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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1answer
18 views

Show that $g(z)=\frac{1}{n}\sum_{k=0}^{n-1} f \left(\xi^{k}\sqrt[n]{z}\right)$ is an entire function.

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f(\xi z)=f(z)$ for all $z\in \mathbb{C}$ and consider the ...
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0answers
14 views

Integral of action (quantum field theory, prescription)

I am struggling to show: $$\int_{0}^{w} \int_{2M}^{2(M-w)} \frac{-drdw}{1-\sqrt{\frac{2(M-w)}{r}-\frac{Q^2}{r^2}}}=2\pi[{2w(M-\frac{w}{2})-(M-w)\sqrt{(M-w)^2-Q^2)}+M\sqrt{M^2-Q^2}}]\\$$ A hint is ...
2
votes
1answer
28 views

Complex derivative numerically using real $h$ and imaginary $h i$?

I want to find numerically (the functional expression might become too complicated) the derivative of a complex function (to use it in a Newton-algorithm). Can I simply do something like $$ \frac ...
-1
votes
0answers
16 views

Proof involving complex limits

Prove that $\lim_{n \to \infty } \left | z_{n} - z \right | = 0$ if and only if $\lim_{n \to \infty } Re(z_{n}) = Re(z)$ and $\lim_{n \to \infty } Im(z_{n}) = Im(z)$. I understand the epsilon delta ...
2
votes
1answer
36 views

Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that…

In reviewing complex analysis, I stumbled upon the following problem: Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that $g(x,y)=3$ when $x^2+y^2=1$ and $g(x,y)=8$ when ...
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0answers
28 views

How to find branch points for complex functions?

I'm looking for a standard way I can approach problems where I am tasked to find the branch points and branch cuts of a complex function. For instance, $$ f(z) = e^{(z^2+1)^{1/2}}$$ or $$ f(z) = ...
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0answers
28 views

There is an entire functión $g$ such that $f(z)=g\left(z^{n}\right)$.

Let $f$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f\left(\xi z\right)=f(z)$ for all $z\in \mathbb{C}$. Show that there is a entire function $g$ ...
-2
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3answers
22 views

Use complex numbers to deduce triple angle formulas [on hold]

How to prove $\cos{3\theta}=\cos^3{\theta}-3\cos{\theta}\sin^2{\theta}$?
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2answers
31 views

Disc of convergence of a power series

Find the disc of convergence: $$\sum_{n=3}^\infty \left(1-\frac{1}{n^2}\right)^{-n^3}z^n$$ I have been manipulating the power series and I am pretty sure it has something to do with $e$ but I cannot ...
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2answers
22 views

Show if one series converges absolutely then so too does the other.

Task at hand: Let $a_n$ and $b_n$ be nonzero complex numbers for $n=1,2,3...$ . Suppose $\lim_{n\to \infty} \left|\frac{a_n}{b_n}\right|=l$ exists, and $l\neq0,\infty.$ show that if one of the series ...
0
votes
1answer
24 views

Show that entire function $f$ is a polynomial of degree at most $n$

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ ...
1
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1answer
20 views

Show that the function $\phi (x,y)=\arctan{\frac{2x}{x^2+y^2-1}}$ is harmonic by considering $w(z)=\frac{i+z}{i-z}$.

Show that $\phi (x,y)=\arctan{\frac{2x}{x^2+y^2-1}}$ is harmonic by considering $w(z)=\frac{i+z}{i-z}$. I know that if $\phi$ is harmonic then it satisfies Laplace's equation but I don't see how ...
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votes
2answers
23 views

how could calculate $ \int_{C} \frac{1}{\sin(z)} \, dz $ when $C=C(0,1)$

i am trying calculate $$ \int_{C} \frac{1}{\sin(z)} \, dz $$ when $C=C(0,1)$ by complex methods, its said, by residues, some one could help me?
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1answer
16 views

Equality of analytic functions equal on a diverging sequence of complex

I ask this question as a subsequent of following one. Suppose that $f$ and $g$ are two analytic functions defined on $\mathbb C$ and that $(a_n)_{n \in \mathbb N}$ is a sequence of complex numbers ...
3
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2answers
47 views

Proving limit of $|1-z|^2$ as $z \to i$ is 2

First off, apologies for my formatting. This is my first post and I'm still unfamiliar with MathJax and Latex, so I'm doing the best that I can. So I'm trying to prove that the limit of $|1-z|^2$ ...
1
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1answer
36 views

Questions 4 and 5 from section 4.3 of Conway's complex analysis book

I'm reading the Conway's complex analysis book and I'm trying to solve theses exercises on page 80: 4.Prove that $e^{z+a}=e^ze^a$ 5.Prove that $\cos(a+b)=\cos a\cos b-\sin a\sin b$ The ...
3
votes
1answer
39 views

Calculate $\int_{\left|z-1\right|=2}z^{n}\sin\left(z\right)dz$ for $n\in \mathbb{Z}$

Calculate $$\int_{\left|z-1\right|=2}z^{n}\sin\left(z\right)dz$$ for $n\in \mathbb{Z}$ My attempt: According to the following result which was presented at my course as Cauchy's integral formula for ...
2
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2answers
25 views

How could I calculate $\int_{C} ze^{\frac{1}{z-1}}$ when $C=C(1,\frac{1}{2})$

I have to solve if $C=C(1,\frac{1}{2})$ $$I=\int_{C} ze^{\frac{1}{z-1}}$$ I know that $I=2\pi i \operatorname{Res}(f(z), 1)$, but I do not know how could I calculate that residue. What I did: ...
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2answers
28 views

If $x\in \mathbb{R}$ then show that $\{z\in \mathbb{C}: \Im(z) < x\} =A$ is open.

THE RED LINE IS $ \Im(z) = x$ Now, my proof is as follows, Let $z' \in A$, then take $\epsilon = x - \Im(z')>0$ Now let $w \in D_{\epsilon}(z')$ and suppose $w \notin A$ then $$\epsilon > ...
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0answers
22 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
22 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
17 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
27 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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0answers
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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0answers
22 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
43 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 ...
2
votes
1answer
41 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
40 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 ...
1
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4answers
69 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 ...
0
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1answer
28 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 ...
2
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2answers
36 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. ...
-1
votes
1answer
17 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 ...
0
votes
0answers
45 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
38 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) ...
0
votes
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 ...
1
vote
1answer
35 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
22 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
26 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
41 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
19 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
35 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 ...
2
votes
2answers
104 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
41 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
48 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 ...
1
vote
3answers
74 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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0answers
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 ...
0
votes
0answers
32 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 ...
0
votes
1answer
22 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 ...
1
vote
1answer
85 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. ...