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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14 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. When $|z|$ is large in magnitude and $-\pi < \arg(z) ...
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1answer
11 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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2answers
29 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 ...
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1answer
12 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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1answer
51 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 ...
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1answer
22 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; ...
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2answers
88 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 ...
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0answers
20 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
72 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 ...
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0answers
33 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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36 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: ...
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2answers
37 views

solving system of equations involving imaginary numbers

What are the values of $a,b,c$ given the system of equations given below: $a+b+ab=i$ $b+c+bc=2i$ $c+a+ac=3i$
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2answers
43 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 ...
2
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1answer
30 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 ...
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1answer
23 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 ...
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1answer
15 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} - ...
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0answers
22 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= ...
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2answers
42 views

Complex analysis using definition of the derivative [on hold]

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 ...
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1answer
10 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 ...
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0answers
18 views

Complex Valued Sequence $\{g_n(z)\}$ and $g(z)$ Achieve $\beta \ k$ Times

Define $\{g_n(z)\}$ to be a sequence of complex valued functions analytic on domain $R$, converging normally to $f(z)$. If $\{g_n(z)\}$ attains each value $\beta \in \mathbb{C}$ at most $k$ times ...
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0answers
41 views

Conformal Mapping: Is this correct?

I have the following two circles in the complex plane, $z=x+iy$, which bound a region, $R$. The equations for the circles are given as follows: $$x^2+(y−1)^2=1\\x^2+(y−2)^2=4 $$ What I now want to do ...
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0answers
35 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 ...
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1answer
36 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 ...
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2answers
63 views

Laurent Series expansion without geometric series

There are several functions in complex analysis which I have not been able to get the Laurent expansion for, both of which are very different from the examples I see online and in the (4) textbooks I ...
2
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1answer
30 views

Simple linear map question

I am struggling with a very simple question. I need to show that any linear map $\mathbb{C}\to\mathbb{C}$ has the form: $$f(z)=\alpha z+\beta\, \bar z,\quad \alpha,\beta\in\mathbb{C}$$ I thought of ...
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1answer
38 views

Show that $\log(z)$ is real if z is real and positive.

Question The problem is this: Show that $\log(z)$ is purely imaginary (i.e. $\operatorname{Re\, Log} z$ $=$ $0$) if $|z|=1$. Show that $\log(z)$ is real if $z$ is real and positive ...
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1answer
13 views

Meromorphic functions on unbounded domains

Suppose $D\subset\mathbb{C}$ is a bounded domain and $f$ is a meromorphic function on the exterior domain meaning on $D_+=\hat{\mathbb{C}}\setminus\overline{D}$. Moreover $f(\infty)=0$ and $f$ has ...
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2answers
23 views

Analyze the Complex Function by using the Principal log Branch

I am trying to analyze the function $\sqrt{1-z^2}$, where the square root function is defined by the principal branch of the log function. I want to locate the the discontinuities. I know the ...
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1answer
25 views

Using ellipse for Cauchy integral.

I am currently working on a homework problem where I need to evaluate the following: $\int_{\gamma}(z^2+2iz)^{-1} dz$ where $\gamma(t) = 2\cos t+i \sin t, 0\le t\le 2\pi$ I was thinking of using the ...
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0answers
42 views

Prove that any holomorphic map between Riemann surfaces is either constant or an open map [on hold]

Part a) Prove that any holomorphic map $\textit{f: X $\rightarrow$ Y}$ between Riemann surfaces, with $\textit{X}$ connected, is either constant or an open map, meaning f(any open set) is open. Hint: ...
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1answer
35 views

How to conclude this proof that real and imaginary parts of holomorphic functions are harmonic.

I want to prove that if $f$ is holomorphic on an open set $\Omega$, then both the real and imaginary parts are harmonic, so I have proved that: $$4\frac{\partial}{\partial z} \frac{\partial}{\partial ...
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0answers
25 views

Curl of a vector in the complex plane

Let there be a vector $u(z)$ in the complex plane. Are these two statements equivalent? $$\nabla\times\overline u=\overline{\nabla\times u}$$ If not, why? I think they should be equal since $\nabla$ ...
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54 views

$R(\alpha) = \frac{f(\alpha)}{g(\alpha)}$ achieves $\beta \in \mathbb{C}, n$ times.

Let $R(\alpha) = \frac{f(\alpha)}{g(\alpha)}, \ \alpha \in \mathbb{C}$. $R$ is rational and $f(\alpha), g(\alpha)$ are polynomials with no factors in common. Define $n=$ deg$(R(\alpha)) = $ max deg ...
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1answer
39 views

Proof: Raising a complex number to a rational power

The problem from the textbook is: Prove that if (a complex number) $z$ is a number on the unit circle, then $z$ has finitely many distinct powers $z^n$ if and only if the argument of $z$ is a ...
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2answers
40 views

Proofs of Liouville Theorem

Are there proofs of Liouville theorem (bounded functions holomorphic in $\mathbb{C}$ are constants) without using the Cauchy theorem?
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2answers
68 views

The complex version of the chain rule

I want to prove the following equality: \begin{eqnarray} \frac{\partial}{\partial z} (g \circ f) = (\frac{\partial g}{\partial z} \frac{\partial f}{\partial z}) + (\frac{\partial g}{\partial \bar{z}} ...
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5answers
50 views

Is it true that $|e^z|\le e^{|z|}$ for all $z \in \mathbb C$?

Is it true that $|e^z|\le e^{|z|}$ for all $z \in \mathbb C$? I believe so as $x \le \sqrt{x^2+y^2}$ but the way the question is worded suggests it is not the entire complex plane.
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1answer
15 views

Mapping a simple region using an exponential function

I have the following region in the complex plane bounded by the two lines: $$ x = y \quad\text{and}\quad x = 2y$$ It is plotted as follows: Region Plot. I am required to map the region under the ...
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44 views

Mapping circles in the complex plane

I have the following two circles in the complex plane, $z=x+iy$, which bound a region, $R$. The equations for the circles are given as follows: $$x^2+(y−1)^2=1 \\ x^2+(y−2)^2=4 $$ That is, I believe, ...
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1answer
34 views

Locally bounded $\Leftrightarrow$ compactness

I am trying to prove the following A set $\mathcal{F}$ in $H(G)$ is locally bounded if and only if for each compact set $K\subset G$ there is a constant $$|f(z)|\leq M$$ for all $f$ in ...
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50 views

Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
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0answers
18 views

theorems or statement that guarentees a function to be a polynomial or constant specially complex functions

This is just a question out of curiosity since i came across that in complex analysis there is a lot of emphasis and stress to prove that certain function is const or a polynomial .can i learn some ...
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0answers
17 views

Conway, continuity and homotopy

Let $G$ be an open set in $\mathbb{C}$ and let $\gamma$ be a closed smooth rectifiable curve in $G$ such that $\gamma$ is homotopic to a constant curve,$\gamma_o$ ; furthermore let $ \Gamma(s,t): ...
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0answers
27 views

Function $\phi (x,y) \mapsto \phi (u,v)$ [ie:…] under the transformation $f(z) = u(x,y) + v(x,y)$ where $f(z)$ is analytic & $d_z \,f(z)$ is not $0$

I am just having a bit of trouble understanding what I am being asked. The ie: in title says $\phi [x(u,v), y(u,v)]$ Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write ...
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1answer
33 views

Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if $e^{iθ} \bar c + c = 0$.

Consider the isometry $ϕ : \Bbb C → \Bbb C$ with equation $ϕ(z) = e^ {iθ} \bar z + c$ where $θ ∈ \Bbb R$ and $c ∈ \Bbb C$. Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if ...
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1answer
33 views

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$.

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$. I already have a proof for this but I would like an explanation ...
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2answers
42 views

Analytic function $f,$ such that $f(0) = 1$ and $f'(z) = zf(z),$ for all $z \in \mathbb{C}$

I'm trying to find an example of an analytic function $f$ satisfying the IVP $$ f'(z) = z\,f(z), \quad f(0) = 1, $$ and for all $z \in \mathbb{C}$, but I'm somewhat at a loss of the best way to ...
1
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0answers
21 views

Decomposing complex function in even and odd parts

It is known that for real functions, we can express any function $f(x)$ as a sum of an even function and an odd one, so that $$f(x)=f_e(x)+f_o(x)$$ where $$f_e(x)=\frac12[f(x)+f(-x)]$$ ...
0
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1answer
43 views

Evaluating the limit of a complex function

$$f(z) = \begin{cases} \displaystyle{\frac{\bar z^2+4}{z^2+4}|z+2i|^2},&z\neq\pm 2i\\ 0,&z = \pm 2i\end{cases}$$ 1. Evaluate $\displaystyle\lim_{z\to 2i} f(z)$ if it exists. 2. Evaluate ...
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0answers
23 views

Can a function be analytic if it is only differentiable on a line?

I was given an function and used the $CR$ equations to determine that $f$ was differentiable only when $y=2x$. Does this mean that it is nowhere analytic since there wouldn't be an ...