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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how to slove T(n)=T(n/3+5)+T(2n/3+7)+c

How can I solve big O of this problem? Plz let me know.. code is following... int test(n) { If (n<=50) then return n; else return (test(n/3+5)+test(2n/3+7); }
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8 views

Finding a bound on a specific value of a holomorphic function

Let $f$ be a holomorphic function on $\overline{D(0,1)}$ such that $|f(z)| \leq M,$ if $|z|=1$ and $Im(z) \geq 0$ and $|f(z)| \leq N, $ if $|z|=1$ and $Im(z) <0.$ Could anyone advise me how to ...
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1answer
13 views

Determining the order of a pole?

I have the function: $$f(z) = \frac{1}{(1+z^2)^{n+1}}$$ where $n \geq 1$. I see that the poles of $f$ are $i$ and $-i$. I've looked online and sources tell me that the pole $i$ is of order $n$, not ...
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13 views

Contour integration with poles

I am trying to solve: $$\int_{-\infty}^{\infty}\frac{x\sin{x}}{x^2 + a^2}dx$$ and show that it equals $\pi e^{-a}$ for all $a > 0$. I translate it to its complex equivalent: $$lim_{R ...
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15 views

Showing a particular part of toy contour tends to 0?

I am trying to solve: $$\int_{-\infty}^{\infty}\frac{x\sin{x}}{x^2 + a^2}dx$$ I translate it to its complex equivalent: $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = ...
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3answers
20 views

Finding poles of $\frac{1}{1 + z^4}$?

$$f(z) = \frac{1}{1 + z^4}$$ has poles when $z^4 + 1 = 0$ $\implies (z^2 + i)(z^2 - i) = 0$ $\implies (z^2 - i) = 0$ when $z = \sqrt{i}$ or $z = -\sqrt{i}$. But how do I solve the equation $(z^2 + ...
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1answer
19 views

At what point is this piecewise function continuous?

let $$f(z) = \begin{cases} z & |z|\leq 1 \\ |z|^{2} & |z|> 1 \end{cases}\ \text{where}\ z\in \mathbb{C}$$Does anyone could help me ?Thanks!
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1answer
14 views

How to prove the existence of a second complex derivative…

I'm trying to prove that if $h(t)=|\gamma(t)|^2$, then $h''(t)$ exists at a point $t_0$. I know that $\gamma(t_0)=0$, $\gamma'(t_0)$ is nonzero and that $\gamma(t)=x(t)+iy(t)$ on $I=[a,b]$. I know I ...
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7 views

(complex variables) How to prove property of $\limsup$ function in complex plane

Can someone show a proof of the following in a textbook anywhere? Assuming that $a_n$ are the coefficients of a convergent power series $\sum_0^\infty a_nz ^n$? \begin{align*} ...
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8 views

Complex analysis, poles and singularities and boundedness

So I am on the following problem: Prove that an isolated singularity of $f(z)$ is removable as soon as either $\text{Re}f(z)$ or $\text{Im}f(z)$ is bounded above or below. The hint is to use a ...
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20 views

Use mathematical induction to prove Σ n,k=1 (1/k(k+1)) = (n/n+1) for all n in Natural numbers?

This is how far I can get: p(n): nΣk=1 (1/k(k+1)) = (n/n+1) p(1): 1Σk=1 (1/(1+1)) = (1/1+1) => 1/2 = 1/2 p(1) is true. Assume that p(k) is true. p(k) = kΣk=1, (1/k(k+1)) = k/k+1 Show ...
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10 views

Confused about pochhammer contour?

I know some theorems about complex analysis such as the argument principle. But I do not get the pochhammer contour. I read about it on the wiki page of the beta function , but I do not understand a ...
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15 views

How to select the integration contour

In the following two figures which describe sets, How many possible integration contour we have for the figure 1 and how many integration contour we have for figure 2.
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27 views

Problem on Complex numbers involving a point on a Circle

Question: The Complex number $z$ is represented by the point $T$ in the Argand Diagram.Given that $$z =\frac{1}{3+it}$$ where $t$ is a variable, show that i) as $t$ varies, $T$ lies on a circle, and ...
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1answer
16 views

Is this set not simply connected?

As an example of a set that is not simply connected the lecturer gave us this picture: of a set that is not simply connected: Have I misunderstood something? It looks simply connected to me. I mean ...
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24 views

Analytic map from punctured half plane to annulus

This is an old comp question. Consider the punctured upper half plane $U$ and the annuals $A$ given by $U:=\{z\in\mathbb{C}:Im(z)>0\backslash{\{i\}}\}$ and $A:=\{z\in\mathbb{C}:1<|z|<2\}.$ ...
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0answers
23 views

Contour integrals for $f(z)= e^{3z}$

Integrate $f(z)=e^{3z}$ along line segment from $(0,0)\to(1,1)$ parabola $y=x^2$ from $(0,0)\to(1,1)$ circle $|z|=3$ once around its arc (positive $360^o$) First I parametrized with $z(t)=t+it$ ...
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25 views

A simple test for degenerate eigenvalues of a holomorphic matrix-valued function?

Consider a symmetric $n\times n$ matrix $H(z)$ whose entries are holomorphic functions of a complex parameter $z$ and real on the real axis. It's known, from Analytic structure of the eigenvalue ...
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20 views

A necessary and sufficient condition for a polynomial to be univalent in the unit disk

I want to prove: Show that a polynomial $p(z) = z + a_2z^2+\cdots+a_nz^n$ is univalent in $\Bbb D$ if and only if its associated polynomials ...
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1answer
29 views

problems on Analytic function

I have a problem with this Let S be the disk $|z|<3$ in the complex plane and let $f:S \rightarrow C$ be an analytic function such that $f(1+\frac{\sqrt 2}{n}i)=-\frac{2}{n^2}$ for each natural ...
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29 views

Mobius Transformation

Is there a Mobius transformation mapping the upper half plane onto itself that interchange two preassigned points in the upper half plane? If so , how many such Mobius transformations are there? ...
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1answer
14 views

Application of Schwarz Lemma from Exercise of Gamelin

Suppose $f$ is analytic for $|z| \lt 1$ & satisfies $|f(z)| \lt 1$ , $f(0) = 0$ & $|f'(0| \lt 1$ . a) Let $r \lt 1$. Show that there is a constant $c \lt 1$ such that $|f(z)| \lt c|z|$ for ...
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1answer
32 views

A version of Cauchy's integral formula

I'm trying to prove this version of Cauchy's integral formula that is little more general than what appears in Stein and Shakarchi "complex analysis": [Let $\rho>0$ and assume that $f$ is ...
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22 views

“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be ...
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0answers
21 views

justification of step in complex integration

What is the justification for the step with the red square next to it, how do we change the integrator like this?
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1answer
50 views

Reading Griffiths Harris: Quick question

Why is a meromorphic section without zeros and poles on a compact Riemann surface necessarily a constant? Thank you very much.
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20 views

product of the lengths of the segments $|AC|\bullet |BC|$ will be maximal.

Let there be a segment $AB$ the Diameter of the circle $S(0,1)$. Find all the points $C$ that belong to the closed circle $D^-(0,1)$ such that the product of the lengths of the segments $|AC|\bullet ...
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1answer
30 views

Show that $\Sigma_{n=1}^\infty |z_n| $ converges.

Assume $z_k = |z_k|e^{i\alpha_k}$ are complex numbers and that exists $0<\alpha<\pi/2$ s.t $\forall k -\alpha < \alpha_k<\alpha$ assume that $\Sigma_{n=1}^\infty z_n $ converges. we ...
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0answers
17 views

Show that $f'_n \rightarrow f'$ uniformly on every compact set. [duplicate]

Assume that $f_n,f:D(0,1)\rightarrow C$ are holomorphic. and $f_n \rightarrow f$ uniformly on every compact set. we need to show that $f'_n \rightarrow f'$ uniformly on every compact set. I kind of ...
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0answers
7 views

Find all Mobius transfers $T$ s.t $T(\Re)=\Re$ and $T(i\Re)=i\Re$

The question is: Find all Mobius transfers $T$ s.t $T(\Re)=\Re$ and $T(i\Re)=i\Re$ Now I've done some calculations and got that $T(z)=1/z$ but that is not enough..there are more and I'm not sure how ...
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1answer
21 views

Equilateral Triangle from three complex points

I need some help proving this, I've seen it proven in the other direction (prove the formula if it is an equilateral) but cant figure out how to prove it this way around. Any help would be much ...
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1answer
44 views

Let $u(x,y) = x^2 + 2axy + by^2$, where $a$ and $b$ are real, when is $u$ the real part of an analytic function and what's the imaginary part?

Does my approach here seem agreeable? (Revised) Attempt: If $u$ is the real portion of an analytic function in the complex plane, it must satisfy the Cauchy-Riemann equations $\frac{\partial ...
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0answers
15 views

f(iy)=1/(y-1) , what is the set of the points M(f(z))? [on hold]

when y changes in R - (1) , what is the set of the points M(f(z)) when z changes in iR We have f(iy)=1/ (y-1) , i have no ide what does this make , its not a circle , what is it
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1answer
48 views

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$ May I verify if my solution is correct? Thank you. Consider ...
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1answer
29 views

proof of Laurents extension theorem

I found this theorem of Laurents expansion theorem online, but there is one thing I do not understand. How can he just say that $\gamma$ is a union of two cicles? Does he make some kind of ...
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1answer
28 views

(complex variables) Expand $\frac{2z+3}{1+z}$ in a power series of $z-1$ and comment on its convergence

Question: Expand $\frac{2z+3}{1+z}$ in a power series of $z-1$. What can we say about its convergence? Attempt: First, notice $ \frac{2z+3}{1+z} = \frac{2z+3}{1} \frac{1}{1+z}$. Let $w = 1 -z$. Using ...
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2answers
64 views

Expressing $1 + \cos(x) + \cos(2x) +… + \cos(nx)$ as a sum of two terms

Question in title, my progress: let $z = \cos(x) + i\sin(x)$ then $1 + \cos(x) + \cos(2x) +\dots + \cos(nx) = Re(1 + z + z^2 +\dots + z^n) = Re\left (\dfrac{1-z^{n+1}}{1-z} \right)$ by geometric ...
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0answers
23 views

contour integral over line segment and parabola

Let $f:z \mapsto \bar{z}^2$ Calculate integral along line segment parabola $y=x^2$ From origo $z=0$ to $z=1+i$ The first one I parametrized $z(t)=t+it, t\in[0,1]$ and $z'(t)=1+i$. Then used ...
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2answers
48 views

How do you evaluate tan(1+i)

I'm a little confused how to evaluate the tangent function with complex numbers. I know $\tan(1+i)=\sin(1+i)/\cos(1+i)=(e^{-1+i}-e^{1-i})/(i(e^{-1+i}+e^{1-i})$.The book says the answer is ...
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1answer
17 views

How to describe two integration contours as set? [on hold]

Friends I need support to understand how one can describe two integration contours as set? can anyone please explain it with the help of a example?
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1answer
78 views

What word means “the property of being holomorphic”?

As in the title, I am looking for a single word meaning "the property of being holomorphic". The obvious candidates are "holomorphy" and "holomorphicity" but both look wrong to my eye. ...
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35 views

What is integration contour and how to discribe it? [on hold]

We knew that an integration contour can be described as a set of points. How one can describe the two integration contours as sets.? Can anyone help me with examples.
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0answers
42 views

Lower bound for $ F(z) = \sum_{n=1}^\infty d(n)z^n $ near radius of convergence.

In Stein and Sharkarchi Problem 2.7.2 one is asked to find a lower bound $$ |F(z)| \geq c\frac{1}{1-r}\log\left(\frac{1}{1-r}\right) $$ for the function $$ F(z) = \sum_{n=1}^\infty d(n)z^n $$ near the ...
2
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3answers
31 views

Cauchy's Inequality $|f(z)|\leq k|z|^2$

Let $f(z)$ be an analytic function satisfying $|f(z)|\leq k|z|^2$ for some positive constant $k$ and all $z$. Show that $f(z)=az^2$ for some constant $a$ Attempt: I have proved that $f(z)$ has ...
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1answer
22 views

Question about a bijection on the upper half plane from Greene, Krantz's “Function Theory of One Complex Variable”

Let $U=\lbrace z\in\mathbb{C}:\text{Im} z>0\rbrace$. I am trying to show that if $$u(z)=\frac{az+b}{cz+d}$$ where $a,b,c,d\in \mathbb{C}$ and $u:U\to U$ a bijection, then $a,b,c,d$ are real (after ...
3
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1answer
26 views

Explicit Conformal Mapping

The Question is: Find an explicit conformal mapping mapping the region : {$z \in \mathbb C | |z| \lt 2 , Im(z) \gt 0$} to the upper Half-plane: $\mathbb H^{+}$ ; such that under that mapping $f$ ...
2
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1answer
48 views

Representation integral for some branch of arcsin(w)?

Let $G:=\mathbb{C}-\lbrace t\in \mathbb{R}\vert$ $\vert t\vert\geq1\rbrace$. Shows there is a representation for all $w\in G$ $$\arcsin(w)=\int_{\gamma(w)}\frac{1}{\sqrt{1-\zeta^{2}}}d\zeta$$ ...
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0answers
21 views

Power series expansion of a meromorphic function and limit of ratio of coefficient [duplicate]

Suppose f is analytic function in $D_{1+\delta}(0)-{z_0}$, where $\delta>0,|z_0|=1.$ And f has a pole at $z_0$. Show that if $$f(z)=\sum_{n=0}^{\infty}a_n z^n,$$ in the unit disk, then ...
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1answer
24 views

General Cauchy theorem aplication

Let $a \neq b\in \mathbb{C} $ and $U := \mathbb{C} -[a,b] $ Let $\Gamma$ be a cycle in $U$. The following equality is true? $$\int_{\Gamma} \frac{1}{(z-a)(z-b)}dz=0$$ I saw it some notes of a ...
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27 views

Analytic continuuation of an integration of an analytic function with quadratic decay

let f be an analytic function in the right half plane $H=\{Re (t)\geq 0\}$ and $$|f(t)|\leq \frac{C}{1+|t|^2},t\in H$$, C is some constant. Suppose that for some ...