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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a question on bounds of complex derivative at particular point

Could any one tell me which of the following are correct? $1$. There exists a holomorphic function $f:\mathbb{D}\to\mathbb{D}$ with $f(0)=0$ and $f'(0)=2$ $2.$ There exists a holomorphic function ...
4
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1answer
261 views

How to better understand where the circles and lines go under fractional linear transformations?

Today I encountered the transformation $f(z) = \frac{z}{z-1}$. It has the following property: As the point $z$ makes a counter-clockwise revolution around the unit circle beginning at $1$, the point ...
6
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2answers
304 views

Find the value of $\large i^{i^{.^{.^.}}}$

Find the value of $\large i^{i^{.^{.^.}}}$ ? How should we start to solve it ? Also you can see this one if it helps. Thanks
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1answer
213 views

How to find the bound for this function?

According to journal entitled Certain subclass of starlike function by Gao and Zhou (2007), it was proven that $-\frac{r}{1+tr} \leq Re \{\frac{z}{1-tz} \}\leq \frac{r}{1-tr}$ where $|z|\leq r<1$ ...
3
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1answer
42 views

Analytic continuation of one parameter subgroup: group property preserved?

Let $(\mathcal{A},\alpha)$ be a C* dynamical system, i.e. $\mathcal{A}$ is a unital C*-algebra and $\{\alpha_t\}_{t\in \mathbb{R}}$ a strongly continuous one-parameter group of *-automorphisms. For ...
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1answer
229 views

Modification of Schwarz-Christoffel integral

I found two different formulations of the Schwarz-Christoffel formula (e.g. Link1, p.20 and Link2, p. 9). The first is \begin{align*} ...
5
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1answer
52 views

Analyticity of C*-algebra valued functions

Let $\mathcal{A}$ be a unital C*-algebra and consider a function $f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that $f$ is analytic, i.e. can be locally ...
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1answer
98 views

Strange application of the mean value theorem

I recently came across the following lemma. Lemma. Suppose $\Omega_n^N = [-\pi n, \pi n]^N$ and $f \in L^1(\Omega_n^N)$ is $2\pi n$-periodic. Let $A$ be a subset of $[-\pi n, \pi n]$. There ...
28
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4answers
885 views

Conjecture: Every analytic function on the closed disk is conformally a polynomial.

Here is my conjecture, any proof, counter-example, or intuitions? If $f$ is analytic on $\text{cl}(\mathbb{D})$ (that is, analytic on some open set containing $\text{cl}(\mathbb{D})$), then there is ...
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1answer
363 views

The sum of the residues of a meromorphic function on a Riemann surface

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $ \Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.
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1answer
111 views

questions about a fiber bundle

I am reading the book From holomorphic functions to complex manifolds by Klaus Fritzsche and Hans Grauert. I have a question about a fiber bundle. On page 186, the last line. How to show that $$ ...
5
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2answers
216 views

Primes and the Unit circle.

Consider the "prime spiral" $f(z) = \sqrt{z}\exp(2\pi i \sqrt{z})$, for integer $z$. It has been shown that the intersections of $f$ with some quadratic curves contain a significantly disproportionate ...
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1answer
76 views

Reconstruction a complex-valued function from its real part on the boundary

This is my question. Let $a(z)$ be a holomorphic function on $\mathbb{C}^{+}$, and the value of $|a|$ on $\mathbb{R}$ is known. Now how can one reconstruct $a$ out of the boundary condition? My plan ...
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1answer
335 views

Uniform convergence of sequence to the exponent function

Let $f_n(z) = (1-z^2/n)^n$, and let $f(z)=\operatorname{exp}(-z^2)$. I need to show that $f_n$ converges uniformly to $f$ in any closed disc. I saw this: Uniform Convergence of an Exponential ...
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1answer
48 views

Why is $\overline{\mathbb{C}\setminus\left\{0\right\}}=\overline{\exp(\mathbb{C})}$?

Why is $\overline{\mathbb{C}\setminus\left\{0\right\}}=\overline{\exp(\mathbb{C})}$? I do not know how one can see that...
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0answers
124 views

Problem about normal family

Let $D\subset \mathbb{C}^n$ be a region and $a\in D$. If $F$ is the set of all holomorphisms $f$ on $D$ such that $\mathrm{Re}(f)>0,f(a)=1$, Prove that $F$ is a normal family on $D$. ...
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1answer
157 views

Show that $F$ is normal family

Let $D$ be the open unit disk and $H(D)$ be the set containing all holomorphisms $h:D\rightarrow D$. If $F\subset H(D)$ and there exists $g\in H(D)$ such that for all $f\in F$ and $k\geq 0$, ...
3
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1answer
83 views

Prove that $f(z)=z$ on region $D$ [duplicate]

Given that $D$ be a bounded region containing $0$ and $f:D\rightarrow D$ be a holomorphic mapping such that $f(0)=0,f'(0)=1$ prove that f(z)=z for all $z$ in $D$ This problem reminded me of the ...
2
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0answers
32 views

Find elliptic function $f$ satisfying the condition

Find an elliptic function $f$ with period lattice $\Omega=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ such that $\deg(f)=2$ and $f$ has $2$ poles of order $1$, then express $f$ in term of the Weierstrass ...
4
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1answer
68 views

Does This Condition Characterize $e^z$?

The following is a question from a Complex Analysis qualifying exam I was studying from: Does there exist an entire function $f$, distinct from $e^z$, such that $f(0)=1$ and $f'(n)=f(n)$ for all ...
4
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2answers
116 views

Prove there are infinitely many $z$ satisfying $|f(z)|=|g(z)|$

Let both $f$ and $g$ be non-constant holomorphic functions on $\mathbb C$ and $f(0)=0,g(0)=1$ Prove that there are infinitely many $z$ satisfying $|f(z)|=|g(z)|$. Is there any other theorem like ...
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2answers
232 views

Laurent series expansion of a function

The question is : Show that $$e^{{c\over 2}(z-{i \over z})}={\sum_{n=-\infty}^\infty} a_nz^n$$ The question solved using Laurent's Series expansion where $$a_n={1\over 2\pi i}\int_cf(z){dz \over ...
5
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1answer
212 views

Good texts in Complex numbers?

I have asked some members on chat about good text to study complex numbers , they recommended for example , "Visual Complex Analysis" by Needham and "complex analysis" by Steins. But, I look for a ...
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0answers
71 views

Infinite product of an analytic function on the right half plane

I want to know whether $G_t=\prod_{i=1}^\infty G(s)^i$ is defined and analytic on the right half plane for an analytic $G(s)$ on the right half plane. If so, is $L^{-1}\left(G_t\frac{1}{s}\right)$ ...
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2answers
211 views

Holomorphic functions with polynomial real part

$f:\mathbb{C}\rightarrow \mathbb{C}$, $f(x+iy)=u(x,y)+iv(x,y)$ is a holomorphic function, its real part $u$ is a harmonic polynomial, i.e. $u\in \mathbb{R}[x,y]$ and $\frac{\partial^2 u}{\partial ...
2
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1answer
329 views

Mean value property for a harmonic function

This is an exercise from Ahlfors' Complex Analysis text. I need to show that the mean value property holds for the function $u=\log|1+z|$ in the circle with center $z_0=0$ and radius $r=1$. The ...
2
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2answers
94 views

An apperent contradiction to 0's of holomorphic function are isolated!

I have a few questions about functions of this form: $\displaystyle f(z) =\sum _{n=0}^{\infty} \frac{a_n}{Z-b_n} $ If I put $a_n = 2^{-n}$, $b_0 = 0$, and $b_n =1/n$ for $n>0$, then we get a ...
35
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10answers
2k views

What's the difference between $\mathbb{R}^2$ and the complex plane?

I haven't taken any complex analysis course yet, but now I have this question that relates to it. Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and ...
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1answer
132 views

Complex analysis Laurent series evaluated on unit circle

Let $f(z)$ be a function analytic on an annulus that includes the unit circle $z=e^{i\theta}$. By taking that circle as the path of integration for the coefficients in the Laurent series, show that $$ ...
4
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1answer
285 views

Residues at poles

What is the residue of $$f(x)=\frac{1}{(x^2+1)^a}$$ at $x^2=\pm i$, where $0<a<1$ ? My intuition tells me that there must be a non-zero residue, but my attempts to compute tells me the residue ...
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0answers
541 views

difficulties in proving Rouche's theorem

This problem came from Schaum's complex variable p.156 Suppose $f(x)$ and $(g)$ are analytic inside and on a simple closed curve $C$ and suppose $|g(z)|\lt |f(z)|$ on $C$. Then $f(z)+g(z)$ and ...
2
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1answer
201 views

Show $\lvert f\rvert\leq\lvert g\rvert\Rightarrow f=cg$ [duplicate]

Let $f$ and $g$ be entire functions with $\lvert f\rvert\leq\lvert g\rvert$. Show that then $$ f=cg $$ for a constant $c\in\mathbb{C}$. Can I just do it like this: $\lvert ...
3
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1answer
37 views

Extension of family of operators

Let $A(z)$ where $z\in \mathbb{R}$ be a family of (bounded) operators on some Hilbert space. Assume we know these operators have a meromorphic extension to all of $\mathbb{C}$. Assume moreover that we ...
3
votes
1answer
613 views

Solving a problem using Cauchy's residue theorem, is there more to it?

Let $z_1,...z_n$ be distinct complex numbers. Let $C$ be a circle around $z_1$ such that no other $z_j$ is in $C$ for $j>1$. Let $$f(z) = (z-z_1)(z-z_2)...(z-z_n)$$ Find ...
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2answers
167 views

Cauchy integral formula for meromorphic functions

Let $D$ be closed unit disk in $\Bbb C $, and $C$ be the unit circle. Let $a\in\text{interior}(D)$. For any continuous function $f : D -> \Bbb C$ We define a function $\displaystyle ...
4
votes
1answer
431 views

Dirac Orthonormality Proof - Can't Make Sense of Complex Integral

I'm having trouble rationalizing a particular statement that is, surely, present in many quantum mechanics textbooks. The following statement comes from the orthnormalization condition for ...
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1answer
47 views

Proving a statmenet about convergence of complex sequence [duplicate]

Let $x_k \in \mathbb C$ for $k \in \mathbb N \cup {0}$ and let $y_k = \frac{(x_0 + x_1 + ... + x_k)}{k+1}$. We want to prove that if $x_k$ converges to $x$ ($x \in \mathbb C$) as $k \rightarrow ...
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0answers
117 views

Use Möbius Transformation Normal Form to prove Lambda

I'm just completely lost on how to answer this question: Let $$\frac{Tz-p}{Tz-q}=\lambda \frac{z-p}{z-q}$$ be the normal form of a Möbius transformation with two fixed points. Prove that $\lambda$ = ...
2
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1answer
65 views

What does $f \in H^\infty$ mean?

I am reading this research paper about polynomials with non-negative coefficients. Can some one tell what does the notation $f \in H^\infty$ mean so that I can research about this function class?
11
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2answers
281 views

Intuition Behind an Identity

I'm currently studying for a complex analysis prelim. exam in August, so I'm working through some of the exercises in Titchmarsh. One of the exercises has us evaluate the integrals ...
3
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2answers
151 views

Calculating $\int\frac{e^{iz}}{z}\, dz$ on the semi-circle given by $re^{i\theta}$ where $\theta:\,0\to\pi$

As a part of an exercise I need to calculate $$ \lim_{r\to0}\int_{\sigma_{r}}\frac{e^{iz}}{z}\, dz $$ Where $$ \sigma_{r}:\,[0,\pi]\to\mathbb{C} $$ $$ \sigma_{r}(t)=re^{it} $$ I know ...
4
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1answer
113 views

Holomorphic function zeros on the circle

I'm learning to use some methods of complex analysis, solving some problems. Could you give me a hint to solve the following problem? $f$ is holomorphic in $D^2=\{z: |z|<1\}$ and continious in ...
17
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5answers
1k views

Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$

In old popular science magazine for school students I've seen problem Prove that $\quad $ $\dfrac{1}{\cos^2 20^\circ} + \dfrac{1}{\cos^2 40^\circ} + \dfrac{1}{\cos^2 60^\circ} + ...
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0answers
56 views

Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...
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2answers
1k views

What is a harmonic complex function?

So, as far as I have learned, a complex function $f(u)$ is considered harmonic if and only if it satisfies the undermentioned equation: $$ \frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 ...
5
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2answers
45 views

$f(z)$ and $f(z+z^2)$ have the same singularities at 0

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an analytic function in a punctured neighborhood of 0 then $f(z)$ and $h(z)=f(z+z^2)$ have the same singularity at $z_0=0$. I was able to show that every ...
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1answer
84 views

Contour Integration of Complex Numbers — question 1

I make up my own questions and contours and solve the integration problem without knowing whether the answer is right or not. Please verify the below question for me: $f(z) = z^2 + e^z$ and ...
1
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1answer
488 views

Complex Analysis: Fundamental Theorem of Calculus

I was given the following question on a complex analysis exam but didn't answer it correctly: Evaluate $$\int \sqrt{z-1} dz$$ about the the unit disk $|z|=1$ using only the Fundamental Theorem of ...
0
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1answer
65 views

Complex Analysis Proof2 [duplicate]

Let $f(z)$ be an entire function satisfying $|f(z)|\leq k|z|$ for some positive constants $k$ and all $z$. Show that $f(z)=az^2$ for some constant $a$.
3
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2answers
388 views

To show an analytic function is one-to-one on the unit disk

Let $\displaystyle f(z) = \sum_{n=0}^\infty a_nz^n$ be analytic in the unit disk $D_1(0)$ with $f(0) = 0$ and $f'(0) = 1$. Prove that if $\displaystyle \sum_{n=2}^\infty n|a_n| \le 1$, then $f$ is ...