All questions with tag [math: complex-analysis]


Field Extension problem beyond $\mathbb C$

There are great deals of areas in between $\mathbb C$ and also Meromorphic Functions on $\mathbb C$. As an example set of "All Even Meromorphic Functions on $\mathbb C$" is a subfield in between $\mathbb C$ and also Meromorphic Functions on $\mathbb C$. Inquiry: How to classify such subfields? I have no suggestion whether someone res...
2022-07-25 20:47:10

Finding the number of analytic functions which vanish only on a given set.

Let $S = \{0\}\cup \{\frac{1}{4n+7} : n =1,2\ldots\}$. Just how to find the variety of analytic features which disappear just on $S$? Alternatives are a: $\infty$ b: $0$ c: $1$ d: $2$
2022-07-25 20:46:18

How to integrate $\int_{\gamma_1} \frac{dz}{z(z-i)}$ with $\gamma_1 = Re^{it}$, $R>1$?

I am stuck calculating the integral $$\int_{\gamma_1} \frac{dz}{z(z-i)}$$ over $\gamma_1 = Re^{it}, R>1$. If I had to integrate over $\gamma_2 = re^{it}, r < 1$, I could just expand the integrand into a power series (using the geometric series) around $z=0$, but with $R>1$ this approach won't work. I quite frankly don't have...
2022-07-25 20:46:07

Limit conditions of a subharmonic function imply that it is constant

Let $u$ be a subharmonic function on $\mathbb{C}$. Intend that $$\limsup_{z\to \infty} \frac{u(z)}{\log|z|}=0$$ I'm attempting to confirm that this indicates $u(z)$ is constant. I sense that it might concern Hadamard is Three Circles Theorem and/or the maximum concept for sub/superharmonic features, yet I'm not obtaining anywhere.
2022-07-25 20:43:44

Translation of entire functions along the real axis

Given a whole function $f(z)$, and also $0\neq a\in \mathbb R$. We specify the translation driver: $$T_{a}f(z)=f(z-a).$$ What buildings the new function $f(z-a)$ could have? It is whole function! What concerning the absolutely nos of $f(z-a)$? I recognize it is an open inquiry, yet anything you recognize can aid me.
2022-07-25 20:42:13

Holomorphic Automorphism Group

By a domain I mean an open connected subset of ${\mathbb C}$. If $D$ is a domain, let $\operatorname{Aut}(D)$ denote the collection of holomorphic bijections $f:D\to D$. It is well-known that if $f$ is holomorphic, so is its inverse, so $\operatorname{Aut}(D)$ is actually a group. We can give a topology to $H(D)$, the set of holomorphic maps wit...
2022-07-25 20:40:42

How to show set of all bounded, analytic function forms a Banach space?

I am trying to prove that set of bounded, analytic functions $A(\mho)$, $u:\mho\to\mathbb{C}$ forms a Banach space. It seems quite clear using Morera's theorem that if we have a cauchy sequence of holomorph functions converge uniformly to holomorph function. Now i am a bit confused what norm would be suitable in order to make it complete .
2022-07-25 20:40:31

Rectangular form of a complex number?

Why does rectangle-shaped kind act as an exact summary of an intricate number? Why not $a * bi$ (reproduction) or an additional procedure? Why does enhancement define the partnership in between the actual component and also intricate component? As an example, polar kind defines the relationship on a fictional and also actual aircraft. What princ...
2022-07-25 20:23:55

Jensen's Inequality for complex functions

Jensen's inequality states that if $\mu$ is a probability measure on $X$, $\phi$ is convex, and $f$ is a real-valued function, then $$ \int \phi(f) \, d\mu \geq \phi\left(\int f \, d\mu\right).$$ Is there a variant on this inequality for complex-valued functions? Namely, if $\phi$ is a function from $\mathbb C$ to itself such that $$\left|\phi\l...
2022-07-25 20:23:29

Residue at $z=\infty$

I'm a little bit perplexed at when to make use of the estimation of a deposit at $z=\infty$ to compute an indispensable of a function. Below is the instance my publication makes use of: In the favorably oriented circle $|z-2|=1$, the indispensable of $$\frac{5z-2}{z(z-1)}$$ returns 2 deposits, which offer a value of $10\pi i$ for the indispensa...
2022-07-25 20:20:55

Multiplying complex numbers in polar form?

Could a person clarify why you increase the sizes and also add the angles when increasing polar works with? I attempted increasing the polar kinds ($r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot r_2\left(\cos\theta_2 + i\sin\theta_2\right)$), and also expanding/factoring the outcome, and also wind up increasing the sizes yet can not appear...
2022-07-25 20:20:22

Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number

Let $f$ be analytic on the closed unit disk centered at the origin and $|f(z)| < 1$ for $|z| = 1$. Show that $f$ has exactly one fixed point inside the open unit disk. That is, there exists a unique number $z_0$ with $|z_0| < 1$ such that $f(z_0) = z_0$. We must prove 1 there exists at least on solution, 2 there is at most one solu...
2022-07-25 20:17:21

contour integral with rational and cosh

Here is a fun looking integral. $$\int_{0}^{\infty}\frac{1}{(4x^{2}+{\pi}^{2})\cosh(x)}dx=\frac{\ln(2)}{2\pi}$$. I rewrote it as $\frac{2e^{z}}{(4z^{2}+{pi}^{2})(e^{2z}+1)}$ It would appear there is a pole of order 2 at $\frac{\pi i}{2}$. This is due to it being a zero of cosh and the rational part. I think the residue at $\frac{\pi i}{2}$ is ...
2022-07-25 20:14:36

Application of Jensen's formula

While studying for an upcoming complex analysis qualifying exam, I found the following problem in Conway's Functions of One Complex Variable (XI.1 exercise #2). Let $f$ be an entire function, $M(r)=\sup\{|f(re^{i\theta})|:0\leq\theta\leq2\pi\}$, $n(r)=$ the number of zeros of $f$ in $B(0;r)$ counted according to multiplicity. Suppose that $f(0)...
2022-07-25 20:13:55

Complex Analysis: Finding the level curves of a function?

Consider the function $f(z)=z^2$. Confirm that degree contours of $Re(f(z))$ and also $Im(f(z))$ at $z=1+2i$ are orthogonal per various other. I am not exactly sure just how to use degree contours or shape lines for intricate variables. Regarding actual variables go, I realize that for a function like $f(x,y) = \sqrt{x^2+y^2}$, the degree conto...
2022-07-25 20:13:30

Roots of unity?

The $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even: The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = -w^j$. Squaring them produces the $\frac{n}{2}$nd roots of of unity. Could someone explain the first statement? I understand why the $n$ roots are plus-minu...
2022-07-25 20:13:26

Difference of two subharmonic functions

Is it real that for a smooth actual - valued function $h(z)$ on some area of the closure of a bounded domain name, that $h$ can be shared as the distinction of 2 smooth subharmonic features? If so, just how? If I took into consideration $u(z)= h(z) + C|z|^2$, just how would certainly that aid? I would certainly value any kind of input I can get...
2022-07-25 20:07:00

Question regarding an analytic function and a meromorphic one

Is it feasible to have an analytic function on the device disk $\mathbb{D}$ that has definitely several separated absolutely nos? What is an example? I presume then that would certainly make this analytic function nontrivial, deal with? Additionally, what is an instance of a meromorphic function on the facility aircraft with straightforward pos...
2022-07-25 20:06:00

Estimates involving a holomorphic function on the unit disc

Assume that $f$ is an analytic function on the unit disc $\mathbb{D}$ and continuous up to the closure. Therefore $f(z)=\sum\limits_{n=0}^\infty c_nz^n$ for all $z \in \mathbb{D}$. If $f$ have $m$ zeros in $\mathbb{D}$ how can you prove that $$ \min\{|f(z)| : |z|=1\}\leq |c_0|+\ldots+|c_m| $$ To begin with, the minimum of the funcion in $|z|=1$ ...
2022-07-25 19:54:21

Characterization of Cauchy-Riemann operator

Let $U \subset \mathbf C$ be an open part of the facility aircraft and also intend we have a differential driver of order 1, $L: \mathcal C^{\infty}(U) \to C^{\infty}(U)$ such that $Lu = 0$ if and also just if $u$ is holomorphic in $U$. Is it real that $L$ must be the Cauchy - Riemann driver $\frac{\partial}{\partial \bar z}$?
2022-07-25 19:45:26