n = |z|^2$, then $|z| = 1$?

We have $z^{n-1} = \bar{z}\ \forall\ n > 2$ which gives us $z^n = |z|^2$, but I dont see why that means $|z| = 1$?
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1answer
32 views

Examples of holomorphic, complex differentiable, always positive functions

I am looking for classes of functions which are: 1) holomorphic 2) |f(z)|>0 for all z 3) complex differentiable (i.e. f(z)=mod(z) is not valid) ...
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1answer
23 views

solve and skecth $\log{|z|}=-2\arg(z)$

Ive asked this question a week ago, but nobody managed to answer but it is doing my heading from then. I know usually You demand some initial work done on the question but I just dont know how to ...
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1answer
32 views

Proving that $h(z)=\overline{h(\overline{z})}$ for all $z \in \mathbb{C}$ assuming that $h$ is holomorphic and the real line maps itself.

I have trouble proving that $h(z)=\overline{h(\overline{z})}$ for all $z \in \mathbb{C}$ under the assumption that $h$ is holomorphic and the real line maps itself or in other words: $h(z) \in ...
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3answers
19 views

Finding disc of convergence

Find the disc of convergence $$\sum_{n=0}^\infty z^{n^{3}}$$ I have applied the ratio test but I can not seem to come up with a conclusion.
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2answers
23 views

A function that satisfies $|f(z)-\bar z|<0.9$ is not analytic in the unit circle.

I've came accros this excersize: Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$ suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where ...
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4answers
39 views

Prove this integral is analytic

Let $\phi$ be a continuous (complex valued) function on the real interval [−1, 1] inside C, and define $$f(z)=\int_{-1}^1\frac{\phi(t)}{t-z}dt$$ Show that f is analytic on C less the interval [−1, ...
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1answer
16 views

Find a power series centered at the origin that satisfies the Bessel

Find a power series centered at the origin that satisfies the Bessel differential equation $$zf''(z)+f'(z)+zf(z)=0$$ with initial condition $f(0)=1$. Show that this series converges for all z in C. I ...
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1answer
24 views

Using the Maximum Modulus Principle to prove that every holomorphic function on a compact Riemann surface is constant

I have read in a number of sources (including here) that a holomorphic function on a compact Riemann surface must be constant. The reason given has always been the Maximum Modulus principle, but ...
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2answers
96 views

Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$

I'm having trouble integrating $$\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$$ where $\alpha$ is a real number and $i = \sqrt{-1}$. I'm guessing that I ...
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1answer
38 views

Complex Continuity [on hold]

Is the function $f$, defined by $$ f(z) = \begin{cases} \frac{z^2+2iz-1}{2z^2+iz+1} & \text{ if } z \not \in \{-i\}\\ 0 & \text{ if } z = -i \end{cases}$$ continuous at $−i$? Explain your ...
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1answer
40 views

Prove Complex Limits from first principle definition [on hold]

Show from first principles, that is using the definition of limit, that $$\lim_{z\to i}\frac{z-1}{z^2+1} = \infty$$ Please can someone actually show me the procedure, struggling to understand it ...
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0answers
23 views

Prove equivalence between two Bessel functions relations

Given the following equation $$\frac{J_{n - 1} (u)}{uJ_n (u)} - \frac{K_{n-1}(w)}{wK_n(w)} = 0$$ (where $J$ is the Bessel function of the first kind, $K$ is the modified Bessel function of the ...
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3answers
71 views

Does the function $z+\frac{1}{z}$, $z\in \mathbb{C}$, have real-world applications?

The function $$z+\frac{1}{z}$$ seems to play a role in complex analysis. However, does it have any applications besides what can be deduced in pure mathematics?
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26 views

Complex exponentials [duplicate]

How do I solve: $$ e^{4z}+e^{3z}+e^{2z}+e^z+1=0 $$ I'm getting lost on where to start. I tried using the definition $$ e^z=e^x(\cos(y) +i\sin(y)) $$ But that doesn't seem to do me any good. I also ...
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30 views

Show that $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a complex number)

How do I prove this? Suppose that $a, b$ and $c$ belong to $\mathbb C$ and that $$\lim_{z\to z_0} f(z)=a$$ and $$\lim_{z\to z_0} g(z)=b.$$ a - $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a ...
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1answer
20 views

Winding number, Conway text

I have a question about this statement in the Cauchy's Integral Formula in Conway text. In the Integral formula, it states that " Let $G$ be an open subset of the plane... If $\gamma$ is a closed ...
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1answer
82 views

Solve $e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0$.

Solve $$e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0.$$ I have attempted this problem with the usual definition by writing $z=x+iy$ and using $e^z = e^x(\cos y + i \sin y)$ but got a large unsolvable mess. ...
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1answer
37 views

Are there complex numbers whose sines are zero?

I recently learned that $\sin(z)$ has an extension into the complex plane, namely: $$\frac{e^{iz}-e^{-iz}}{2i}$$ Is there any complex number $z=a+bi$, with $b≠0$ such that $\sin(z)=0$ ? I am ...
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1answer
13 views

Prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$

I'm asked to prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$ I know that for Big O, I need to show: $f(n) <= c*g(n)$ But I'm not sure how to show this, since it involves theta. Any help would ...
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1answer
36 views

How to bound this complex number from below?

I am doing an $\epsilon-\delta$ proof ($z \rightarrow i, f(z) \rightarrow \infty$) and currently have the absolute value $$|f(z)|=\left|\frac{z-1}{z^2+1}\right|$$ and I wish to make a statement about ...
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0answers
28 views

how to prove this function has zeros interlacing and including those of Riemann zeta

Let $\chi (t) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left( \frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right) \right) \zeta (t)^3 - 48 \zeta (t) \dot{\zeta} (t) ...
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1answer
38 views

Zeroes of Complex Cosine

Find the zeroes of $\cos z=2$. Attempt: $\cos z=\cos(x+iy)=\cos(x)\cos(iy)-\sin(x)\sin(iy)=\cos(x)\cosh(y)+\sin(x)\sinh(y)=2$ I don't know how to proceed form here...
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0answers
24 views

what is $\int_{\gamma}\frac{2}{(z+2)^2}dz$ with $\gamma(t)=t+it\sin(\frac{\pi}{t})$ for $t>0$?

Again a question about integration. Consider the integral $$\int_{\gamma}\frac{2}{(z+2)^2}dz,$$where $\gamma:[0,1]\to\mathbb{R}$ such that $\gamma(0)=0$ and $\gamma(t)=t+it\sin(\frac{\pi}{t})$ if ...
2
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0answers
59 views

Help with finding the numerical average of $x^x$ from $(-4,-2)$.

I wanted to find the approximate average of all defined points in $(x)^{x}$ from $[-4,-2]$ To first solve this I found the following defined sets when $x<0$. ...
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2answers
25 views

Conformal mapping $z+\frac{1}{z}$, how to see the mapping to hyperbolas?

http://www.webassign.net/zillengmath4/20.2.pdf p.2. The conformal map $z+\frac{1}{z}$ maps circles $|z|=r$ to ellipses and $arg(z)=\theta$ to hyperbolas. I believe one can display both using the ...
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1answer
54 views

Is $\int_{|z|=2}\frac{z}{(z-3)^2}dz=0?$

I have a question. What is $$\int_{|z|=2}\frac{z}{(z-3)^2}dz?$$ In my optinion it must be zero, because the singularity $3$ is outside $\{z\in\mathbb{C}:|z|<2\}$, is it correct? Regards
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2answers
39 views

Complex differentiability and differentiability in R2

In $\mathbb R$ for a derivative to exist (or a limit generally) it is necessary that the limit be the same in both directions (from below and above) and this is the same in $\mathbb C$ where for a ...
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2answers
30 views

How do I evaluate this sum :$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$?

I'm interesting to know how do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$$, I have tried to evaluate it using two partial sum for odd integer $n$ and even integer $n$ ...
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1answer
36 views

page $102$ from Ahlfors.

He talks about a function $f(a)$ for which all the derivatives vanish. He shows inside a circle within our domain $\Omega$, for any circle $C$ we take, there $f$ is identicaly zero. Then he shows ...
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1answer
32 views

Complex numbers as linear operators?

If it is valid to interpret multiplication by a complex number as a dilative rotation, does that mean that it can be viewed as a function $$f: R^2 \rightarrow R^2$$ making it a linear operator?
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30 views

What general mobius transformation maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$.

What is the most general mobius transformation that maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$. I want to find the most general form of such a linear transformation, I'll denote it $T$. ...
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0answers
27 views

Find maximum value or upper bound of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$ [duplicate]

If $|z_{1}|=2,|z_{2}|=3,|z_{3}|=4$,then find maximum value of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$. My attempt:I considered 3 circles having centre origin and radii as $2,3,4$. Then I ...
6
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4answers
141 views

How many values does $\sqrt{\sqrt{i}}$ have?

Wolfram says, there are only two roots, but $\sqrt{i}$ already gives two roots. So if we express them in Cartesian form we can take square roots of them separately and end up with four roots. ...
2
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0answers
37 views

Is there any generalization of Riemann Mapping theorem?

Given any two regions in complex plane when can we say they are conformally equivalent? I mean does there exists some "complex-geometric" invariant which determines whether two regions are conformally ...
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1answer
30 views

Disc of convergence involving logs

Find the disc of convergence: $$\sum_{n=2}^\infty \frac{z^{n}}{n(log(n))^p};(p>0)$$ I have tried geometic series, ratio test, root test... What would be your thought on the best test to use?
6
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2answers
131 views

Computation of an iterated integral

I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not ...