# All questions with tag [math: complex-analysis]

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## 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
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## 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
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## 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 $\gamma_1 = Re^{it}, R&gt;1$. If I had to integrate over $\gamma_2 = re^{it}, r &lt; 1$, I could just expand the integrand into a power series (using the geometric series) around $z=0$, but with $R&gt;1$ this approach won't work. I quite frankly don't have...
2022-07-25 20:46:07
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## 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
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## 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
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## 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
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## 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
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## 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
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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 $$\left|\phi\l... 2022-07-25 20:23:29 1 ## Residue at z=\infty I'm a little bit perplexed at when to make use of the estimation of a deposit at z=\infty to compute an indispensable of a function. Below is the instance my publication makes use of: In the favorably oriented circle |z-2|=1, the indispensable of$$\frac{5z-2}{z(z-1)}$$returns 2 deposits, which offer a value of 10\pi i for the indispensa... 2022-07-25 20:20:55 2 ## Multiplying complex numbers in polar form? Could a person clarify why you increase the sizes and also add the angles when increasing polar works with? I attempted increasing the polar kinds (r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot r_2\left(\cos\theta_2 + i\sin\theta_2\right)), and also expanding/factoring the outcome, and also wind up increasing the sizes yet can not appear... 2022-07-25 20:20:22 1 ## Prove that there is an unique z s.t. f(z) = z where z is a complex number Let f be analytic on the closed unit disk centered at the origin and |f(z)| &lt; 1 for |z| = 1. Show that f has exactly one fixed point inside the open unit disk. That is, there exists a unique number z_0 with |z_0| &lt; 1 such that f(z_0) = z_0. We must prove 1 there exists at least on solution, 2 there is at most one solu... 2022-07-25 20:17:21 1 ## contour integral with rational and cosh Here is a fun looking integral.$$\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 1 ## 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 1 ## 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 0 ## Roots of unity? The nth 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 nth 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 1 ## 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 1 ## 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 1 ## 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
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## 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