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
30 views

Show sums of complex $\sin$ and $\cos$ series

By considering the series $\sum_{n=0}^\infty r^ne^{in\theta}$ for $0<r<1$ show that $$\sum_{n=1^\infty}r^{n}\cos(n\theta)=\frac{1-r\cos(\theta)}{1-2r\cos(\theta)+r^2} \text{ and } ...
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1answer
49 views

Show that the open disk $D(a,r)$ is connected.

Let $a\in\mathbb{C}$ and $r>0$. Show that the open disk $D(a,r)=\{z\in\mathbb{C}\colon \vert z-a\vert<r\}$ is connected. The disk is connected if there exists a path between any two points ...
1
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1answer
49 views

Show that $g$ is analytic and discuss the properties of $g$

Let $f$ be analytic in $\overline{B}(0; R)$ with $f(0)=0$, $f'(0) \neq 0$ and $f(z) \neq 0$ for $0<|z| \leq R$. Put $\rho=\min\{|f(z)|:|z|=R\}>0$. Define $g: B(0; \rho) \rightarrow \mathbb{C}$ ...
2
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3answers
85 views

Analyticity of $\tan(z)$ and radius of convergence

Define $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$ Where is this function defined and analytic? My answer: Our function is analytic wherever it has a convergent power series. Since (I am assuming) $\sin(z)$ ...
0
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2answers
385 views

Complex differentiable iff map is Frechet differentiable and Cauchy Riemann equations hold

Suppose $\ f: U \rightarrow \mathbb{C}$ is complex differentiable at $z=x+iy \in U$. Then $\lim_{v \rightarrow 0} \frac{\left|\ f(z+v)-f(z)-f'(z)\ \right|}{|v|}=0$. Let $v=v_1+iv_2$. $f'(z)\ ...
1
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1answer
44 views

Power series for complex exponential

Let $z=x+iy$ where $x,y\in\mathbb{R}$. The exponential function is $$e^z=e^x(\cos{y}+i\sin{y}).$$ Using the power series of $e^x$, $\cos{y}$ and $\sin{y}$, find a power series representation for ...
1
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1answer
50 views

Cauchy's integral formula more then one pole?

Cauchy's integral formula says that: $$f(z_0)=\frac{1}{2\pi i}\oint_\Gamma \frac{f(z)}{(z-z_0)}dz$$ Where $\Gamma$ is a closed contour in the positive sense. Does this still hold if $\Gamma$ contains ...
0
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1answer
31 views

Am i evaluating this contour integral correctly?

Let $\gamma$ be the circle of radius 2 centered at the origin. $$ \int_{\gamma} \frac{1}{z^2+i}dz $$ I tried factoring the denominator out to where $ \int_{\gamma} \frac{1}{z^2+i}dz $ = $ ...
2
votes
1answer
33 views

Complex Analysis - Argument - Need Explanations [closed]

Can anyone explain this solution? How did we get $ - i(\pi/2 + 2n\pi) $?
1
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1answer
24 views

Complex power series centered at w

For any $w \in \mathbb{C} \setminus \{1\}$, find a power series for $$f(z)=\frac{1}{1-z}$$ centred at $w$ and give the radius of convergence. Further, find a power series for $f$ ...
2
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0answers
36 views

Cauchy's Residue Theorem and Cauchy's Theorem

Cauchy's theorem in short says for a holomorphic function $f$ which is holomorphic on and inside a path $\gamma$ the path integral is $0$ I have calculated a path integral around a path where there ...
1
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1answer
36 views

The limit of $\frac{1}{z}$ in complex number

How to find $\lim_{z\to z_0}\frac{1}{z}$? I want to make a $\delta-\epsilon$ argument to prove $\lim_{z\to z_0}\frac{1}{z}=\frac{1}{z_0}$. Let $\epsilon>0$. There is a $\delta = ...
1
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1answer
47 views

Show that each biholomorphic function $f:U\setminus M \longrightarrow U\setminus M$ has a biholomorphic extension $g:U \longrightarrow U$.

Let $U\subseteq \mathbb{C}$ be an open and bounded set without isolated points on the border. Let $M\subset U$ be a set without limit points in $U$. Show that each biholomorphic function $f:U\setminus ...
0
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2answers
48 views

Does a compact set with non-empty interior have a limit point?

My Question: Let $U\subseteq \mathbb{C}$ open and $K\subset U$ be a compact set with nonempty interior $K^{o}$, then $K$ must have a limit point in $U$. Remark: I think that the statement is true. I ...
3
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1answer
49 views

Question about non trivial zeros of Riemann zeta function

Riemann zeta function is $$\zeta(s)=\sum\frac{1}{n^s}$$ I read at wiki that the first nontrivial zero is located at $14.134725\ldots$ As far as I understand it means ...
0
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0answers
21 views

Plane integral for continuous curves

I'm trying to understand complex path integral $\int_C f(z)dz$ for continuous closed curve $C$. Is it necessary that $C$ is rectifiable and not just generally continuous? Do we get all the ...
2
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1answer
47 views

Let $f$ holomorphic funcion in $U$ such that $\left|f\right|$ constant on the border of $K$. Show that $f$ is constant or $f$ have a zero in $K^{0}$.

Let $U\subseteq\mathbb{C}$ be an open and connected set and $K\subset U$ a compact subset with nonempty interior $K^{o}$. Let $f:U\rightarrow \mathbb{C}$ holomorphic funcion such that $\left|f\right|$ ...
0
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2answers
31 views

Does a closed set not discrete have a limit point?

My Question: Let $U\subseteq \mathbb{C}$ open and $A\subset U$ be a close set not discrete in $U$, then $A$ must have a limit point in $U$. Remark: I do not know if the statement is true. I know that ...
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1answer
50 views

Prove the roots of $p(z)+z^n\bar{p}(\frac{1}{z})$ lie on the unit circle

I have to prove the following question from "A course in complex analysis and riemann surfaces": Let $p(z)=\sum_{i=0}^na_iz^u$ be a polynomial with all roots inside the (open )unit disk. Denote by ...
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1answer
35 views

Manipulating a sum and showing it has an upper bound

Let $x_1, \ldots ,x_n$ and $y_1, \ldots , y_n$ for $n \in \mathbb{N}$ be complex numbers such that $$\left| \sum_{k=1}^n x_k y_k \right| \leq 1. $$ Further I know that $$ \sum_{k = 1}^n |x_k|^2 \leq ...
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1answer
74 views

Largest subset on which a function is continuous

Let $f: \mathbb{C} \to \mathbb{C}$ a function with $$f(x) =0, ~~~ \text{if} ~~ x = 0 $$ and $$f(x) = (e^x - 1)/x, ~~~\text{if} ~~x \neq 0$$ I want to determine the largest subset $A \subset ...
2
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1answer
58 views

Riemann zeta-function functional equation proof

I'm reading through Titchmarch's "The Theory of the Riemann Zeta-Function" and there's a part in the functional equation proof number 3 that I haven't figured out. He defines a function ...
0
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1answer
35 views

Calculating the residue of $\frac{1}{f(z)}$ at the simple pole $z=z_0$.

I need to show that the residue of $\frac{1}{f(z)}$ at it's simple pole $z=z_0$ is $\frac{1}{f'(z_0)}$. I have tried using the residue theorem together with Cauchy's integral theorem but not really ...
2
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2answers
356 views

Convergence or divergence of infinite power towers of complex numbers $z^{z^{z^{z{…}}}}$

Let $s$ be any complex number, $t = e^s$ and $z = t^{1/t}$. Define the sequence $(a_n)_{n\in\mathbb{N}}$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n \geq 0$, that is to say $a_n$ is the sequence ...
2
votes
1answer
72 views

Schwarz' lemma: Prove that the inequality is strict unless the function f is of a certain form.

the question (not homework) I am trying to answer is, in part: Let $f$ be an analytic function that maps the open unit disk $D$ into itself and vanishes at the origin. Prove that the inequality ...
3
votes
1answer
61 views

A strange identity related to the imaginary part of the Lambert-W function

Working on a problem in QFT, i was stumbeling about some expressions containing the Lambert-$W$ function. Playing around, i discovered experimentally that the following statement seems to be true ...
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0answers
47 views

If $c_{n}$ coefficient of the expansion of $f$. Show that $\sum_{n=0}^{\infty}\left|c_{n}\right|^{2}r^{2n}$ is an expression determined by an integral [duplicate]

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
0
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1answer
35 views

Basic Complex Analysis Question: simplification of $e^{2\pi i\left(17-\frac{1}{3}\right)}$

I have the following detail of a derivation: $$ \begin{align} [\dots] &= 2^{50}\cdot e^{2\pi i\left(17-\frac{1}{3}\right)} \\ &= 2^{50}\cdot e^{-\frac{2\pi}{3} i} \end{align} $$ See source ...
2
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0answers
31 views

If $f(z)=\sum_{n=0}^{\infty}c_{n}(z-z_{0})^{n}$, then $c_{n}r^{n}=\frac{1}{2\pi}\int_{0}^{2\pi}f\left(z_{0}+re^{-2nt}\right)dt$

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
8
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1answer
117 views

An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$

Let $J_{0}(z)$ be the Bessel function of the first kind of order zero, and assume that $\alpha$ and $\beta_{m}$ are positive real parameters. If $0 < \arg(z) < \pi$ and $|z| \to \infty$ at a ...
0
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1answer
39 views

Example of maximum modulus principle

As it's known , an holomorphic($\neq constant$) function $f:G\subseteq\mathbb{C}\rightarrow \mathbb{C}$ has maximum modulus on $\partial G$ . I wuold an example of a function holomorphic on a disk ...
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3answers
50 views

Complex function in terms of z

let $$f(x, y) = x^3 - 3xy^2 + i(3x^2y - y^3)$$ How to express $f(x, y)$ in terms of $z$ ? given that $$z = a + ib$$ $$a = x^3 - 3xy^2$$ $$b = 3x^2y -y^3$$ I need to find $f(z)$ and $f'(z)$ and ...
0
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1answer
20 views

Bounding the Roots of a Complex-Valued Function

Roots: $Z_1$= $\frac{v(1+ \alpha)+ \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ $Z_2$= $\frac{v(1+ \alpha)- \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ It is clear that $|Z_2| \leq|Z_1|$ However I'm stuck on ...
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votes
1answer
64 views

What is the mathematical nature of $i$? [duplicate]

It is well known that $i$ is unit imaginary part of any complex number, but many uses of $i$ show that has others mathematical properties, for example in integration area, if I want to compute ...
1
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1answer
48 views

Don't understand how Hurwitz's Theorem implies its corollaries

I have the following statement of Hurwitz's Theorem which I understand how to prove: Let G be a region and suppose the sequence $\{f_n\}$ in $H(G)$ converges to $f$. If $f\not\equiv0$, $\bar{B}(a; ...
5
votes
2answers
117 views

What is the derivative of $z^{-1}$ with respect to $\bar{z}$?

I asked a question here a few days ago but it wasn't answered and, as often happens with me, in trying to answer it myself I just confused myself out of understanding what I thought I knew. What is ...
1
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0answers
51 views

Absolute Convergence of Infinite Weierstrass Product

I am really stuck on something. I need to show the following: Let $U$ be a domain in $\mathbb{C}$. If $f_n: U \to \mathbb{D}$ are analytic functions satisfying that $\sum |f_n - 1|$ converges ...
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3answers
102 views

Cauchy's Residue Theorem contradiction?

Consider the contour integral: $$I=\oint_\Gamma \frac{1}{\sqrt{z^2-1}}$$ Where $\Gamma$ is a circle at infinity and we have taken the branch cut to be between $z=\pm 1$. Now this function does not ...
10
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0answers
105 views

Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$ −\psi''(x) − (ix)^ N \psi(x) = E\psi(x). $$ where $N$ can be any real number, the boundary condition ...
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0answers
51 views

radius of convergence of $\Sigma a_nz^n$ and $\Sigma a_{2n}z^n$

Suppose $\Sigma a_nz^n$ has radius of convergence $r$, Find the radius of convergence of $\Sigma a_{2n}z^n$. I think this question is not appropriate. Since we can construct a series like this: ...
0
votes
2answers
63 views

solving system of equations with constant $i$ [closed]

What are the values of $a,b,c$ given the system of equations given below: $a+b+ab=i$ $b+c+bc=2i$ $c+a+ac=3i$
2
votes
2answers
67 views

Why can entire function be written as exponential, and why is it bounded in this way?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose now that $f(x) \in \sigma(x)$ for every $x \in A$ where $\sigma(x)$ denotes the spectrum of $x$. Now, let $x\in A$ and ...
1
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1answer
42 views

Hyperbolic area

Define Hyperbolic area of a subset $E$ of the unit disk $D$ to be $\displaystyle 4\int \int_E \frac{dx dy}{(1-|z|^2)^2}$. Show that the hyperbolic area is invariant under conformal self maps of ...
0
votes
1answer
32 views

Limit Evaluation of a Function in the Complex Field

Given the sequence \begin{equation} z_n=\frac{1}{2n\pi}, \quad n \in \mathbb{N} \end{equation} try to evaluate the following limit: \begin{equation} \lim_{z \to z_n} f(z) \end{equation} where ...
1
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1answer
17 views

Application of z

In which lines of the $w$ plane is the line $|z| = \frac{1}{2} $ transformed with the function $w= \frac{1}{z} $? $$| z | = \frac{1}{2}$$ $$\sqrt{x^2 + y ^2} = \frac{1}{2}$$ $$x = \sqrt{\frac{1}{4} - ...
0
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0answers
27 views

Two different answers with applying the Cauchy integral formula and parametrizing.

I have the integral $\int_{|z|=1}\frac{cosz}{z}dz$ By applying the Cauchy Integral Formula, I get that this equals $2\pi i*cos(0) = 2\pi i$ Is this correct? If I parametrize the integral with $z= ...
0
votes
2answers
51 views

Complex analysis using definition of the derivative [closed]

Question: $f(z) = z + 2iz^2 \operatorname{Im}(z)$ Is the function differentiable at $z = 0$? Where is $f(z)$ analytic? Is there any way to do this using the definition of a ...
0
votes
1answer
15 views

Subspace of $C^3$ that spanned by a set over C and over R

Given $A=$ $\left\{ {(1,2 + i,i),(1,3 + i,3 - i),(i,3i,4 + i)} \right\}$ Let $SP_CA$ be the linear space spanned by A over $C$ Let $SP_RA$ be the linear space spanned by A over $R$ what is the ...
0
votes
0answers
40 views

The partial sums of the form $\zeta _N(s)= \sum_{k=1}^{N} \frac{1}{k^s} $

My question concerns the partial sums of the form: $$\zeta _N(s)= \sum_{k=1}^{N} \frac{1}{k^s}$$ Is there an analytic continuation to the entire complex plane such sums as the complex functions of s ...
1
vote
1answer
39 views

Kernel of a map $\phi: \mathbb{C^*}^3 \to \mathbb{C^*}^2$

I cannot understand which is the kernel of the following map $\phi: \mathbb{C^*}^3 \to \mathbb{C^*}^2$ with $$ (t_1,t_2,t_3) \mapsto \left(\frac{t_2}{t_1}, \frac{t_3}{t_1}\right) $$ In other words I ...