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
0answers
19 views

Make a conform map g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i) = 0$

make a conform transformation g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i)$. I actually solved it by taking the inverse of $f(x) ...
1
vote
0answers
34 views

Confusion About Inversion of z-Transform

I got two method to take inversion of $X(z)$ 1. Inversion Integral Formula $$ X(z) = x(0) + x(T)z^{-1} + x(2T)z^{-2} + ... + x(kT)z^{-k} $$ where we try to isolate $x(kT)$ as a residue of ...
0
votes
1answer
229 views

Find a branch of $\log (2z - 1)$ that is analytic at all points in the plane

Find a branch of $\log (2z - 1)$ that is analytic at all points in the plane except those on the following rays a) {$x + iy : x \leq \frac{1}{2}, y = 0$} Definition: $F(z)$ is said to be a branch ...
0
votes
2answers
130 views

If $f$ is an entire function then what about the set $S=\{Ref(z)+Imf(z) :z\in D\}.$

Let, $f$ be an entire function on $\mathbb C$ and let $D$ be a bounded open subset of $\mathbb C$. Let, $$S=\{Ref(z)+Imf(z) :z\in D\}.$$ Which of the following(s) is(/are) necessarily true ? (a) $S$ ...
2
votes
2answers
51 views

Integrals of fractions(Complex)

I'm a bit clueless about some (presumably basic) complex integrals. How would I integrate (over a circle centered at the origin, let's say of radius 2) things like $\frac{1}{z^2+z+1}$ or $\frac ...
1
vote
1answer
380 views

How to integrate $e^z/z^2$?

This may be a very basic question. How to compute the integral $ \int_\gamma \frac{e^z}{z^2} \, dz$, where $\gamma$ is the unit circle? I did it with Cauchy's integral formula for $\int_\gamma ...
2
votes
0answers
36 views

What are some general strategies to build measure preserving real-analytic diffeomorphisms?

One could prove the following theorem in the smooth setting: Theorem Let $(M,m)$ be a $d$ dimensional $C^\infty$ manifold with smooth volume $m$. Let $\{F_i\}_{i=1}^k$ and $\{G_i\}_{i=1}^k$ be ...
7
votes
2answers
162 views

What are some good sources to learn about real analytic manifolds?

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade ...
2
votes
1answer
51 views

Cauchy's integral polynomial roots

If we have $A(z)$ as the polynomial where none of its roots lie on the contour $\gamma$, how do we show that $$\frac{1}{2\pi i}\int_{\gamma}\frac{A'(z)}{A(z)}\,dz=N, $$ where $N$ is the number of ...
0
votes
1answer
36 views

Limits and Continuity of Complex Functions

I'm looking to prove the following: Let $w_0$, $z_0$, $\in \mathbb{C}$ and let $f$ be a function defined on a deleted neighborhood of $z_0$: If $lim_{z \rightarrow z_0}$ $f(z) = w_0$, then $lim_{p ...
2
votes
2answers
55 views

suppose $|a|<1$, show that $\frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.

Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself. To make such a mobius transformation i tried to send 3 points on the edge ...
2
votes
1answer
48 views

Interchanging summands among infinitely many infinite series

I am reading the following lecture notes concerning analytic number theory: http://www.math.uiuc.edu/~hildebr/ant/main4.pdf On the pages 111/112 the partial product $P_N(s)$ is defined. Then some ...
2
votes
2answers
36 views

Show this function is holomorphic

Let $f:ℂ→ℂ$ be a holomorphic function such for $x, y ϵ ℝ$, $\operatorname{Im}f(x + iy) = x + y$ Find f and check that it is indeed holomorphic. Firstly I have put $f = u+iv$ and $u = x + y$, and ...
-1
votes
1answer
149 views

Prove that a function is not holomorphic

Prove that the function $(Im z)^2$ is not holomorphic is any open subset of $C$. Please help!
3
votes
1answer
161 views

Connection between algebraic geometry and complex analysis?

I've studied some complex analysis and basics in algebraic geometry (let's say over $\mathbb C $). We had been mentioned GAGA but nothing in detail. Anyway, from my beginner's point of view, I already ...
0
votes
1answer
43 views

Question about Abel theorem application

About the application I found in my book that $\log 2=1 -\frac{1}{2}+\frac{1}{3}...$ and the explanation is $\log 2 =\lim_{r\to 1^-} \log (1+r)$, this is straight forward $\sum \frac ...
3
votes
2answers
188 views

Showing $\lim_{r \to 1^-} \sum a_nr^n=A$ using Abel's theorem

Theorem: Let $\sum a_n(z-a)^n$ have radius of convergence $1$ and suppose that $\sum a_n$ converges to $A$. Prove that $\lim_{r \to 1^-} \sum a_nr^n=A$ I found this answer Abel's Theorem, ...
-1
votes
1answer
123 views

Show that the complex function is non-holomorphic everywhere…

Can someone help me with this question: Show that complex function $f(z) = (z^2)*\overline{z}$ is non-holomorphic everywhere.
1
vote
0answers
140 views

Complex integral over sphere in polar coordinates

I have trouble evaluating the integral: $$\int_{B(0,\frac{3R}{|h|})} \frac{1}{(r e^{2i a}-e^{i a})}dr da$$ In fact I just need to estimate it from above in terms of $|h|log (\frac{1}{|h|})$, where ...
2
votes
1answer
61 views

Showing $\Bigg\vert \frac{f(z)-f(i)}{f(z)-\overline{f(i)}}\Bigg\vert\le \Bigg\vert \frac{z-i}{z+i}\Bigg\vert$

Let $\mathbb{H}=\lbrace z\in \mathbb{C}: \text{Im}z>0 \rbrace$. If $f:\mathbb{H}\to \mathbb{H}$ is analytic, then $$\Bigg\vert \frac{f(z)-f(i)}{f(z)-\overline{f(i)}}\Bigg\vert\le \Bigg\vert ...
0
votes
1answer
269 views

How to extend a conformal map from a rectangle to the upper half plane to the entire plane meromorphically

I'm taking a look at Ahlfors's Complex Analysis, Third Edition. In Section 2.3 "Mapping on a Rectangle", the author talks about how to extend a conformal map from a rectangle to the upper half plane ...
3
votes
2answers
110 views

Computing Residue for a General, Multiple-Poled function?

I'm trying to compute the residue of the following function at $a$. I'm having a little trouble seeing which poles are relevant: Compute $\,Res_f(a)$ for the following function: $$f(z) = ...
0
votes
1answer
127 views

Degree and Ramification points of an holomorphic map between Riemann Surfaces

The question is the following: we have an holomorphic map from $\Bbb P^1$ to $\Bbb P^1$, defined by $f(z)=z^3-3z$. I need to find the degree and the ramification points and their orders, then verify ...
2
votes
2answers
74 views

Laurent series expansion for $\lvert z\rvert >1$.

I have a simple complex function like this: $$\frac{z+1}{z-1}$$ When I expand it by its Maclaurin series: $$\frac{z+1}{z-1} = \frac{z-1+2}{z-1} = 1 - \frac{2}{1-z} = 1 - 2\sum_{k=0}^{\infty}z^{k} ...
0
votes
1answer
49 views

Are the sets $Re(z^2)>0$ and $\left | arg(z) \right |<\pi /4$ domains?

So I was doing some practice problems on complex analysis and got confused on this problem. I know that a domain is an open connected set, which in other words means a set with every point being ...
3
votes
4answers
65 views

Complex Analysis. Analytic functions

How could I solve this problem?: "Supose an open set A $\subset$ $\mathbb C$ , so that $A^*= \lbrace z \in \mathbb C : \bar{z} \in A \rbrace$. If f is an analytic function in A, demonstrate that ...
0
votes
1answer
37 views

Suppose that $F$ is analytic in a convex domain $G$ and $Re(F')>0$. Prove or disprove: $F$ is univalent in $G$

Suppose that $F$ is analytic in a convex domain $G$ and $Re(F')>0$. Prove or disprove: $F$ is univalent in $G$ My professor gave this example in class and gave us the solution hand out without any ...
0
votes
0answers
28 views

Find all functions $f(z)$ holomorphic on $\mathbb{C}-\{z_0,z_1\}$ such that $f(z)\leq|z-z_0|^a|z-z_1|^b$ for $a,b\in\mathbb{R}$

Question: Find all functions $f(z)$ holomorphic on $\mathbb{C}-\{z_0,z_1\}$ such that $f(z)\leq|z-z_0|^a|z-z_1|^b$ for $a,b\in\mathbb{R}$. you may assume $z_0\neq z_1$. I found that if $a,b$ are ...
0
votes
0answers
65 views

Question about branches of functions (complex power)

I have the following question, I really appreciate if someone can help me to clarify ideas and I apologize if is a stupid question: This is from Conway's complex analysis book: Let $f: G \to ...
4
votes
3answers
63 views

holomorphic complex function such that $f(\frac{1}{n})=n\space$ but $f$ is not identically $1/z$

Question: Find a function $f(z)$ holomorphic on $\{0<|z|<1\}$ such that $f(\frac{1}{n})=n\space$ for each integer $n >1 $, but so that $f$ is not identically $1/z$. I attempted to solve ...
0
votes
1answer
123 views

Differentiability implies continuity in $R^2$

Let F be a function from $R^2$ to $R^2$. F is differentiable at a point (a,b) in $R^2$, prove that F is continuous at this point. Can i write F(x,y)= F(a,b)+ c(x-a)+ d(x-b)+e where c,d,e are real ...
2
votes
0answers
77 views

Strange Holomorphic Function

Let $f$ be holomorphic function on unit disk and it is continuous on boundary of the disk. It is well-known that $f$ is constant and equal to zero if $f$ is vanishing on sub-arc of boundary (Maximum ...
1
vote
0answers
49 views

Prove that $\frac{1}{z+i} +\sin(z)=0$ has infinite solutions over $\mathbb{C}$

Prove that $\frac{1}{z+i} +\sin(z)=0$ has infinite solutions over $\mathbb{C}$ Can someone give me a clue?
0
votes
1answer
28 views

On Compactness in Runges theorem

Let $f$ be holomorphic function in an open set $\Omega$ in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions, converging uniformly to $f$ on $\Omega$. For each $f_n$, let ...
1
vote
0answers
70 views

Can $\int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2-x^2})}e^{ikx}dx$ be found in closed form?

I am trying to see if it is worth pursuing to try to calculate the following integral analytically: \begin{align} \int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2 ...
1
vote
0answers
49 views

Analytical evaluation of complex root at real extrema

The following is based on my belief/sense: If a Circle radius R is shifted up along y-axis to $ (0,h) \; [ h>R $] , then two complex roots exist due to imaginary intersection of Circle with ...
1
vote
1answer
50 views

Show that $\sum_{n=1}^{\infty}n^{i(z^2+a)}$ represents an analytic function.

Let, $a\in \mathbb R$ be fixed. Find the set of $z\in \mathbb C$ for which $$\sum_{n=1}^{\infty}n^{i(z^2+a)}$$ represents an analytic function. I know that if the radius of convergence of a power ...
6
votes
3answers
234 views

Does there exist an analytic function $f$ such satisfying the following three conditions?

Does there exist an analytic function $f:\{z\in \mathbb C:|z|<1\}\to \{z\in \mathbb C:|z|<1\} $ such that, $f(0)=1/2$ , $f(1/2)=1/3$ , $f(1/3)=1/4$ ? I tried through the Schwarz-Pick lemma ...
2
votes
1answer
74 views

evaluating $\int_0^{k!}e^{i\frac{t^k}{k!}} dt$

How to evaluate the following integral?$$ \int_0^{k!}e^{i\frac{x^k}{k!}} dx $$ Here $k$ can be any positive integer. When $k=2$, I can square it and use polar coordinates. But I've no idea about the ...
2
votes
3answers
60 views

Evaluate $(-1)^x$ by hand.

How to calculate the value for $(-1)^x$ for any $x$ by hand. Using Mathematica I kind of figured $(-1)^x=\cos(\pi x)+i\sin (\pi x)$. but how can I prove this. This is my first question here. Sorry ...
2
votes
4answers
119 views

How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$

I need to show the following: $$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$ For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
3
votes
1answer
33 views

Is this right? Difference between evaluating and expressing to cartesian form $z= 1 + \sqrt{3} i $

Am i going right with this? $$z= 1 + \sqrt{3} i $$ i need to i) evaluate $z^9$ and ii) express in cartesian for $z^5$... Which i'm a bit confused with. First what i did was find the polar form... ...
0
votes
0answers
99 views

Recursion Formula Euler Numbers

I am trying to derive the formula $$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}E_k = \displaystyle\sum_{k=0}^{n}{n\choose k}^2E_k=0$$ Where $E_k$ are the Euler Numbers. The approach that I have taken ...
0
votes
1answer
45 views

help proving function is constant

Let $f\in O(C)$ be entire function. Assume there is a disk $D_r(z_0)$ such that $f(C)\cap D_r(z_0)=\emptyset$ prove that f is constant. I need help starting the proof. I tried to using Liouville ...
3
votes
1answer
88 views

Question about Eremenko's paper on iteration of entire functions

I have two questions about Eremenko's paper On the iteration of entire functions On the second page, it says "The family $\{f^n\}=\{\underbrace{f\circ f\circ\ldots\circ f}_{n}:n\in \mathbb{N}\}$ is ...
0
votes
2answers
102 views

Consider $\left(\frac{z-2}{z}\right)^4 = 1$ then solve $z^3-3z^2+4z-2=0$

Consider $\left(\frac{z-2}{z}\right)^4=1$ then solve for the roots of $z^3-3z^2+4z-2=0$ First consider $$\left(\frac{z-2}{z}\right)^4=1$$ Let $$w^4=\left(\frac{z-2}{z}\right)^4=1$$ $$w^4=1$$ Now ...
1
vote
1answer
76 views

Prove that $f=u+iv$ is differentiable if and only if $\lim_{r→0} \frac{1}{πr^2 } \int_{C(z_0,r)}f(z)dz=0$

Suppose that $u,v$ are real-valued function that having continuous partial derivative of first order in the neighborhood of $z_0=x_0+iy_0$ . Prove that $f=u+iv$ is differentiable if and only if ...
0
votes
1answer
62 views

laurent series convergence

Find the Laurent Series representations in powers of $z$ for i) $\frac{ \cos{z}}{z}$ ii) $z^4 \cosh{\frac{1}{z^2}}$ Where do they converge? I found the Laurent Series for each of the functions, ...
2
votes
1answer
72 views

Suppose $\lim\limits_{z\to z_0} g(z) = B \neq0$. Prove that There exists $\delta > 0$ such that $|g(z)|> 1/2|B|$ for $0 < |z-z_0|< \delta$

Suppose $\lim\limits_{z\to z_0} g(z) = B \neq0$. Prove that There exists $\delta > 0$ such that $|g(z)|> 1/2|B|$ for $0 < |z-z_0|< \delta$ I wanted to know if my proof is correct. ...
1
vote
1answer
45 views

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$ Theorem: Let $f: G \to \mathbb C$ be analytic, suppose ...