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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82 views

Determine all the analytic functions on $\Bbb{C}$ such that $f(z)=f(-z)$; $\forall $ $z$ in $ \Bbb{C}$.

find all the analytic functions on $\Bbb{C}$ such that $f(z)=f(-z)$; $\forall $ $z$ in $ \Bbb{C}$.
0
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1answer
61 views

Maclaurin expansions

Let $f(z)=\sum_{j=0}^{\infty}a_jz^{j}$ be the Maclaurin expansion of a function $f(z)$ analytic at the origin. Prove each of the following: a.) $\sum^{\infty}_{j=0}a_jz^{2j}$ is the ...
1
vote
1answer
58 views

Elementary converge and diverge

Why does the geometric series $\sum^{\infty}_{j=0}c^j$ converge when $|c|<1$, but diverge when $|c|\ge 1$? Since the geometric series is $= \frac{1}{1-c}$, which means it is undefined at $c=1$, but ...
6
votes
2answers
254 views

A ‘strong’ form of the Fundamental Theorem of Algebra

Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial $$ p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in ...
0
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1answer
157 views

Let $f(z)$ be an analytic function on a open connected subset $\Bbb{G}$ of $\Bbb{C}$ with $|f(z)|= z_{0}$ for some fixed $z_{0}$ [duplicate]

Let $f(z)$ be an analytic function on a open connected subset $\Bbb{G}$ of $\Bbb{C}$ with $|f(z)|= z_{0}$ for some fixed $z_{0}$ then prove that $f(z)$ is a constant function.
5
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0answers
167 views

Is there a closed form expression for this integral?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
1
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1answer
242 views

Real part of holomorphic function cannot have a maximum

I am trying to prove that if $f$ is a holomorphic function from a domain $U$ to $\mathbb{C}$, and the real part has an interior local maximum at a point $a$ in $U$, then $f$ is a constant. I am new to ...
2
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1answer
290 views

A complex function that is harmonic but not analytic

Can someone give a simple example of a complex function that is harmonic but not analytic? Thanks. D.
2
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1answer
261 views

Can we prove the open mapping theorem using the maximum modulus principle?

Can we prove the open mapping theorem using Maximum Modulus Principle? I myself can prove the other way.
1
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1answer
51 views

$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq \int_{-\infty}^{\infty}|f|^2 dx$?

Suppose complex function $f$ in the Schwartz Space, its definition see http://en.wikipedia.org/wiki/Schwartz_space how can we argue that $$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq ...
0
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1answer
77 views

A function not in $L^2(\mathbb{R}^3)$

From this equation $$ (p^2-\alpha)\hat{f}(p)=\frac{e^{-ip\cdot y}}{p^2+\lambda}$$ where $\hat{f}$ is the Fourier transform, $\alpha,\lambda>0$ e $y$ a fixed point in $\mathbb{R}^3$ can I conclude ...
2
votes
1answer
95 views

A Growth Inequality on $\mathbb{C}$-Polynomials

For which class of polynomials over $\mathbb{C}^{n}$ does the following growth inequality hold? For any multi-index $\alpha$, there are positive constants $A, B, C, D < \infty$ such that ...
4
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0answers
244 views

Möbius transformations form a simple group

How to show the group $M$ of Möbius transformations is a simple group? I know: $SL_2(\mathbb C)/\{+I,-I\}\cong M$ then if $A \lhd M \implies \phi^{-1}(A) \lhd SL_2(\mathbb C)/\{+I,-I\}$. So if ...
1
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1answer
180 views

Harmonic functions proofs

a.) Show that if $v(x,y)$ is a harmonic conjugate of $u(x,y)$ in a domain $D$, then every harmonic conjugate of $u(x,y)$ in $D$ must be of the form $v(x,y)+a$, where $a$ is a real constant. ...
3
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1answer
449 views

Integration of Complex Logarithm

I want to prove that $$\int_0^{2\pi}\log|1-ae^{i\theta}|\, d\theta=0$$ for all $|a|\leq 1$. I can prove it easily for $|a|<1$ via power series expansion for $\log|1+(-a)e^{i\theta}|$ and then ...
3
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1answer
57 views

Question about an identity

Is it true that: $$ \coth^{-1}(z) = \tanh^{-1}\left(\frac{1}{z}\right), z\in \mathbb{C} $$ I used this identity: $$ \coth{z} = \dfrac{-1}{\tanh{z}} $$ To obtain such a result.
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1answer
311 views

Derivatives of univalent functions must converge to derivative of univalent function?

This is probably something basic that I am missing. I am reading the article Normal Families: New Perspectives by Lawrence Zalcman, and in one of his examples he makes the following assertion (I am ...
0
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2answers
277 views

Möbius transformations on $D$ such that $f(D)=D$

I need to find all Möbius transformations on unit disk such that $f(D)=D$, please help!
18
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5answers
895 views

Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$

I need to show that $$ \int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3} $$ I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result ...
4
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2answers
193 views

Find an analytic bijection function ${f(z)}$ on $\Bbb{C}$ such that there exist only one $z_{0}$ such that ${f(z_{0})} = z_{0}$.

Find an analytic one-one onto function ${f(z)}$ on $\Bbb{C}$ such that there exist only one $z_{0}$ such that ${f(z_{0})} = z_{0}$.
2
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2answers
248 views

If $f(z)^3$ is analytic then $f(z)$ is analytic.Is it true?if yes prove it.otherwise give counterexample.

If $f(z)^3$ is analytic on $\Bbb{C}$ then $f(z)$ is analytic.Is it true?if yes prove it.otherwise give counterexample.
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1answer
211 views

Wirtinger derivatives $\mathbb{C}$-linear

May i ask you for a little help about the following problem: Prove that the Wirtinger derivatives $\frac{\partial }{\partial z}$ and $\frac{\partial }{\partial \bar{z}}$ are: 1) ...
1
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0answers
164 views

few short question about branch-cut and branch points

Q. Find branch points and construct branch lines for the functions $\displaystyle (a) f(z) = \left( \frac{z}{1-z} \right)^{\frac 1 2 } $ $\displaystyle (b) \left( z^2 - 4\right)^{\frac 1 3 } ...
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1answer
27 views

$(y^2+2iy)^{m-\frac{1}{2}}=y^{m-\frac{1}{2}}(2i)^{m-\frac{1}{2}}+O(y^{m+\frac{1}{2}})(0\leq y \leq 1)?$

This question comes from stein's book Introduction to fourier analysis on euclidean space,page 159. when $m > 1/2$, ...
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1answer
135 views

There is no function $f$ on the open unit disk, defined by a convergent power series, such that $f(1/n)=(-1)^n/n^2$

Prove that there is no function $f$ on the open unit disk, defined by a convergent power series, such that $f(1/n)=(-1)^n/n^2$. I'm not sure how to start... any hints would be appreciated!
3
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1answer
468 views

how to calculate gamma function in programming language? [duplicate]

how to calculate gamma function in programming language? I also need to support complex numbers (complex numbers are in the programming language) and negative numbers. thank you :)
5
votes
1answer
439 views

A question about normal families [duplicate]

This is Ahlfors q. 1, p. 227. Prove that in any region the family of analytic functions with positive real part is normal. Under what added condition is it locally bounded? Hint: Consider $e^{-f}$. ...
4
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1answer
87 views

branch of logarithm of $z^2$

Here's one of the recommended exercises from our complex analysis class. Prove or disprove: There is no analytic $f$ on $\mathbb{C} \setminus 0$ such that $exp(f(z)) = z^2$ for all nonzero $z \in ...
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1answer
36 views

What's wrong with this computation?

that's an idiot question, but I have to ask. If $f: \Omega \subset \mathbb{C} \longrightarrow \mathbb{C}$, and $f(z(x, y)) = x² + y² +2xyi $, then $f'(z) = 2z$ by computing with the Cauchy-Riemann ...
1
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1answer
232 views

Uniform Convergence: Complex Analysis

To show that $f_k(z) = \frac{z^k}{k}$ converges uniformly for $|z| < 1 $ and that $f'_k(z)$ does not converge uniformly for $|z| < 1$, what must be done? What other things can be said about the ...
1
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2answers
74 views

Is this Analytic functions

Why is the function not analytic in the complex plane? I believe it is analytic on real plane. $e^{(-\frac{1}{z^2})}$ where $z\in\mathbb{C}$. Well a complex function should be infinitely ...
3
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1answer
404 views

What is the Lebesgue measure on $\mathbb{T}$?

This is a rather basic question in complex analysis. Let $\mathbb{T}$ be the complex unit circle. Now, I am trying to understand what is the Lebesgue probability measure on $\mathbb{T}$. To my ...
3
votes
2answers
129 views

Entire function invariant under different rotations is constant?

How can we reason to show that an entire function that's invariant under two rotations of the plane, must be constant ? Assume the rotations are around different axes, and by rational multiples of ...
1
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1answer
52 views

Clarity on the Notion of Branch: A Visual Consideration

Below is an excerpt from a question in Gamelin's complex analysis textbook: "Let $D = \mathbb{C}$ \ $(-\infty, 1]$, and consider the branch of $\sqrt{z^2 - 1}$ on $D$ that is positive on the ...
4
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1answer
153 views

Finding a trigonometric polynomial

I'm trying to solve exercise 5 in chapter 14 of Rudin's Real & Complex Analysis: Suppose $f$ is a trigonometric polynomial, $$f(\theta) = \sum_{k=-n}^n a_k e^{ik\theta}$$ and $f(\theta) ...
1
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0answers
112 views

Image of unit disc of holomorphic one-to-one function

In Real & Complex Analysis, exercise 7 chapter 14, we are asked to find $a$ so that $$f_a(z) = \frac{z}{1+az^2}$$ is one-to-one on the unit disc $U$ and then describe $f_a(U)$. I easily solved ...
2
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1answer
83 views

Local integrability of a Cauchy transform in the plane

Let $\mu$ be a Radon measure in the plane, typically with support included in a small neighborhood of the origin. Let $h(z)=\int \frac{d\mu(y)}{z-y}$. I am wondering when it can be said that ...
4
votes
2answers
730 views

condition for roots of quartic equation to be purely imaginary

(a) Show that the roots of equation $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ where $a_1, a_2, a_3, a_4$ are real constants different from zero, has a pure imaginary root if $a_3^2 + a_1^2 a_4 = ...
1
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1answer
43 views

Proving $\arg(a)\equiv \alpha\Longleftrightarrow z_1z_2 \in \mathbb{R}$

Suppose the complex equation $iz^2+(2-i)az-(1+i)a^2=0$ as $a\in \mathbb{C}^{*}$. $z_1$ and $z_2$ are the solution of this equation and we have also $z_1*z_2 = a^2(i-1)$. How can I prove that ...
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0answers
65 views

$C^{\infty}$ 1-form on a Riemann surface is unique.

Let $X$ be a Riemann surface and $\mathcal{A}$ be a complex atlas on $X$. Suppose that $C^{\infty}$ 1-forms are given for each chart of $\mathcal{A}$, which transform to each other on their common ...
3
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2answers
811 views

contour integration and branch points

An exercise in a textbook says to evaluate $\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos (ax) \cos^{b} (x) \ d x \ (a > b > -1)$ by letting $\displaystyle f(z) = z^{a-1} ...
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0answers
43 views

Can you come up with any more examples?

I want to look at the set of functions $\{f : \mathbb{C} \rightarrow \mathbb{C}\}$ such that proving $f(s) = 0 \implies s+s^* = 1 $ ($\Re(s) = 1/2$) is easy. Here are two examples: $$ f(s) = s - 1/2 ...
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1answer
152 views

Factoring $x^{n} - y^{m}$ as a Product involving Roots of Unity

The following relation is well-known: For odd $n$, \begin{align} x^{n} - y^{n} = \prod_{k = 0}^{n-1} ( \zeta_n^{k} x - \zeta_{n}^{-k} y) \qquad \zeta_n =e^{2\pi i/n}. \end{align} Is there a similar ...
3
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1answer
291 views

show that Joukowski transform is one-to-one in the upper half outside the unit disk

I have a problem on showing how the Joukowski transform $w=J(z) = .5(z + 1/z)$ takes the upper half plane, $|z| \gt 1$, one-to-one into the w upper half plane. I have shown how the unit disk itself ...
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3answers
44 views

How can I find $\lim_{|z|\to 0}|\frac{1-e^{2iz}}{z}|$?

In calculation of a contour integral, I need to find $$\lim_{|z|\to 0}\bigg|\frac{1-e^{2iz}}{z}\bigg|.$$ Let $z=re^{i\theta}$. Then $$ \lim_{|z|\to ...
6
votes
2answers
494 views

How many ways to calculate: $\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}$ where $u \not \in \Bbb{Z}$

Today I have encounter a series: $$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$ where $u \not \in \Bbb{Z}$ . I have known a method to computer it (by Residue formula): ...
13
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1answer
1k views

What is the intuition behind the Wirtinger derivatives?

The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I ...
1
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1answer
329 views

Laurent expansion with essential singularity

I am doing a multiple choice test for complex analysis, and I am stuck a bit at the following one. Let $f$ be holomorphic with an essential singularity at $0$. Then for every $z_0\in \mathbb{C}$ the ...
0
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1answer
194 views

Is $\text{arc2sinh}(\dots(\exp(2\sinh(\dots z))$ an entire function?

Let $^{*n}$ denote the $n$-th iteration and $z$ be a complex number. Let $n$ be a positive integer. Let $2\sinh(z)$ be $\exp(z)-\exp(-z)$ and $\text{arc2sinh}$ its functional inverse. Is the limit for ...
4
votes
1answer
887 views

How to prove Schwarz reflection principle?

Suppose that $f$ is non-vanishing and continuous on a closed unit disk that is holomorphic in the interior $D$. Show that if $\lvert f(z) \rvert = 1$ whenever $\lvert z \rvert = 1$ then $f$ is ...