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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129 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) ...
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votes
3answers
4k views

How to find a Laurent Series for this function

How do I give a Laurent Series on various ranges of $|z|$? I need to find the Laurent series expansion for $$f(z)=\frac{1}{z(z-1)(z-2)}$$ for the following ranges of $|z|$: $0<|z|<1$ ...
12
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6answers
1k views

Natural derivation of the complex exponential function?

Bourbaki shows in a very natural way that every continuous group isomorphism of the additive reals to the positive multiplicative reals is determined by its value at $1$, and in fact, that every such ...
10
votes
3answers
384 views

Ring of holomorphic functions

Am I correct or not? I think that a ring of holomorphic functions in one variable is not a UFD, because there are holomorphic functions with an infinite number of $0$'s, and hence it will have an ...
8
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2answers
479 views

A Complex approach to sine integral

this integral: $$\int_0^{+\infty}\frac{\sin x}{x}\text{d}x=\frac{\pi}{2}$$ is very famous and had been discussed in the past days in this forum. and I have learned some elegant way to computer it. ...
7
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1answer
741 views

On the growth of the Jacobi theta function

So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5: Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function $$\theta(z | r) = ...
6
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1answer
325 views

Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial.

My question is in the title. I'm a little lost in how to solve this problem. There is a hint associated with the problem that states the following: Use a line integral to show that $ |f(z)| \leq A + ...
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2answers
773 views

Integration (Fourier transform)

$\mathrm{Re}(i) = 0$, but the fourier transform of $f(x) = e^{-ix^2}$ is $g(\alpha) = \sqrt{\pi\over i}\times e^{i\alpha^2 \over 4}$, is it not? Is there an easy to show that it is so, knowing the ...
5
votes
3answers
675 views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
5
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2answers
645 views

Is a meromorphic function always a ratio of two holomorphic functions?

Suppose $D$ is a region (connected open set) in complex plane, and $f$ is a meromorphic function on $D$. Question: Does there always exist two holomorphic function $g$ and $h$ such that ...
5
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1answer
1k views

Stone-Weierstrass Theorem in $\mathbb{C}$

I am having difficulty understanding how to prove the Stone-Weierstrass Theorem for complex valued functions defined on the closed unit disc $\mathbb{D}\subset\mathbb{C}$. Here is a version I have ...
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votes
2answers
151 views

Integration of exponential and square root function

I need to solve this $$\int_{-\pi}^{\pi} \frac{e^{ixn}}{\sqrt{x^2+a^2}}\,dx,$$ where $i^2=-1$ and $a$ is a constant.
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votes
2answers
887 views

Complex part of a contour integration not using contour integration

A propos of a user's comment on this question, quoting Feynman to the effect that some integrals are only possible using contour integration, I wonder what the simplest example of such an integral ...
3
votes
2answers
87 views

How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
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1answer
335 views

Characterization of Harmonic Functions on the Punctured Disk

The following is an old qual problem I came across. If $h$ is harmonic on $D-\{0\}$, where $D$ is the unit disk, show that $h(z) = \Re(f(z)) + c \log|z|$ for where $f$ is analytic on $D- \{0\}$. ...
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votes
1answer
697 views

Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.

Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$ I've ...
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2answers
600 views

Proving theorem connecting the inverse of a holomorphic function to a contour integral of the function.

I am asked to prove this theorem: If $f:U \rightarrow C$ is holomorphic in $U$ and invertible, $P\in U$ and if $D(P,r)$ is a sufficently small disc about P, then $$f^{-1}(w) = \frac{1}{2\pi i} ...
3
votes
1answer
270 views

Riemann Zeta Function Manipulation

The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$ (i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
3
votes
1answer
409 views

Relation between linearity and injectivity of an entire function

Given $f$ entire function on $\mathbb C$ and $f$ one-one. Is it true that $f$ is linear? At least among polynomials the only such functions are linear!
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votes
2answers
271 views

Branch point-what makes a closed loop around it special?

I am having difficulty understanding the concept of a branch point of a multifunction. It is typically explained as follows:branch point is a point such that the function is discontinuous when going ...
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4answers
119 views

Values for $(1+i)^{2/3}$

This question might seem easier than I'm making it seem. But how many values are there for $(1+i)^{2/3}$? Do I let $z=(1+i)^{2/3}$ so that $z^3=2i$? I'm asked to write each in polar coordinates and in ...
2
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1answer
79 views

Order of growth of $ \prod_{n=1}^{+\infty} (1-e^{-2\pi n}\cdot e^{2\pi i z})$

The order of an entire function $f$ id defined as $$ord ( f) = inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ I have $F(z) = ...
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2answers
222 views

Making a cube root function analytic on $\mathbb{C}\backslash [1,3]$

I am still not convinced by the post that the function$$\sqrt[3]{(z-1)(z-2)(z-3)}$$ can be defined so it is analytic on $\mathbb{C}\backslash [1,3]$. We define for each $z\in \mathbb{C}\backslash ...
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1answer
85 views

What is the value of $\int_C\dfrac{f(z)}{z-z_0}dz?$

Suppose that $f$ is analytic inside and on a simple closed contour $C$,and $z_0$ lies outside $C.$ What is the value of $\int_C\dfrac{f(z)}{z-z_0}dz?$
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1answer
120 views

Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is

Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, the value of $f (i) $ is (A) $ 1 $ (B)$-1$ (C) $i$ (D) $-i$ how can I able to ...
2
votes
3answers
418 views

Analytic function in the punctured plane satisfying $|f(z)| \leq \sqrt{|z|} + \frac{1}{\sqrt{z}}$ is constant

I saw this question on my book (Complex Analysis/Donald & Newman): Let $f(z)$ be an analytic function in the punctured plane $\{ z \mid z \neq 0 \}$ and assume that $|f(z)| \leq \sqrt{|z|} + ...
2
votes
1answer
152 views

First derivative bounded by supremum of difference of values in disc

Need a little help in the following: Let $f(z)$ analytic function on $D = \{z\in\mathbb C: |z| < 1\}$. Define $\displaystyle d = \sup_{z,w \in D} |f(z) - f(w)|$. Prove that $|f'(0)| \leq ...
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2answers
479 views

On the growth order of an entire function $\sum \frac{z^n}{(n!)^a}$

Here $a$ is a real positive number. The result is that $f(z)=\sum_{n=1}^{+\infty} \frac{z^n}{(n!)^a}$ has a growth order $1/a$ (i.e. $\exists A,B\in \mathbb{R}$ such that $|f(z)|\leq ...
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vote
3answers
416 views

Laurent series - $f(z)=\frac{1}{z^2-4}+\frac{1}{6-z}$

We have $f(z)=\frac{1}{z^2-4}+\frac{1}{6-z}$. I want to expand this as a Laurent series in $z_0=2$ on $\{4<|z-2|<\infty\}$. The partial decomposition is: ...
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vote
2answers
132 views

Prove that partial sums of $\sum_{n=1}^{\infty}{z^n}, z \in \mathbb{C}, |z|=1$ are bounded

It's obvious that if $arg(z)=2\pi \cdot q,\ q \in \mathbb{Q},\ q=\frac{a}{b}$ then $\sum_{n=1}^b{z^n}=0$. Thus the partial sums are bounded by $\sum_{n=1}^{b-1}{z^n}$. But what to do with the case of ...
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3answers
289 views

Evaluating this integral for different values of a constant

As you helped me so well last time, I might as well ask a final question! Today I'm trying to prove this: $$ \int_0^\infty \frac{x^{p}}{ 1+x^{2}}dx = \frac{\pi}{2}\cos\left(p\frac{\pi}{2}\right) $$ ...
9
votes
3answers
273 views

Can the the radius of convergence increase due to composition of two power series?

When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of ...
8
votes
2answers
901 views

Can any harmonic function on $\{z:0<|z|<1\}$ be extended to $z=0$?

After looking at this question for quite some time, I've asked a couple of other students, and they also couldn't seem to come up with an answer. This is from an old qualifying exam at our ...
6
votes
2answers
293 views

Prove the open mapping theorem by using maximum modulus principle

The open mapping theorem says a non constant analytic function maps open sets to open sets. The maximum modulus principle says if $f$ a non constant analytic function on an open connected set ...
6
votes
4answers
256 views

Calculate the value of the integral $\int_{0}^{\infty} \frac{\cos 3x}{(x^{2}+a^{2})^{2}} dx$ where $a>0$ is an arbitrary positive number.

Question: Calculate the value of the integral $ \displaystyle \int_{0}^{\infty} \frac{\cos 3x}{(x^{2}+a^{2})^{2}} dx$ where $a>0$ is an arbitrary positive number. Thoughts: I don't know how to ...
6
votes
2answers
3k views

Does the complex conjugate of an integral equal the integral of the conjugate?

Let $f$ be a complex valued function of a complex variable. Does $$ \overline{\int f(z) dz} = \int \overline{f(z)}dz \text{ ?} $$ If $f$ is a function of a real variable, the answer is yes as $$ \int ...
6
votes
2answers
219 views

a question about an infinite series calculation.

I want to prove that for $y >0$, $ x \in \mathbb R$, $$ \sum_{n=-\infty}^\infty \frac{y}{(x+n)^2 + y^2} = \frac{1}{2} \frac{1 - e^{-4 \pi y }}{1 - 2 e^{-2 \pi y} \cos ( 2 \pi x ) + e^{-4 \pi ...
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3answers
2k views

Infinity plus Infinity

Let $a \in \mathbb{C}$. Ahlfors says we let $a + \infty = \infty$ and $a \cdot \infty = \infty$. But we cannot define $\infty + \infty$ without violating the laws of arithimetic (i.e. field axioms). ...
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3answers
364 views

Expressing the area of the image of a holomorphic function by the coefficients of its expansion

I have the following problem. Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
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2answers
1k views

Isolated zeros on closure of a domain

Let $f$ be an analytic function on the open unit disk domain $D$. Suppose also that $f$ is bounded. Since $f$ is bounded I beleive that $f$ can be continuously extended to the closed unit disk. I ...
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2answers
275 views

Ahlfors “Prove the formula of Gauss”

He says: Prove the formula of Gauss: $$ (2\pi)^\frac{n-1}{2} \Gamma(z) = n^{z - \frac{1}{2}}\Gamma(z/n)\Gamma(\frac{z+1}{n})\cdots\Gamma(\frac{z+n-1}{n}) $$ This is an exercise out of Ahlfors. ...
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3answers
129 views

Calculating $\int_{0}^{\infty} x^{a-1} \cos(x) \ \mathrm dx = \Gamma(a) \cos (\pi a/2)$

My goal is to calculate the integral $\int_{0}^{\infty} x^{a-1} \cos(x) dx = \Gamma(a) \cos (\pi a/2)$, where $0<a<1$, and my textbook provides the hint: integrate $z^{a-1} e^{iz}$ around the ...
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3answers
600 views

Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $

can you please guide me on evaluating this integral using residue theorem and binomial theorem $$ \int_0^\pi \sin^{2n}\theta\, d\theta $$ for $n = 1,2,3$ Honestly, I do not even know where to start, ...
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1answer
139 views

Understanding the definition of the order of an entire function

Let $f: \mathbb C \to \mathbb C$ be an entire function. The order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r}, $$ where $$M(r)=\max_{|z|=r} |f(z)| .$$ The ...
5
votes
1answer
257 views

$f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic

Prove that the functions $f(z)$ and $\overline{f(\overline{z})}$ are simultaneously holomorphic. I take this to mean that $f(z)$ is holomorphic if and only if $\overline{f(\overline{z})}$ is ...
5
votes
4answers
393 views

Graphing Compex Functions 3D (x,y,i axes) Instead Of Color-Coded (SAGE).

Following this guide to Sage: and using Sage Online produced the following graphs: Graphing $\frac{1}{1-z}$ that way yeilds: Graphing $\frac{1}{1-z^2}$ that way yields: It would be nice to see ...
5
votes
2answers
1k views

Prove that the zeros of an analytic function are finite and isolated

Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$. The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...
5
votes
3answers
474 views

If $f^2$ and $f^3$ are analytic prove that $f$ is analytic at every point of $\mathbb{C}$. [duplicate]

Let $f : \mathbb{C} \to \mathbb{C}$ be continuous. If $f^2$ and $f^3$ are analytic prove that $f$ is analytic at every point of $\mathbb{C}$. if $f^2$ has no zero then $f=f^3/f^2$ and then it is ...
5
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1answer
512 views

Rational function which is bijective on unit disk

What is the general form of a rational function which is bijective on the unit disk? I'm stuck on this problem. If I let $R(z) = \frac{a_0(z-a_n) \cdots (z-a_1)}{(z-b_m)\cdots (z-b_1)}$, then exactly ...
5
votes
1answer
680 views

injective holomorphic functions

If I have a bounded, connected, open subset of the complex plane, and a function that is holomorphic on it, continuous on its closure, and injective on its boundary, is my function necessarily ...