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, ...

learn more… | top users | synonyms (2)

0
votes
1answer
38 views

Is it valid to divide over this factor?

Is it permissible for me to perform this operation: $$z^nf(1/z) = {c}$$ $\implies$ $$f(1/z) = \frac{c}{z^n}$$ This is in the context of complex analysis, analyzing poles at infinity / zero. If ...
3
votes
1answer
32 views

single valued analytic branch of multivalued function

Consider $f(z)=\sqrt{z\sin z}$. Can $f(z)$ be defined near the origin as a single valued analytic function? How do we choose the branch cut. The answer is here ...
7
votes
4answers
317 views

Why do we say “radius” of convergence?

In an intuitive sense, I have never understood why a power series centered on $c$ cannot converge for some interval like $(c-3,c+2]$. Also, I have had a few professors casually mention that a series ...
3
votes
2answers
49 views

Why is $z^2$ a conformal mapping?

It's not a one-to-one mapping, by the Fundamental Theorem of Algebra. $e^z$ is one-to-one, when restricted to a horizontal strip of width = $2\pi i$. Is it a similar argument for $z^2$? Thanks, ...
3
votes
1answer
100 views

$\lim\limits_{z \to \infty} f(z) = \infty$, show that $f$ is a polynomial.

I had an approach to the following problem which now I'm not sure will work: If $f$ is entire, and $\lim\limits_{z \to \infty} f(z) = \infty$, show that $f$ is a polynomial. Case 1: there exist ...
0
votes
0answers
8 views

When proving that the most general conformal mapping of the UHP is a LFT, can I start the proof with a linear polynomial?

I'm having some difficulty with starting a proof that proves the most general one-to-one conformal map from the UHP onto itself is of the form az+b/cz+d, with a,b,c,d real and ad-bc =1. My idea is to ...
1
vote
2answers
57 views

Find the image of $|z| \leq 1$, with the function $f(z) = \frac{z}{z^2 + 1}$. [closed]

Find the image of $|z| \leq 1$, with the function $f(z) = \frac{z}{z^2 + 1}$. I hope that anyone can help me. Thanks in advance.
1
vote
2answers
44 views

showing the exsitence of fixed point in $\bar{D} = \{z:|z| \leq R\}$

we have close circle $\bar{D} = \{z:|z| \leq R\}$ and analytic function $f$ in $\bar{D}$ that: $f(\bar{D})\subseteq \bar{D}$. we need to show that there is $z_0 \in \bar{D}$ that $f(z_0) = z_0$. ...
4
votes
2answers
39 views

help verify a conformal map between regions

Find a conformal map from the set $\{z \in \mathbb{C}: |z|>1\}\setminus (-\infty,-1)$ onto the set $\mathbb{C}\setminus(-\infty,0]$. Here is my thought, but I'm not sure if it is correct, can ...
4
votes
1answer
59 views

Regarding a complex analysis problem.

I'm trying to do this problem from Gamelin's book: Let $f_n(z)$ be a sequence of analytic functions on a domain (= open connected set) $D$ such that $f_n(D) \subset D,$ and suppose that $f_n$ ...
-1
votes
0answers
19 views

Conformal mapping on line segment

How do we map a line segment to the upper half plane? How would we map the complement of a line segment to the upper half plane? Let say the line segment is $\{iy | y\in[-1,0]\}$. All the maps are ...
1
vote
1answer
61 views

radius of convergence of $\sqrt{\cos z}$

Let $f(z)=\sqrt{\cos z}$ and pick the branch such that $f(0)=1$. Consider the power series of $f$. Find the radius of convergence of power series. I claim that the radius of convergence is at least ...
0
votes
2answers
55 views

Let $ \gamma $ be the unit circle then $ \int_\gamma \frac {dz}{z^2 − 2z} = -\pi i$

Definition: If $ f $ is holomorpic in $G$ and gamma $\gamma$ is $G$-homotophic to a point then gamma is G-contratible and if gamma is G-contractible then $ \int_\gamma f = 0 $. By splitting the ...
1
vote
2answers
25 views

where I'm wrong in calculating Residue at $\infty$

Im looking at the function $f(z) = \dfrac{e^z}{z}$. trying to calculate integral on , say, $|z| = 1$. the answer is $2\pi i*res(f,0) = 2\pi i$ so far, so good, but , when I try to calculate the ...
2
votes
2answers
58 views

Proof of $u\circ f$ is harmonic.

If $u$ is a harmonic function defined on the complex plane and $f$ is entire then show that $u\circ f$ is harmonic. Actually I don't understand the question properly. As, $f$ is entire so, ...
2
votes
1answer
51 views

Poles and Singularity

Let $f(z)$ be an analytic function in the whole complex plane apart from simple poles at $z_1,...,z_m$. Moreover $f(1/z)$ has a simple pole at $0$. Show that $f$ is a ratio of two polynomials. To ...
12
votes
3answers
494 views

Does this proposition from complex analysis depend on AC?

I was reading III vol. of Princeton lectures on analysis. Proposition 1.4: "If $\Omega_{1}\supset\Omega_{2}\supset\ldots\supset\Omega_{n}\supset\ldots $ is a sequence of non-empty compact sets in ...
5
votes
2answers
73 views

Fourier transforms of $f(t)=\frac{\sin{at}}{t}$

I want to derive the following pair of Fourier transforms: First: $$f(t)=\dfrac{\sin{at}}{t}$$ $$F(\lambda)= \begin{cases} \sqrt{\dfrac{\pi}{2}}, & \text{if } |\lambda|<a \\ 0, & ...
1
vote
1answer
31 views

Showing f(0) is bound above by geometric mean of supremum over intervals?

So I am working on the following problem. Suppose that $f$ is entire and $n$ is a fixed positive integer. If $$I_k:=\left[\frac{2(k-1)\pi}{n},\frac{2k\pi}{n}\right],$$ for $k=1,2,\dots,n$ and ...
0
votes
2answers
53 views

Analytic continuation of primality function

(There is my initial question, but by advice of @Charles I'm splitting it) For integers we have a primality function: $$ isprime(n)=\begin{cases}1,&\text{$n$ is prime}\\0,&\text{$n$ is not ...
0
votes
0answers
61 views

Does it make sense to talk about the “degree” of a function, if the function is not a polynomial?

Let f and g be one-to-one conformal mappings. Show that f/g must be a linear fractional transformation az+b / cz+d. Proof: look at h(z) = f(z) + w*g(z). If either f or g were of degree greater ...
4
votes
1answer
39 views

Maximum Modulus path

Consider any entire, non constant function $f:\Bbb C\to \Bbb C$. Choose any $z\in\Bbb C$ and define $m(r)\in\overline D(z,r)$, for any $r\ge 0$, with this property: $$|f(m(r))|\ge|f(w)|\;\forall w\in ...
2
votes
0answers
15 views

Determining if an equation represents (?) a Riemann surface

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...
1
vote
1answer
43 views

Is an analytic one-to-one function on the whole plane necessarily a polynomial? (Can it be disproved?)

I had to show what a one-to-one analytic function from the plane to itself could possibly be. So, I studied the behavior of such a function at infinity: Case 1: Such a function cannot have no ...
-1
votes
1answer
100 views

Proof of why there are infinite solutions of $i^x=x$

I recently created the problem $i^x=x$ and was enlightened to the fact that there were infinite solutions. Now I want to know why. Thank you for any help.
1
vote
1answer
25 views

Characterize all analytic functions satisfyinf the given condition

Characterize all analytic functions $f(z)$ in $|z|<1$ such that $|f(z)|\le |\sin(1/z)|$ , for all $0<|z|<1$. I can't understand from where I will start ?
0
votes
1answer
48 views

Characterization of analytic functions

First, see this link on the alternative characterizations of analytic functions. I want to prove a version of 3) for complex-analytic functions. In particular: If $f$ is a complex-analytic function ...
4
votes
1answer
55 views

Inverse image of $[-2,2]$ under cosine.

I solved the following problem: Let $g(z) = \cos z$. Find $g^{-1}[-2,2]$. but my solution was kind of long. I was wondering if there was a faster way to do this problem. Here's my solution: ...
1
vote
0answers
29 views

Image of a complex region

Let $A$ be the complex region that satisfies: $1\leq|z|\leq2 \wedge 0\leq\operatorname{Arg}z\leq\frac{1}{3}\pi$. Draw $A$ in the complex plane and describe the image of $A$ under $z\mapsto z^2$. ...
0
votes
0answers
36 views

single valued holomorphic functions.

Do there exists singled valued analytic function $f(z)$ such that $$f(z)=\int_0^\infty e^{-tz}\sqrt{\sin t} dt$$ More precisely, can we find domain $D$ such that the $f$ is single valued and analytic? ...
2
votes
1answer
54 views

Path integration in the complex plane

Problem: Show that $$\int_\gamma e^{iz-z^2}dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. I got stuck in the middle of the calculation when I write : ...
14
votes
1answer
202 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
2
votes
0answers
22 views

Schwarz Inequality of function from upper half plane to disc

So I've been working on this problem and I have everything nailed down (I think) except for the very end. In particular I get a bound, but I can't seem to reduce it down to the one the question is ...
9
votes
2answers
141 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$

Edit: This problem 1 is a 2014 Sydney mathematics competition problem (8th grade). It seems difficult to solve. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such ...
1
vote
0answers
22 views

Holomorphicity and modularity of Jacobi forms

I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = ...
9
votes
1answer
115 views

Non-computable function having computable values on a dense set of computable arguments

A rational complex number is a complex number whose both real and imaginary parts are rational numbers. Note that a rational complex number is a finitary object that can be an input or an output of an ...
0
votes
1answer
43 views

differentiability of a complex function and its (real) vector-valued equivalent

Let f be a complex-valued function of a complex variable, defined as $$f(z)=u(z) + iv(z),$$ where $z=x+iy$. Let $g:R^2 \rightarrow R^2$ be its (real) vectored-valued equivalent, i.e. ...
10
votes
1answer
182 views

Show that $\lim\limits_n \frac{a_n}{a_{n+1}} = z_0$

I was wondering if anyone could give me a hint on the following problem: Let $f$ be meromorphic on the unit disc with only a simple pole at $0 \neq z_0 \in D$. Let $a_0 +a_1z + \cdots$ be a power ...
3
votes
0answers
73 views

Is my proof of linear fractional transformations correct?

a) Prove that the most general $1-1$ conformal map of the upper half-plane onto itself is of the form $$z \to \frac{az+b}{cz+d}$$ where $a,b,c,d \in \mathbb{R}$ and $ad-bc =1$. b) Let $f$ be a $1-1$ ...
0
votes
1answer
48 views

Is it true that every open subset of $\mathbb{C}$ is a countable union of disjoint connected open sets?

Is it true that every open subset of $\mathbb{C}$ is a countable union of disjoint connected open sets? That is if $A\subset\mathbb{C}$ then $A=\bigcup _{i=1}^{\infty}U_{i}$ where $U_{i}$ are ...
4
votes
1answer
60 views

Problem in exercise of Complex Analysis

I have a Complex Analysis exam in 2 days. The last exam had, among other exercises, the following: Let $f$ be a function holomorphic in $\mathbb{D}\smallsetminus\{0\}$ that does not have a ...
0
votes
0answers
28 views

Normal convergence of complex series

I have troubles with this task: Let $\mathbb{R}\_$ be the set of non-positive real numbers and $U = \mathbb{C}\backslash \mathbb{R}\_$ For $n \ge 0$, consider a function $f_n$$:U \rightarrow ...
0
votes
1answer
34 views

how to show $f(iz) = if(z)$ under the following conditions

If I have a $f$ isomorphism that map unit square to the unit circle in the complex plane, and also $f(0) = 0$ the question is how to prove that $f(iz)$ = $if(z)$ for every $z \in$ unit square. I ...
0
votes
2answers
75 views

Integrate along the vertical strip

I want to show that some integration with vertical line is bounded. function $f(\mu)$ is given by $$ f(\mu)=A^{-\sqrt{\mu}} \frac{(B_1-\sqrt\mu)}{(B_2-\sqrt\mu)(B_3+\sqrt\mu)} $$ where $f$ is defined ...
0
votes
1answer
48 views

Infinitely (countably) many essential singularities

Let $f$ be a holomorphic function on $\mathbb{C}\setminus A$, where $A$ is a set of isolated singularities. I know it is possible, that $A$ contains infinite (countable) number of poles, but: 1) Is ...
2
votes
1answer
25 views

Applying Rouche's Theorem to $f_\epsilon(z) = f(z) + \epsilon g(z)$ on $\vert z \vert \leq 1$, where $f(0) = 0$.

Suppose f and $g$ are holomorphic in a region containing $|z|\leq 1$. Suppose that $f$ has simple zero at $z=0$, and vanishes nowhere else in $|z|\leq 1$. Let $f_\epsilon(z)=f(z)+\epsilon g(z)$. ...
0
votes
1answer
28 views

Function with removable singularity

This is an exercise in stein: Suppose $f$ is holomorphic in the punctured disk around the origin, suppose also $|f(z)|\leq A|z|^{-1+\epsilon}$ for some $\epsilon>0$ and all $z$ near 0. Show that ...
0
votes
0answers
48 views

On Riemann's extended zeta function

Let $\xi$ be Riemann's extended zeta function. Can any one calculate $\xi(-\frac{2\pi k }{\ln 2}i)$ for $k \in \mathbb Z \setminus \{0\}$? P.S. It really seems that $\xi(-\frac{2\pi k }{\ln ...
2
votes
0answers
46 views

Schwarz's Lemma Type Application

So I have the following question: Let f be analytic in an open set which contains the closed unit disc $\overline{\mathbb{D}},$ and assume $M:=\max\{\textrm{Re}(f):|z|=1\}\geq0.$ Prove that for ...
2
votes
2answers
80 views

What can be a Fourier transform's domain?

As I understand it, Fourier inversion theorem states that, for a Schwartz function $f: \mathbb{R} \to \mathbb{C}$, Fourier transform $$\mathcal{F}f(\omega) = \int_{\mathbb{R}} f(t) e^{i \omega t} dt ...