5
votes
1answer
71 views

To prove this complex polynomial has all zeros on unit circle

I'm trying to prove a self-inversive polynomial $P(z) = \sum\limits_{n=0}^{N-1}a_nz^n$ has all its roots on the unit circle. The coefficients are such that $ a_n = e^{j(n-\frac{N-1}{2})\pi u_0} - ...
0
votes
0answers
19 views

Proof Strategy for Proving an Inequality Involving Products

I'm working on a proof in my complex analysis course that involves showing that $$ A \cdot B \leq C \cdot D $$ ($A$, $B$, $C$, and $D$ are expressions involving the moduli of complex numbers). My ...
0
votes
1answer
40 views

show analytic function such $f(z)={\operatorname{Log}(z+5)\over z^2+3z+2}$

Show that $f(z)=\dfrac{\operatorname{Log}(z+5)}{z^2+3z+2}$ is analytic everywhere except at the point $-1,-2$ and on the ray $\{(x,y):x\le -5,y=0\}$. i think that separate denominator and ...
1
vote
1answer
67 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
0
votes
2answers
38 views

Usage of Maximum Modulus Theorem?

I have a function $f(z)=ze^z$, I want to find the maximum value of $|f(z)|$ as $z$ varies over the region $D=\{x+iy: x^2+y^2\leq 4, x,y\geq 0\}$. I was thinking that this is case where I can use the ...
0
votes
1answer
25 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
0
votes
4answers
51 views

Induction of logarithmic derivatives of complex functions?

I am trying to use induction to prove the logarithmic derivative of a complex function (called $P(Z)$ here). I define a function $P(z) = (z-z_1)(z-z_2)...(z-z_n)$ and then I want to use induction on ...
3
votes
1answer
90 views

Complex integral $ \int_{\partial D_R} \frac{\exp\bigr( \pi i (z - 1/2)^2 \bigl)}{1 - \exp(-2\pi i z)} \mathrm{d}z $

I have been working on the following problem from Gamelin VII.1 problem 6. Consider the integral $$ J = \int_{\partial D_R} \frac{\exp\bigr( \pi i (z - 1/2)^2 \bigl)}{1 - \exp(-2\pi i z)} ...
1
vote
1answer
27 views

Is this an appropriate strategy for evaluating a complex limit?

$$\lim_{z\rightarrow 0} \frac{\overline{z}^2}{z}=\lim_{z\rightarrow 0}\frac{r^2e^{-2i\theta}}{re^{i\theta}}=\lim_{z\rightarrow 0}re^{-3i\theta}=0$$ I wanted to avoid an $\varepsilon$-$\delta$ proof.
4
votes
1answer
98 views

holomorphic function with bounded real part on punctured neighborhood $\dot{D}_{\epsilon}(z_0)$

I've seen here that a holomorphic function with bounded imaginary part on a punctured neighborhood of $0$ has a removable singularity at $0$. I just wanted to know if this result could be also ...
3
votes
3answers
176 views

All possible values of $i^{-2i}$ - NBHM $2013$

Question is to write down all possible values of $i^{-2i}$ I know that $e^{i\theta}=\cos(\theta)+i\sin (\theta)$ So, I can write $i=e^{i.\frac{\pi}{2}}$ then I would have : ...
2
votes
1answer
132 views

Analytic continuation of zeta is meromorphic on $\mathbb{C}$ with simple pole at 1

We have the following identity: For some contour $\gamma$ and $\forall s \in \mathbb{C} $ Re $s > 1$: $$-2i\sin(\pi s) \Gamma(s)\zeta(s)= \Large\int_{\gamma} \frac{(-z)^{s-1}}{e^z-1}dz$$ The ...
0
votes
1answer
86 views

Prove meromorphic function can be written as product of holomorphic and rational function

I'm not able to prove this. Any help would be welcomed ! Let U be a simply connected domain and let $f$ be a meromorphic function on U with only finitely many zeroes and poles. Prove that there is ...
0
votes
1answer
34 views

Complex proof problem

Let be P(z) a complex polynomial with a degree of n>=1 and |(p(z)| <= a|z| then there exist a complex number c such that p(z) = cz.
1
vote
0answers
90 views

Determine where the function $f(z)=\operatorname{Log}(z^3+2i)$ is analytic.

I need to know if my intuition is correct here. Idea: I would use De'Moivre's formula to find all third roots of -2i and exclude one of these rays because these are the values that give $z^3+2i=0$ ...
2
votes
0answers
208 views

Cauchy's theorem for integral homotopic closed curve in $G\subset\mathbb{C}^n$.

Recall Cauchy's theorem (third version in the Conway's book "Function of one complex variable", thm 6.7. page 90 in the second edition): Let $f$ be an analytic function on $F\subset\mathbb{C}$ and ...
2
votes
1answer
140 views

Can I use Schwartz's Lemma to prove that $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ implies $f(z)=0$ for all $z\in\mathbb{C}$?

Problem. Suppose that $f(x)$ is an entire function satisfying $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ as $|z|\rightarrow \infty$. Show that $f(z)=0$ for all $z\in \mathbb{C}$. The ...
1
vote
1answer
310 views

Derive Poisson's integral formula for Im z>0

How to derive Poisson's integral formula for $\text{Im }{z}>0$ given that for $|z|<1 $ we have ...
1
vote
1answer
80 views

A theorem about contour integration in $\mathbb C$.

The following theorem is stated in a book: If $f$ is continuous on the arcs $\gamma_r=\{a+re^{i\theta}\,:\,\theta_1\leq\theta\leq\theta_2\}$ where $a,\theta_1,\theta_2$ don't depend on $r$, and if ...
24
votes
6answers
1k views

Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were ...
3
votes
3answers
1k views

Proof of Cauchy-Schwarz inequality - Why select s so that so that $||x-sy||$ would be minimized?

I was looking at a number of different proofs of the cauchy schwarz inequality in an inner product space ($\mathbb{R}^n$ or $\mathbb{C}^n$). All of them used the idea of $||x-sy||$ where $s$ was ...
8
votes
2answers
280 views

How to prove a property regarding periodicities of points in the Mandelbrot set?

While studying a visual representation the Mandelbrot set, I have come across a very interesting property: For any point inside the same primary bulb (a circular-like 'decoration' attached to the ...
1
vote
1answer
241 views

How to show it is convex?

From a journal entitled Certain subclass of starlike functions by Gao and Zhou in 2007, they mentioned that " since $ k(z)=\frac{z}{1-zt}$ is convex in open unit disk $E,z:|z|<1$, $k(\bar{z})= ...
0
votes
1answer
266 views

Proof of Uniform Convergence

Prove that the sequence of functions $\{f_{n}(z) = (1+nz)^{-1}\mid n=1,2,...\}$ converges uniformly to $f(z)=0$ for $|z| \geq r > 0$. To answer the question, for a given choice of $\epsilon > ...
0
votes
1answer
100 views

Easiest/shortest proof of the following theorem

If $f$ is a rational function defined on the complex plane. Then the number of the zeros is equal to the number of the poles (counting multiplicity) and considering points at infinity. I can imagine ...