1
vote
1answer
29 views

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$ I am trying to prove the triangle inequality of this norm. So far I have that: \begin{align} ...
-1
votes
2answers
37 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
0
votes
2answers
82 views

Real and imaginary parts of a complex-valued function

How do you get a complex-valued function $ f(z) = f(x+iy) = \frac{z^{s-1}}{e^{-z}-1}, $ where $s$ is a constant complex number and $z$ is a complex variable, into the form: $ f(x+iy) = a(x,y) + ...
0
votes
1answer
72 views

Separable Function: Alternative Representation

How does one get the following function $$ f(u) = f(x+iy) = \frac{u^{z-1}}{e^{-u}-1}, $$ where $z$ is a constant complex number and u is a complex variable, into the form: $$ f(x+iy) = v(x,y) + ...
1
vote
1answer
28 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
2
votes
1answer
155 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
2
votes
1answer
41 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
6
votes
1answer
161 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
0
votes
1answer
35 views

Singularities of a function

Given $f(z):=\operatorname{Log}(\frac{z-2}{z-3})$, $\operatorname{Log}:\mathbb{C}\setminus\mathbb{R_{\le 0}}\to \mathbb{C}$. Is in $z_0=3$ a essential singularity of f? I'm not sure what is correct... ...
0
votes
1answer
31 views

Extension of a holomorphic function in the disc

let $f$ be a continuos function in ${0<|z| \leq r} $ holomorphic in the inner and such that $f(z) $ is real for $|z|=r$. Prove that exist a function $g$ on $\mathbb{C} ^*$ such that $f(z) =g(z) $ ...
1
vote
0answers
28 views

$f(z) = u(x,y) + i\cdot v(x,y)$ holomorphic in a connected open set $D$, such that $a\cdot u(x,y)+b\cdot v(x,y)=c$, is constant

Let $f(z) = u(x,y) + i\cdot v(x,y)$ be a holomorphic function in a connected open set $D$. If $a\cdot u(x,y)+b\cdot v(x,y)=c$ in $D$, where $a,b,c$ are real constants which are not all zero, why ...
1
vote
1answer
23 views

A continuos and holomorphic function on $D^2$ that take pure imaginary values on $S^1$ is costant

Let $D := \{ |z| < 1\}$ and $f : \overline{D} \rightarrow \mathbb{C}$ be a continuos and holomorphic function on $D$ that take pure imaginary values on $\partial D$. Why $f$ is constant? From ...
2
votes
1answer
53 views

An open map from $\mathbb{C} \rightarrow \mathbb{C}$ has open real and imaginary part?

If $f(z) :\mathbb{C} \rightarrow \mathbb{C}$ is an open map such that $f(z) = f_1(z) + if_2(z)$ where $f_1$ and $f_2$ represent respectively his real and imaginary part, we could say that both $f_1$ ...
3
votes
1answer
33 views

Quick question on poles

Consider this function for $0 < a < b$: $$f_{(z)} = \frac{z^4}{z^2(z-\frac{a}{b})(z-\frac{b}{a})}$$ This function has a pole of order $2$ at $z=0$, a pole of order 1 at $z=\frac{a}{b}$, but ...
2
votes
3answers
102 views

Prove that there are no entire function satisfying $|f(z)|\ge |\cos(z)|+|\sin(z)|$ for all $z\in \Bbb C$

Hi. I need to prove that there are no entire function satisfing $$|f(z)|\ge|\cos(z)|+|\sin(z)| \\\forall z\in\mathbb{C}.$$ I think I need to use the Liouville theorem. Appriciate any help, Thanks!
2
votes
2answers
45 views

Find all holomorphic functions on $\mathbb{C}$, except for some singularities, such that $|f(z)|\leq C(|z|^{3/2}+|z-1|^{-3/2}), z\in\mathbb{C}-\{1\}$

First I wrote the Laurent series of $f(z)$ around $z=1$: $$ f(z)=\sum_{n=-\infty}^{-1}c_n(z-1)^n+\sum_{n=0}^{\infty}c_n(z-1)^n. $$ Now if $|z|$ becomes very large, the first sum with the negatives ...
1
vote
1answer
25 views

sequence of analytic functions on an open subset of $\mathbb{C}$ that converges uniformly on compact subsets

Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of analytic functions on $U$. Suppose that $(f_n)$ converges uniformly on any compact subsets of $U$ to a function $f$. Let ...
1
vote
2answers
226 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
0
votes
0answers
18 views

How to write the condition for Image of a function?

If $\Omega_l$ is $\Omega$ with $|x|<l$ and if $\Omega_S$ is the image of $z$ under mapping how we will write the condition for it. Am I right if I write $\Omega_S$ is $\Omega$ with $|S|<l$ or ...
1
vote
1answer
73 views

Conformal map entire domain to a strip with specific branchcuts

I am looking for a conformal mapping function that maps the entire z-plane to an infinite strip. (e.g. T=f(z) & -b < Real(T) > b ) I hope to find a function that cuts open to original domain ...
3
votes
2answers
88 views

Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
4
votes
1answer
102 views

Is this analytic continuation possible?

I'n new to complex analysis and am a little flustered by the following function. I would like help understanding whether or not it is possible to analytically continue it outside of the unit circle. ...
1
vote
2answers
44 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
2
votes
0answers
44 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
5
votes
4answers
334 views

Define a branch of $(z^2 − 1)^{1/2}$ which is analytic in the unit disk.

Hint: $z^2 − 1 = (z + 1)(z − 1)$. I'm really struggling with this question. I understand that for this function to be analytic it has to be differentiable in some neighbourhood, but I have no idea ...
0
votes
0answers
39 views

Limiting an integral

Let us define a function $$h_z(t)= 1/2\pi i (\frac{1}{t-z}-\frac{1}{t-\bar z})$$ for $z,t \in \mathbb H ^+$, the upper half plane. Let $f$ be a bounded analytic function on $\mathbb H^+ \cup \mathbb ...
1
vote
1answer
41 views

Finding the types of singularities

I am working on the following: Let $f : \mathbb C \setminus \{1,2\} \to \mathbb C$ be defined by \begin{align*} f(z) = \frac{\sin(z)-2}{z-2} + \frac{\cos(1/(z-2))}{(z-2)^2}+\frac{\cos(z-1)-1}{z-1}. ...
1
vote
2answers
71 views

Visualizing a complex valued function of one real parameter

I'm looking for a way to capture/graph or visualize it in my head, but I can't find how.. a 2-dimensional path won't do, because it doesn't reveal the rate-of-change.. 2 1-dimensional graphs on top ...
0
votes
1answer
33 views

Singularity structure of function in the complex plane.

Consider a piecewise differentiable function $f(x):\mathbb{R}\to\mathbb{R}$. Now, analytically continue this function ($x\to z$) to complex argument and values $f(z):\mathbb{C}\to\mathbb{C}$. For such ...
2
votes
2answers
45 views

Hunt for a function.

I am looking for any nontrivial function $f(z): \mathbb{C}\to\mathbb{C}$ such that: $f(z)$ is an entire function. A $z_p\in\mathbb{C}$ exists for which $\Re(f(z))\geq\Re(f(z_p))~\forall ...
1
vote
1answer
50 views

When is $|f(x)|$ equivalent to $f(|x|)$

Specifically for functions of a complex variable. Are there any rules of thumb?
2
votes
1answer
95 views

Why isn't $\log(-1)$ defined?

Why isn't $\log(-1)$ defined. It can be defined as being equal to $i\pi$. Why don't we define the $log$ function over Complex Numbers as well.
0
votes
1answer
68 views

Representing a function as a real part of a complex variable?

To represent a simple sinusoidally varying function $V(t)$ let us use $V(t)=Re(\hat V e^{i\omega t})$ where $\hat V$ can be a complex constant.Let $\hat V =-iV_o$ Therefore, $V(t)=V_o \sin(\omega t)$. ...
0
votes
1answer
124 views

Findin the most general harmonic polynomial of the form $ax^2 + bxy + cy^2$

The question says to find the most general harmonic form of $ax^2 + bxy + cy^2$. And I've seen one or two answered questions here on this topic but I couldn't understand $why$ certain steps were took ...
6
votes
2answers
178 views

Does inverse of a nontrivial holomorphic function always have a branch point?

Any nontrivial (i.e. which is not a first order polynomial) entire in $\mathbb{C}$ function I have thought of has a multifunction as its inverse and has a branch point. For example, ...
0
votes
2answers
119 views

For what values of $z$ is $f(z) = e^z$ real? Imaginary?

I feel like I might understand this already, but I just wanted to make sure. I said that $f(z)$ is real is $z$=any real number and $f(z)$ is imaginary if $z=rj$, where $r$ is any real number. Thanks.
0
votes
0answers
30 views

Are these functions defined or used?

Are the following functions defined and/or used in complex analysis? $$f:\Bbb R^2\to \Bbb C, (a, b)\mapsto a+bi$$ It's inverse $$f^{-1}:\Bbb C\to\Bbb R^2, a+bi\mapsto (a, b)$$ The following can also ...
1
vote
0answers
93 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 $z^3+2i=0$ ...
0
votes
1answer
37 views

Finding examples of functions with these properties.

I have this question regarding functions $f:\mathbb{C}\mapsto \mathbb{C}$, asking me to show no function exists such that it possesses both i) $f(z)^2=z$ for all $z\in \mathbb{C}$, and ii) ...
0
votes
1answer
47 views

Complex Analysis

Determine the derivative of the following functions and state where they are analytic. $$f(z) = \log(z^3) \quad \Rightarrow f'(z) = \frac{3z^2}{z^3} = \frac{3}{z}$$ Hence, this function is analytic ...
1
vote
0answers
104 views

Schwarz Reflection Principle for function complex on the real line

The Schwarz reflection principle is usually proved for function real on the real (or a subset of) line. I wonder if the same principle/theorem works for general analytic functions on the real line? ...
0
votes
1answer
27 views

Understanding the function $s^\lambda$

Consider the function $f(s) = s^{\lambda}$ where $s$ is a real variable and $\lambda$ is a complex constant with $\Re \lambda > 0$. I want to understand $f$ when $s \in I=[0,1]$. Is $f$ ...
0
votes
1answer
44 views

A question on the domain of a complex function

Let $\log$ be the main branch. What is the domain of the complex function $f(z)=\log(1/(1-z^{2}))$?
2
votes
0answers
46 views

Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.

Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
11
votes
2answers
285 views

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable? Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
4
votes
4answers
392 views

what is the relation of smooth compact supported funtions and real analytic function?

What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
4
votes
0answers
149 views

How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
3
votes
2answers
367 views

Weierstrass factorization of sine, and related questions

So the idea is that you can represent a function as a product of its zeroes, and there are some fundamental factors that often crop up. I am interested in, give this is the WF of sine : Is it ...
1
vote
0answers
35 views

Relations between complex functions satisfying a specific condition

What is the relation between the following two complex functions: $$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$ and $$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$ ...
0
votes
3answers
91 views

How to show that $\frac{z-a}{a-z}$ has not inverse?

How to show that $\frac{z-a}{a-z}$ has not inverse? I know that $$\frac{z-a}{a-z}=\frac{-1(a-z)}{a-z}=-1 .$$ But state if I'm wrong, that following is true: $$f(z)=\frac{z-a}{a-z} \Leftrightarrow ...