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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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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19 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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0answers
14 views

Nessecceriarly and sufficiantly for univalency

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 ...
2
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1answer
124 views

The Laurent series of $ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1}$

How to find Laurent series of g(z) ? $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1} \hspace{10mm} \begin{cases} n \in N \\ 0<a<1 \end{cases} $$ answer is : $$ ...
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1answer
11 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
28 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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0answers
24 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
43 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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15 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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0answers
18 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
26 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 ...
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40 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 ...
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0answers
19 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
27 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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16 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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0answers
16 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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1answer
36 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
6 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
44 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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2answers
57 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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1answer
92 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let two points $$z_1 = \frac{-1+i\sqrt3}{2}\quad\text{and}\quad z_2 =\frac{-1-i\sqrt3}{2}.$$ I am trying to show that there is no point $w$ on ...
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1answer
17 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
28 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
77 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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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
23 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
45 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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0answers
22 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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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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29 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
91 views
+200

A difficult complex integral question

Let: $$ \alpha = -\frac{u^2}{2}-\frac{iu}{2}+jiu\\ \beta = \lambda-\rho \eta i u - j \rho \eta\\ \gamma = \frac{\eta ^2}{2}\\ r_{\pm}\frac{1-e^{-d\tau}}{1-ge^{-d\tau}} $$ where $j \in {0,1}$ and ...
2
votes
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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3answers
29 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
283 views

Analytic continuation of Fourier transform

Let $$h(u)= \int_{-\infty}^{\infty} g(x) \exp(iux) \, \text{d}x$$ be the Fourier transform. Then let us suppose that for $ |x| \to \infty $ the function $g$ goes as $$g(x) = \exp(-ax) \text{ for ...
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4answers
522 views
+100

Why does being holomorphic imply so much about a function?

I haven't yet started my complex analysis course (soon!), but recently (inspired by you guys) I've been looking into holomorphic functions. And wow, they're cool! There's so much stuff that's true ...
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1answer
21 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 $f$ ...
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0answers
24 views

How find a maximal region for log(Log)?

Find maximal region $U\subset\mathbb{C}$ and some branch for $log$ (logarithm function) such that $log\circ Log$ is well defined where $Log$ is the principal branch of logarithm. The immediately ...
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1answer
173 views

Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
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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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0answers
20 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 ...
234
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22answers
20k views

Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
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1answer
25 views

How to prove that $ - \frac{\pi i}{2n} \sum_{k=0}^{n-1} e^{ \frac{\pi i}{2n} + \frac{k \pi i}{n}} = \frac{\pi}{2n \sin(\pi/2n)} $?

I'd like to prove that $$ - \frac{\pi i}{2n} \sum_{k=0}^{n-1} e^{ \frac{\pi \cdot i}{2n} + \frac{k \cdot \pi \cdot i}{n}} = \frac{\pi}{2n \sin(\pi/2n)} .$$ Up until now, I've tried to do this by ...
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1answer
28 views

Definition of hyperbolic trig functions

I was doing some homework for my complex analysis class and ran into a personal question. I haven't worked a lot with hyperbolic trig functions (e.g. $\sinh (x)$, $\cosh(x)$, etc.) so this question ...
8
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1answer
96 views

Analytic function $f$ in $\overline{\mathbb{D}}$ satisfying $\left\lvert\,f'(\tfrac{1}{2})\,\right \rvert\leq 8.$

Let $f$ be an analytic function on the closed unit disk $\overline{\mathbb{D}}$. On its boundary $\partial \mathbb{D}$ it holds that $\vert\,f(z) -z\rvert < \lvert z\rvert$. I now have to show ...
3
votes
1answer
287 views

Why are $\limsup \sqrt[n]{1/n!}=0$ and $\limsup\sqrt[n]{n!}=\infty$?

I was looking at the power series $\sum\frac{z^n}{n!}$ and $\sum n!z^n$, and wanted to compute their radii of convergence. For the first, $\limsup \sqrt[n]{1/n!})=0$, and for the second $\limsup ...
4
votes
5answers
71 views

If $f=u+iv$ is an entire function such that $u^2\geq v^2,$ then $f$ is constant

Let $f=u+iv$ be an entire function such that $u^2(z) \geq v^2(z), \forall z \in \mathbb{C}.$ Could anyone advise me how to prove $f \equiv$ constant $?$ Hints will suffice. Thank you.
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1answer
28 views

Upper bound for |f(z)|

The question is: Let $f$ be analytic in the unit disc $\mathbb D$ satisfying $f(0)=1$ & $Re (f(z)) \gt |Im(f(z))|$ , $\forall z \in \mathbb D$ . Then show that : $|f(z)|^{2} \leq \frac ...
13
votes
1answer
281 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
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votes
0answers
16 views

functions with positive real part proof

Given $\varphi$ is a function with positive real part i.e. $Re\varphi(z)>0$ how to show $\varphi(e^{i\theta}) \not= 1$ where $\theta\in [0,2\pi)?$ Thanks.