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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20 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
44 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 ...
1
vote
2answers
36 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 ...
-2
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3answers
26 views

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

How to prove $\cos{3\theta}=\cos^3{\theta}-3\cos{\theta}\sin^2{\theta}$?
0
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0answers
45 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) = ...
0
votes
1answer
27 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 $z \in \mathbb{C}$ ...
0
votes
2answers
24 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 ...
1
vote
2answers
29 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?
3
votes
2answers
52 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 $|1-z|^2$ ...
3
votes
1answer
40 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 ...
0
votes
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 ...
0
votes
2answers
30 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 > ...
4
votes
2answers
2k views

Finding the Number of Zeros of a Function in a Given Annulus

Consider $z^6 - 6z^2 + 10z + 2$ on the annulus $1<|z|<2$. By Rouche's Theorem $|f(z) + g(z)| < |f(z)|$ implies that both sides of the inequality have the same number of zeros. I understand ...
2
votes
2answers
30 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: ...
0
votes
1answer
31 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 ...
7
votes
2answers
409 views

Simplest way to determine if a number is a member of the Mandelbrot set?

I'm writing JavaScript code to plot the Mandelbrot set on an HTML5 Canvas element. (That's probably not relevant to the answer to this question). A core part of the problem is to write a simple ...
0
votes
0answers
24 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 ...
0
votes
1answer
37 views

Closed sets and accumulation points

In complex analysis how to prove that if $S$ is closed in $\mathbb{C}$ then it contain all of its accumulation points. If $S$ is closed then $S$ contain all its boundary points.(If $z_{0} $ is a ...
1
vote
1answer
26 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} = ...
1
vote
1answer
40 views

Absolute value of product is less than product of absolute values: $|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$

For a sequence $a_n\in\mathbb{C}$ I want to show that $$|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$$ I think I should show this by induction on $n$. For the base case I'm ...
2
votes
2answers
107 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 ...
0
votes
1answer
43 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 $T$. ...
1
vote
1answer
33 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. ...
1
vote
3answers
78 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?
0
votes
0answers
16 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 ...
1
vote
2answers
45 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 ...
0
votes
0answers
24 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 ...
1
vote
4answers
72 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 ...
2
votes
2answers
37 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. ...
2
votes
1answer
37 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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votes
1answer
21 views

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

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
47 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 $|z| = 1$?
1
vote
1answer
40 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 ...
0
votes
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.
0
votes
2answers
27 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 ...
1
vote
4answers
43 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, ...
0
votes
0answers
27 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
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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votes
1answer
47 views

Complex Continuity [closed]

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 ...
0
votes
1answer
65 views

Prove Complex Limits from first principle definition [closed]

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 ...
0
votes
1answer
39 views

Give the power series expansion of $\log z$ about $i$

I'm reading Conway's complex analysis book and I'm trying to solve the exercise 5 from page 74. In this exercise the author asks for the radius of convergence and power series expansion of $\log z$ ...
1
vote
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
36 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
24 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
88 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. ...
2
votes
1answer
38 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 ...
0
votes
1answer
39 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 ...
0
votes
1answer
17 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 ...
13
votes
2answers
385 views

Regularity of root spacing of $G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$

Define, on $\mathbb{C}$: $$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$ A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs): suggests that the roots of $G(z)$ are ...