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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1answer
86 views

A basis for $\mathbb{C}^n$

What I want to prove: Suppose $\lambda \in (-\pi,\pi]$ are natural frequencies at time $n$. Then for every $\lambda_j$ define a vector $e_j^n = \frac{1}{\sqrt{n}} \left(e^{i \lambda_j},e^{2i\lambda_j},...
4
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1answer
217 views

what is a Möbius transformation with no fixed points?

is there a Mobius transformation with no fixed points? I have the equation $$\frac{az+b}{cz+d}=z\implies cz^2+dz-az-b=0$$ given the fixed points when this is true. So if we set $c\ne 0$ we get two ...
-1
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2answers
69 views

Maximum modulus principle

Is maximum modulus principle obeyed by antiholomorphic functions. I think it should be as they are the functions of z conjugate alone and not z .
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2answers
30 views

Triangle Inequality for complex functions

If $z$ is st. $z=Re^{i\theta}, R > \sqrt2$ how do I show that: |$z^2+2z+2$|$\geq R^2-2R-2$? If I use the traingle inequality I get: |$z^2+2z+2$|$\geq||z^2+2z|-2|$ but I can't proceed further
5
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1answer
198 views

sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$

Does there exist a sequence $a_n$ of complex numbers such that $\sum _{i = 0}^\infty a_n$ converges and the product $\prod _{i = 0}^\infty (1+a_n)$ does not converge to any complex number(not even 0)....
5
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2answers
326 views

Convergence of $\prod_{n=1}^\infty(1+a_n)$

The question is motivated by the following exercise in complex analysis: Let $\{a_n\}\subset{\Bbb C}$ such that $a_n\neq-1$ for all $n$. Show that if $\sum_{n=1}^\infty |a_n|^2$ converges, then ...
0
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0answers
62 views

Define probability measure on the space of multivalued function

Suppose I have a collection of multivalued functions $f:[0,2\pi]\rightarrow\mathbb{R^{3+}}$. It is also known that this space is a vector space. We define distance ...
3
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1answer
152 views

Proving $\int_\mathbb R\frac{\sin(x)}{x}dx = \pi$

I've been searching the web for a way to prove that $\int^{\infty}_{-\infty}{\sin(x)/x} = \pi$ with complex analysis, because I have a problem of consistency. I found two, carried in the following ...
1
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1answer
41 views

Limit as $r$ tends to zero of integral $\int_C \frac{e^{iz}-1}z \mathrm dz$

Let $\mathcal C$ be a semi-circle of center $O(0,0)$ and radius $R$, such that $y \ge 0$. Find the limit as $R$ tends to zero of: $$\int_{\mathcal C} \frac{e^{iz}-1}z \mathrm dz$$ How can I find ...
2
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2answers
88 views

How to do this integral.

I need to do this: $$\int_0^\infty e^{ikt}\sin(pt)dt$$ I already have the solution, I'm just clueless on how to actually calculate it. I've tried several changes,integration by parts and extending ...
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1answer
29 views

How to find the map of an inversion

I want to find the image of the unit disk $D:= \{z: |z|\lt1 \}$ under the Möbius transform. $$f(z)=\dfrac{iz+3}{iz-1}=1+ \dfrac{4}{iz-1}$$ Now, $f$ can be decomposed into $f= f_5 \circ f_4 \circ f_3 ...
8
votes
2answers
56 views

Lower boundary for $ |f(z) - 1/z| $, where $ f(z) $ is holomorphic

I've been trying to prove the following statement: Let $ f:U \rightarrow \mathbb{C} $ be holomorphic with $ \overline{B(0, R)} \subset U$. Suppose $ r < R $. Prove that $$ \sup\limits_{r \leq |z| ...
2
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1answer
39 views

When the singular inner part disappear in inner outer factorization?

I saw this remark in Hoffman's book - "Banach space of analytic function". If $f$ is analytic in a neighborhood of $\bar{\mathbb{D}}$, the closure of $\mathbb{D}$; then in the inner-outer ...
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2answers
55 views

let $f$ be holomorphic and bounded on $\mathbb{C}\backslash K$ with $K = \{0\}\cup\{\frac11, \frac12, \cdots\}$ and further $f(2) = 1$. Find f.

let $f$ be holomorphic and bounded on $\mathbb{C}\backslash K$ with $K = \{0\}\cup\{\frac11, \frac12, \cdots\}$ and further $f(2) = 1$. Determine such f. What we tried Since f is bounded, points ...
1
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1answer
46 views

Holomorphic maps between smooth algebraic curves

I am looking for a reference for the following statement: Let $X$ be a smooth projective curve over $\mathbb{C}$. Every holomorphic function $f: X \to \mathbb{P}^1_{\mathbb{C}}$ is in fact a morphism ...
0
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2answers
33 views

A simpler way of finding all points where $g(z)=(z-3)(\overline z-3)e^{z^2}$ is holomorphic?

A simpler way of finding all points where \begin{equation*} g(z)=(z-3)(\overline z-3)e^{z^2} \end{equation*} is holomorphic? I tried the way of setting $z=x+iy$, in order to use the Cauchy-Riemann ...
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1answer
32 views

Limit of integral of a complex function [closed]

Let $C$ be a semi-circle of center $O=(0,0)$ and radius $R$, $y\geq 0$ , find the limit as $R$ tends to infinity of : $\int_C e^{iz}/z^2 dz $
1
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1answer
54 views

What on earth is $ \mathcal{O} (\mathbb{C})$?

What on earth is $ \mathcal{O} (\mathbb{C})$? I used this in a project I hastily put together and now I cannot remember the definition of it. I checked in as many books as I still had with me and this ...
2
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1answer
172 views

Non-conformal Schwarz-Christoffel integral

Using "conformal" to mean a holomorphic bijection, the Riemann Mapping theorem guarantees the existence of a conformal map from the upper half-plane $\mathbb{H}=\{z=x+iy\in\mathbb{C}:y>0\}$ to the ...
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1answer
142 views

What is a Bi-Analytic function

I want to know what the definition of a Bi-analytic function is. I have tried looking it up online, but all I am able to find are research papers/books on the theory of bi-analytic functions. Can ...
4
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1answer
59 views

Why does the Residue Theorem still hold, when I let my contour get infinitely large?

The theorem (as I know it) only allows for a finite set of isolated singularities. I integrated, along a square box, a function that has simple poles at all the non-zero integers -- and a triple pole ...
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2answers
83 views

Inequality derived Schwarz's Lemma

Let $f$ be a holomorphic function on $U(0,R)$ with $0< R.$ Assume there exists an $M > 0$ such that $|f(z)| \leq M$ for $z \in U(0,R)$ and an $n \in \mathbb{Z}_{\geq 0}$ satisfying $$0 = f(0) = ...
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0answers
19 views

Complex valued smooth fuction and lateral limits

Let $f:(t_0-\varepsilon,t_0+\varepsilon)\backslash\{t_0\}\to\mathbb{C}^*$, $f\in C^{\infty}((t_0-\varepsilon,t_0+\varepsilon)\backslash\{t_0\})$ and $|f(t)|=1, \forall t\in (t_0-\varepsilon,t_0+\...
1
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1answer
38 views

Multivariate Complex Function

Suppose $f(x,w)\not=0$ for all $x,w\in H^+\cup H^-$ (open upper and lower half planes) and $f$ is a multivariate entire function. Must there exist univariate entire functions $\phi_1$ and $\phi_2$ ...
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1answer
34 views

Prove that the line integral on $\beta$ of $f'(z)/f(z) = (A-B)/2 \pi i$ using Rouche's Theorem

Suppose that $\alpha$ is a regular closed contour. $f$, our function, lacks zeros and poles on $\beta$ and if A=the number of zeros of f inside $\beta$ (a zero of order n is counted n times) and B= ...
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0answers
48 views

Harmonic Functions on Connected Open Set

From the maximum principle, any unbounded harmonic function $u : \Omega\rightarrow \mathbb R$ on a connected open set must be surjective. If $\Omega$ is bounded, does there always exist such a $u$?
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1answer
71 views

Prove that the integral of $sin(z)/(z^2+4z+5)$ from negative to positive infinity is $-\pi sin(2)/e$

I think I've made the problem a lot nastier than it supposed to look. Here's what I have so far. First notice that $(z^2+4z+5)$ is equivalent to $(z^2+4z+4)+1$ so our singularities are -2-i and -2+...
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1answer
40 views

Expand $(e^{2x}-1-2x)/x^5$ into Laurent Series on 0<|x|<$\infty$ and classify its singularity

I guess I'm having difficulty with this because its not in the form of a polynomial expression, which is what I've been taught. Nevertheless here's what I did: I know that the expansion for $e^{2x}=\...
3
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2answers
43 views

Not understanding solution to $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz$ computation

Not understanding solution to $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz$ computation. What was shown in class: $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz=\large \int_{|z-2|=2} \frac {5z+...
3
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2answers
83 views

Find the Laurent series about $z=i$

Let $g(z)=\cfrac{3z+1}{(z-i)^3}$. Find the Laurent expansion of $g$ about $z=i$. My idea is first to find the Laurent series of $\cfrac{1}{z-i}$ about $z=i$, and then diferenciate, but I have problem ...
0
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1answer
97 views

Complex analysis exercise - boundary points of nonconstant analytic functions.

The exercise has two parts: a) Suppose $f$ is nonconstant and analytic on $S$ and $f(S)=T$. Show that if $f(z)$ is a boundary point of $T$, $z$ is a boundary point of $S$. b) Let $f(z)=z^2$ on the set ...
3
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1answer
60 views

Is this type of smooth function analytic?

Let $f:(a,b)\to\mathbb{C}$ be a $C^{\infty}$ application, such that for any $t_0\in (a,b)$ there is $g:(a,b)\to\mathbb{C}$, $g\in C^{\infty}((a,b)),\ g(t_0)\neq 0$ and $m\in\mathbb{N}^*$ such that: $f(...
2
votes
1answer
45 views

Question about finding Laurent Series over closed region and classifying singularity

Represent $\sin(\pi x/(x+1))$ Laurent Series about the region $0<|x+1|<2$: Its true that $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ So the $$\sin(\pi x/(1+x))=\sum (-1)^{n-1} \frac{(\pi ...
1
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2answers
78 views

Analytic Functions, Cauchys Integral Formula

Let $f: \mathbb D \to \mathbb D$ be analytic or holomorphic with $f(0)=\frac{1}{2}$ and $f(\frac{1}{2}) = 0$ where $\mathbb {D} = \{ z: |z| \leq 1\}$. Then find $|f^{'}(0)|$ and $|f^{'}(\frac{1}{2})|$....
3
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1answer
152 views

Prove that the integral of $x\cos(x)/(x-2)(x-1)$ from negative to positive infinity is $\pi(\sin1-2\sin2)$. Use an indented contour

To do this I used the Residue Thm but the main issue here is that I cannot get the sine term to appear. Perhaps I'm ignoring something here. We know that the singularity is $x=1,2$ so we should just ...
0
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1answer
29 views

If $p>0$ demonstrate that the $1/2\pi i$ the line integral of $z^p f'(z)/f(z)$ is $\sum (z_k)^p$

This is basically a deviation of Rouche's Theorem from what I can tell. My first instinct was to do this via induction in which we know that $p=0$ we would have Rouche's theorem. But it gets ...
3
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3answers
326 views

How to know if an integral is well defined regardless of path taken.

I can calculate \begin{equation*} \int_0^i ze^{z^2} dz=\frac{1}{2e}-\frac12, \end{equation*} but why can I calculate this irrelevant to the path taken? Is this since it is analytic everywhere - if ...
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0answers
36 views

Integral of function has different values depending on contour?

What can I say if the integral of my function has different values depending on contour? If my function were analytic on a domain it would evaluate to $0$ right? My contours give values like $\pi i,-\...
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3answers
119 views

If $f$ is an entire function, and $\Im(f)$ is bounded below then show $f$ is a constant function

If $f$ is an entire function, and $\Im(f)$ is bounded below then show $f$ is a constant function. The solutions write: suppose $\Im(f(z)) \geq m$ for all $z \in \mathbb{C}$. Then consider the entire ...
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1answer
36 views

Application of Rouche's theorem on $x^4-6x+3$

I'm being asked to find the number of zeros given $|z|<1$ and $1<|z|<2$. So here are my inequalities but I'm not quite sure how to find the number of zeros though based on these inequalities. ...
1
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2answers
66 views

sequence and series (complex analysis) part 2

If a sequence of complex numbers $\{a_n\}$ has the following properties: $$\lim_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right|=\alpha>1,$$ then $$\limsup_{n \rightarrow \infty} \left|\sum_{...
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2answers
589 views

Computing the $n^{th}$ coefficient of the power series representing a given rational function

Is there a easy way to compute the coefficients of the power series which represents \begin{equation*} \frac{x - x^k + x^{k+1}}{1-2x + x^k - x^{k+1}}. \end{equation*} I am currently solving this ...
4
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1answer
45 views

$\int_{-\infty}^\infty\frac{1}{(x^2+a^2)^3}dx=\frac{3\pi}{8a^5}$ for $a>0$

I've been trying to show that $\int_{-\infty}^\infty\frac{1}{(x^2+a^2)^3}dx=\frac{3\pi}{8a^5}$ for $a>0$ using complex analysis methods. But for some reason I can't get it to come out. Perhaps ...
3
votes
2answers
72 views

Using Cauchy Integral Formula $\int_C \frac2{z^2 -1}dz$

I want to understand why I can't use Cauchy Integral Formula for the following problem: $$\int_C \frac2{z^2 -1}dz\text{ on the contour } |z-1|=\frac12$$ Now it says that I need $f$ to be analytic ...
2
votes
1answer
124 views

Show that $f(z)=z+a_2 z^2$ is univalent in $\mathbb{D}=\{z∈\mathbb{C}:|z|<1\}$ if and only if $|a_2 | \leq 1/2.$

Show that f(z)=z+a_2 z^2 is univalent in D={z∈C:|z|<1} if and only if |a_2 |≤1/2. My solution: (If part): Suppose f(z)=z+a_2 z^2 is univalent in D. By definition, we know that f(z_1 )=f(z_2 ) ...
7
votes
0answers
797 views

If $\Re(f)$ is bounded then f is constant.

I have to solve following problem If $\Re (f)$ is bounded above or below for a function $f$ holomorphic on $\mathbb{C}$ then $f$ is constant. My attempt: If there is $M$ such that $\Re(f) \le M$,...
3
votes
1answer
79 views

Is the Gamma Function multivalued??

Consider the definition of the Gamma function $$ \Gamma(s) = \int_{0}^{\infty}\left[x^{s-1}e^{-x} \right] dx $$ Clearly: $x^{s-1}$ may have multiple defined values for $s$ if $s-1$ is rational or ...
2
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2answers
69 views

Is the difference in the hypotheses of these two statements relevant?

The following is a problem from Conway's Functions of One Complex Variable, and my proof: Let $G$ be a region and suppose that $f:G \rightarrow \mathbb{C}$ is analytic. Show that if $f(z)$ is real ...
1
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1answer
125 views

indicator function of an entire function of finite exponential type?

Let $\Phi(z)$ be an entire function of finite exponential type. The indicator function of $\Phi(z)$ is defined as $$ h_{\Phi}(\theta)=\overline{\lim_{r\rightarrow\infty}}\frac{\ln|\Phi(re^{i\theta})|}...
1
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1answer
56 views

$f:D \rightarrow \mathbb C$ holomorphic, $D$ a convex set, $Re(f'(z))>0$. Prove that $f$ is injective [duplicate]

$f:D \rightarrow \mathbb C$ holomorphic, $D$ a convex set, $Re(f'(z))>0$. Prove that $f$ is injective. What I tried: Assume that $f$ is not injective. So there are $a,b \in D$ so that $f(a)=f(b)$. ...