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, ...

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29 views

Let $f(z)$ be a function on the unit disk $\mathbb{D}$ which is meromorphic on $\mathbb{D}$ with only one simple pole at $z = 1$ [duplicate]

Let $f(z)$ be a function on the unit disk $\mathbb{D}$ which is meromorphic on $\mathbb{D}$ with only one simple pole at $z = 1/2$ and which is continuous up to $∂\mathbb{D}$ and $|f(z)| ≡ 1$ along ...
1
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2answers
34 views

Does there exist a function $f$ analytic in the unit disc $\mathbb{D} := \{z ∈ \mathbb{C} | |z| < 1\}$

Here is a past qual problem I am having trouble with. Does there exist a function $f$ analytic in the unit disc $\mathbb{D} := \{z ∈ \mathbb{C} | |z| < 1\}$ such that for any $n ∈ \mathbb{N}$ ...
3
votes
1answer
36 views

The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
6
votes
0answers
60 views

How to recognize when a function is secretely holomorphic

Let $f : M \rightarrow N$ be a holomorphic map between complex manifolds (I'd be interested even in the case $M=N=\mathbb{C}$ which should not be much different). Now take $K$ a compact subset of ...
0
votes
3answers
32 views

Complex solutions of polynomial question

$2z^3-6z^2+mz+n = 0$ $m, n$ are real and $1+\sqrt{ 2} i$ is a solution. Find $m$ and $n$. Attempt to solve : Giving the known theorem $1-\sqrt{2}i$ is also a solution, so we can substitute each time ...
0
votes
1answer
14 views

Cauchy's Integral Formula: conditions vs singularities

I'm sure this is a simple misunderstanding but it was annoying me. So using the version of Cauchy's Integral Formula given on Wikipedia http://en.wikipedia.org/wiki/Cauchy's_integral_formula, it is ...
3
votes
2answers
29 views

Harmonic non-surjective functions are constant

Let $u:\mathbb R^2 \to \mathbb R$ be a non-surjective harmonic function. $(i)$ Show that $u$ is bounded from below or from above. $(ii)$ Prove that $u$ is constant (and therefore any harmonic ...
2
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2answers
50 views

If $f'(z_0)\neq 0$ then $f$ has an holomorphic inverse.

Problem: Let $U\subset\mathbb{C}$ be an open set, $f:U\to\mathbb{C}$ an holomorphic function of class $C^1$ and $z_0\in U$. Prove that if $f'(z_0)\neq 0$ then there exists a neighborhood $V$ of $z_0$ ...
2
votes
1answer
37 views

Letting $r \rightarrow 1$ in $\frac{1}{2\pi}\int_{0}^{2\pi}\log |f(re^{i\theta})|\, d\theta$

Suppose $f$ is continuous on $\{z: |z| \leq 1\}$, analytic on $\{z: |z| < 1\}$, and $f(0) \neq 0$. For $0 < r < 1$, consider the integral $$\frac{1}{2\pi}\int_{0}^{2\pi}\log ...
0
votes
1answer
22 views

Check complex differentiability

I am trying to take a derivative w.r.t $z\in\mathbb{C}$ of the following map: $z\mapsto \sum_{j=0}^{\infty}\lambda_{j} (T(\psi+zh))_{j}$ where $(\lambda_{j})$ is a bounded sequence, $T$ is a ...
2
votes
1answer
31 views

Inequations on holomorphic functions

Let $f : \mathbb{C}\setminus\{0\} \to \mathbb{C}$ an holomorphic function such as $$\exists C_0 > 0 / \forall z \in \mathbb{C}\setminus\{0\} \qquad |f(z)| \leqslant C_0\left( |z| + ...
0
votes
0answers
18 views

Divergence of Euler integral for non-positive arguments

Why is it necessary that $\operatorname{Re}(x),\operatorname{Re}(y) > 0$ for the Beta-function $$B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$$ I suppose it is because the integral diverges when ...
-1
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0answers
21 views

calculating residues using residue formula

I am struck with the question: If $z=\alpha_n=ae^{(2n+1)\pi i /4},n=0,1,2,3$ are poles of the function $$f(z)=\frac{z^6}{(z^4+a^4)^2}.$$ If $\alpha$ is either of the poles then the residues of $f(z)$ ...
0
votes
1answer
12 views

If $f$ is analytics in $z_{0}$ ($z_{0}$ is a zero of order m). Show that $1/f$ has a pole of order m at z

I'm solving the exercises of Churchill was unable to resolve this in particular, any help is welcome. 9. If a function $f$ is analytics in $z_{0}$ and $z_{0}$ is a zero of ordem m of $f$, proves ...
2
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0answers
21 views

A question on normal family

Let $f$ be an entire function. Suppose that the sequence $F_0=\{f(0),f'(0),f''(0),\cdots,f^{(k)}(0),\cdots \}$ is bounded. Show that $F_z=\{f(z),f'(z),f''(z),\cdots,f^{(k)}(z),\cdots \}$ is a normal ...
1
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0answers
34 views

Evaluation of Real Integral (with pole) using Complex

For research, I’m attempting to solve this real integral through the use of complex variables. I’m stuck on the approach of the actual integration because the pole lies on the real axis. I started ...
1
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1answer
15 views

Matrix form for conjugate linear transformation

I knew that every linear transformation has matrix representation. I wonder whether I could have a matrix representation for conjugate linear transformation. By conjugate linear transformation, I mean ...
2
votes
2answers
39 views

Question about Riemann Mapping theorem

The Riemann mapping theorem says that given a simply connected region R not all off $\mathbb{C}$ and $z_0 \in R$ then there is a unique conformal map $f: R \rightarrow \mathbb{C}$ such that $f(z_0) = ...
0
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1answer
62 views

function meromorphic on C

Good evening I have a doubt: let $f$ and $g$ are two functions meromorphic on $\mathbb{C}$ such that $g(w) =f(\frac{1}{w})$. Now g is defined for $w = 0$ (because of all meromorphic $\mathbb{C}$).Can ...
0
votes
1answer
18 views

Show that $lim_{N\rightarrow \infty} \int_0^N f(x)dx$ exists and relate the limit $\lim_{N\rightarrow \infty}\int_0^N f(iy)dy.$

Let $f$ be a continuous function on the quadrant $\{z : \Re z \geq 0, \Im z \geq 0\}$ satisfying $f$ is analytic on $\{z : \Re z>0, \Im z>0\}$; $|f(x+iy)|\leq 10e^{−y}$; for each $y>0$ we ...
0
votes
2answers
52 views

Approximation of reciprocal of a complex number

Let $z\in\mathbb{C}$. Is $\frac{1}{1 + z}\approx 1-z$ if $z$ is small? Is there any proof of this?
0
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1answer
19 views

Numerical evaluation of a complex integral

I have to evaluate numerically $f(z)$ via the Cauchy representation (so via a complex integral), in other words, I have to calculare $f(z)$ performing a complex integral: $\dfrac{1}{2\pi ...
1
vote
6answers
124 views

Meaning of $f(z,\bar{z})$

What is the meaning of having $\bar{z}$ in your description of a function, i.e. $f(z,\bar{z})$? The conjugate is simply a function of the complex number, z, so I don't understand why you need to ...
4
votes
0answers
40 views

Holomorphically simply connected implies simply connected

In my book on complex analysis a "Holomorphically simply connected" set is defined as a set where for any holomorphic function $f $ and any closed path $\gamma _1 $ we have that $\int_{\gamma _1 } ...
4
votes
2answers
138 views

Complex integration, any ideas?

I need to solve the following integral of a function analytic on an open environmemt of the unit circle, but i dont know how to get that answers $$\frac{1}{2\pi ...
1
vote
1answer
41 views

Image of a small open disk under $f(z)= \exp (1/z)$

In Conway's Functions of One Complex Variable I, the function $f(z)= \exp (1/z)$ has an essential singularity at $z=0$. But what would the image of a small punctured open disk $\{z:0<|z|< ...
0
votes
1answer
16 views

Prove that a Möbius transformation $T$ sends the imaginary line to the circle $\{z: |z|=2\}$,

Problem Let $T:\overline{\mathbb C} \to \overline{\mathbb C}$ be a Möbius transformation such that $T(1+2i)=1$, $T(-1+2i)=4$ and $|T(0)|=2$. Show that $|T(bi)|=2$ for all $b \in \mathbb R$. The ...
2
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1answer
29 views

Why is $\overline{f(1/\overline{z})}$ holomorphic in $\mathbb{D}^{c}$?

Suppose $f$ is holomorphic in $\mathbb{D} = \{z : |z| < 1\}$. Why is $\overline{f(1/\overline{z})}$ holomorphic in $\mathbb{D}^{c}$?
4
votes
0answers
29 views

Proving a holomorphic function is constant on an open connected set under certain conditions. [duplicate]

Let $\Omega \subset \mathbb C$ be an open and connected set, such that $\overline{D_1} \subset \Omega$. Let $f:\Omega \to \mathbb C$ be a holomorphic function with $f(z) \in \mathbb R$ for all $z$ ...
2
votes
3answers
57 views

Help with equality of complex integrals

I need to prove this equality of integrals...but i dont know how to begin, so if anyone can give an idea... Let f a continuous function on $\overline{D}=\{z : |z|\leq 1\}$. Then: ...
1
vote
2answers
58 views

Complex analysis - existence of field $\mathbb{C}$

In the following: $\mathbb{F}$ is defined to be a field containing $\mathbb{R}$ and in which the equation $x^{2}+1=0$ can be solved. Then define a set $\mathbb{C}$ to be subset of $\mathbb{F}$ whose ...
1
vote
0answers
26 views

Exchanging Limits in Series inversion

I have the Lagrange Bürmann formula as follows: $$\sum^{m-1}_{n=1}\frac{1}{n!}(w-b)^n\lim_{z \rightarrow a} \frac{d^{n-1}}{dz^{n-1}} \left(f^{'}(z)\frac{z-a}{g(z)-g(a)}\right)^n$$ Where g(z) is the ...
2
votes
2answers
34 views

Idempotent entire complex function problem

Problem statement Find all the entire functions $f:\mathbb C \to \mathbb C$, that satisfy $f(f(z))=f(z)$ for all $z \in \mathbb C$. I have no idea how to attack this problem, I would appreciate ...
0
votes
0answers
37 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
2
votes
3answers
61 views

Application of Liouville's theorem exercise

Let $f$ be a holomorphic function such that $Im(f(z))$ is positive for all $z$. Prove that $f$ is constant. Liouville's theorem states that if an entire function is bounded, then it must be constant. ...
2
votes
4answers
65 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
4
votes
0answers
80 views

Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
0
votes
1answer
38 views

Prove that These Families of Level Curves are Orthogonal

From p. 79 in Brown's and Churchill's "Complex Variable and Application": Let the function $f(z) = u(x, y)+iv(x, y)$ be analytic in a domain $D$, and consider the family of level curves $u(x, y) = ...
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vote
1answer
44 views

Inequality on the unit circle

Is $$\left|np(z)+(\alpha-1)zp'(z)\right| \\\geq\left|np(z)+(\alpha-z)p'(z)\right|$$ on $|z|=1,$ and $|\alpha|\geq 1,$ where $p(z)$ is a polynomial of degree $n?$
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vote
2answers
49 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
1
vote
1answer
43 views

Let $f(z)$ be a function on the unit disk $\mathbb{D}$ which is meromorphic on $\mathbb{D}$ with only one simple pole at $z = 1/2$

Let $f(z)$ be a function on the unit disk $\mathbb{D}$ which is meromorphic on $\mathbb{D}$ with only one simple pole at $z = 1/2$ and which is continuous up to $∂\mathbb{D}$ and $|f(z)| ≡ 1$ along ...
3
votes
1answer
42 views

Prove that the function $F ( z ) = \sum_{n = 0}^\infty \frac{c_n z^n}{n!}$ is analytic on the whole $\mathbb{C}$

Let $f : \mathbb{D} → \mathbb{D}$ be an analytic function with Taylor series $f(z) = \sum_{n = 0}^\infty c_nz^n.$ Prove that the function $F ( z ) = \sum_{n = 0}^\infty \frac{c_n z^n}{n!}$ is analytic ...
0
votes
0answers
27 views

Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
1
vote
0answers
47 views

Number of branch points of holomorphic function on torus [closed]

Let $\Lambda\subseteq\mathbb{C}$ be the lattice $\{m+in: m,n\in\mathbb{Z}\}$ and let $X:=\mathbb{C}/\Lambda$ the associated complex torus. Consider the meromorphic function ...
2
votes
1answer
32 views

Lusin's area integral

I was reading "Steven G. Krantz - Handbook of Complex Variables" and came around a complex surface integral called "Lusin's area integral": If $\Omega \subseteq \mathbb{C}$ is a domain and $\varphi: ...
0
votes
0answers
40 views

Method for evaluating $\int_{|z| = 1} \dfrac{z^2}{\sqrt[4]{P(z)}} dz$

I have a problem where I must evaluate $$\int_{|z| = 1} \dfrac{z^2}{\sqrt[4]{P(z)}} dz$$ Where $P(z)$ is a polynomial with degree at least four and has exactly four roots in the unit circle. I know ...
1
vote
1answer
71 views

Why does an automorphism of the disk that is an involution not have fixed points in its boundary?

I'm reading Milnor's Dynamical Systems in One Complex Variable and I'm stuck with a detail in one of his proofs and would appreciate some help! Let $F$ be an automorphism of $\overline{\mathbb{D}}$ ...
0
votes
2answers
61 views

Branches of the complex logarithm

Problem statement Let $\Omega \subset \mathbb C^*$ be open, we call branch of logarithm of $z$ in $\Omega$ to every continuous function $f:\Omega \to \mathbb C$ such that $e^{g(z)}=z$ for all $z \in ...
0
votes
0answers
57 views

Integral $\int^\infty_{-\infty}\int^\infty_{-\infty}(\frac{(x-x_1)^2+(y-y_1)^2}{s_1^2}+1)^{-a_1-1}(\frac{(x-x_2)^2+(y-y_2)^2}{s_2^2}+1)^{-a_2-1}dxdy$

Under $x_i,y_i\in\mathbb R$, $s_i>0$ and $a_i>0$ for $i=1,2$, is there any good function to express the following integral? $$\int^\infty_{-\infty}\int^\infty_{-\infty} ...
1
vote
1answer
43 views

How to show that a complex-valued function is uniformly continuous?

should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that ...