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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Differentiability complex functions

If u, v : Ω → C are C^1 functions in an open set Ω ⊂ C, and f = u + iv. Then if the Cauchy–Riemann equations is not satisfied at any points of Ω, does it mean that f is not differentiable?
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2answers
72 views

Complex Analysis - Write in x+iy form

Write ($\frac{\sqrt{2}}{(i-1)})^{10}$ in $x+iy$ form. I'm completely stuck and cannot figure it out. Can someone please walk me through the solution?
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45 views

Prove Basic Complex Number Inequalities

Let $$z_1 = a_1 + b_1i$$ $$z_2 = a_2 + b_2i$$ where $$|z_j| = \sqrt{a_j^2 + b_j^2}$$ Prove $$|z_1 + z_2| \le |z_1| + |z_2|$$ $$|z_1 + z_2| \ge |z_1| - |z_2|$$ $$|z_1 - z_2| \ge |z_1| - |z_2|$$ $$...
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4answers
46 views

Prove $|z1/z2| = |z1|/|z2|$ without using polar

Prove $|z1/z2| = |z1|/|z2|$ where $$z_1 = a_1+b_1i$$ $$z_2 = a_2+b_2i$$ $$|z_1| = \sqrt{a_1^2+b_1^2}$$ $$|z_2| = \sqrt{a_2^2+b_2^2}$$ $$RHS = \frac{|z_1|}{|z_2|} = \frac{\sqrt{a_1^2+b_1^2}}{\sqrt{...
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1answer
49 views

Solving a functional equation in complex analysis

Is there any simple (general) way to find the complex solution of a functional-equation? For example, it is given $ f(z) = f\big(\frac{z^2+4}{3}\big) $, and the question is to find all the ...
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0answers
31 views

What is the value of $x^{n}$ when $x\in\Bbb N$ and $n\in\Bbb C$? [duplicate]

How can I calculate $x^{n}$ when $x\in\Bbb N$ and $n\in\Bbb C$ ? Respectively $x^{ni}$
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1answer
20 views

Limit Of A Complex Valued Function

Consider the complex valued function given by- $$f(z)=\frac { Re(z)+Im(z)}{|z|^2}$$ Does $$\lim_{z\to 0} f(z)\ exists? $$ This is how I approached the problem- Consider $z \in \Bbb {C}$ , then $...
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2answers
28 views

Find the Taylor-series expansion of a square of a rational function of a complex variable

I've been trying to find the Taylor-series expansion of the following function: $$ f(z)=\left ( \frac{1+z}{1-z} \right )^2 $$ az the origin : Z0 = 0. also I would like to find the region of ...
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0answers
44 views

Integral over the full complex plane, inconsistent result

I'm trying to integrate a function over the complex plane. The plan is: I first do the angular integration, using some residue method, and then the radial one. However, depending on what point I ...
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1answer
43 views

Which vectors are obtainable by my function?

Imagine a disc with $N$ radially displaceable masses $m_g$. A total imbalance with respect to the center of the disc can be calculated as follows (using the respective radiuses $r_1,...,r_N$): $$\...
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2answers
31 views

Density of complex polynomial in the space of entire holomorphic functions.

Let $\int_\mathbb C f(w)e^{\frac{-|w|^2}{2}} p(w) dA(w)=0$ for all complex polynomial $p(w)$. Then show that $f=0$. Anyone could please help me for this. Thanks in advance.
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1answer
21 views

find the laurent series using z=w+1

Here is the question that my books is asking Find the same Laurent series for $f(z)=1/(z(z-1)^2)$ center at $c=1$ by using the following procedure. Set $z=w+1$, expand the resulting function in ...
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0answers
71 views

Why is analysis over the complex numbers so useful vs say other fields?

First I'll state a statement that I hope is false, but I do not know if it is: "Complex analysis is used a lot compared to analysis over other fields (as in it gives a lot of results like the prime ...
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2answers
31 views

Laurent Series expansion in sigma notation

I am trying to find the Laurent Series expansion in sigma notation of $$\frac{1}{z^3-2z^2+z}$$ where $0<|z-1|<1$. I've tried partial fractions and am still stuck on the approach necessary to ...
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1answer
87 views

Computed wrong the sum $\displaystyle\sum_{n\ge1}\frac{\cos n}{n^2} $ [duplicate]

I've computed (using standard complex analysis techniques) the sum $$ \sum_{n\ge1}\frac{\cos n}{n^2} $$ and I found $\pi^2/6+1/4$ which is strictly greater than $\pi^2/6$, and this is impossibile, ...
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28 views

Extend $g(z)$ holomorphically from $\mathrm{Re}(z) > 0 $ to $\mathrm{Re}(z) \geq 0$.

If I have holomorphic function $g(z)$ defined on $\mathrm{Re}(z) > 0$ and I extend in a holomorphic way to $\mathrm{Re}(z) \geq 0$, how do I get $g(0)$ purely in terms of stuff in the right half ...
7
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1answer
237 views

Arnold's proof of Abel's theorem

I'm seeking help understanding this video. The author considers the equation $ax^5+bx^4+cx^3+dx^2+ex+f = 0$ and shows both the coefficients $a, b$... and solutions $x_1, x_2$... in the complex ...
2
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0answers
45 views

a problem about proving a harmonic function to be a holomorphic function

Let $D\subseteq \Bbb C$ be a simply connected domain ,and let $f$ be a non-vanishing complex-valued harmonic function in $D$. Prove that if there is a branch of $\log f(x)$ which is also harmonic ,...
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1answer
43 views

On a property of a uniformly bounded sequence of holomorphic functions on $D$

Let $\left\{f_{n}:n=1,2,3\ldots\right\}$ a uniformly bounded sequence of holomorphic functions on a unit disk $D$.Suppose there exists a point $a\in D$,so that $\lim_{n\rightarrow\infty}f^{\left(k\...
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1answer
56 views

residue theorem, integral of inverse quadratic not working

$$\int_{-\infty}^\infty\frac{dx}{3x^2+0.4x+10}$$ roots of $3x^2+0.4x+10$ $= z_1,z_2=-\frac{1}{15}\pm 1.825i$ Using $\mathrm{res} = \frac{1}{z_1-z_2}=-0.274i$ ans: $2\pi i\cdot -0.274i=1.72$ the ...
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1answer
72 views

What's wrong with my “proof” that branch cuts are not arbitrary?

Recently, I've been thinking about contour integrals around branch cuts in the complex plane. Now clearly the choice of contour is arbitrary, so long as you don't deform past any poles or cuts, but I'...
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0answers
41 views

Holomorphic extension using reflection principle

I cannot really understand how the reflection principle works. I have found various forms of the theorem, but the notes i use are not so clear on how we actually extend functions and what is the ...
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1answer
76 views

Prove that $u$ is upper semicontinuous on $\Delta(0,\rho)$.

Let $u:\Delta(0,\rho)\rightarrow \mathbb{R}$ be a function such that $u(x+iy)$ is convex in $x$ for each fixed $y$, and convex in $y$ for each fixed $x$. Prove that $u$ is subharmonic on $\Delta(0,\...
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1answer
69 views

Sequence of polynomials converging to zero, but not uniformly on unit disc

I have been trying to solve the following without success so far: Show that there exists a sequence of polynomials satisfying $P_n(z)\rightarrow 0$ for every $z\in \mathbb{C}$, but the convergence is ...
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0answers
26 views

Calculation of a limit including the Dedekind eta function

I do not understand this attached equation, where $\eta$ means the Dedekind eta function and $E_2$ the Eisenstein series of weight $2$. Has someone an idea what happens here?
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1answer
39 views

Complex Analysis: Möbius transformations

Create a transformation from $(z_1,z_2,z_3)=(0,i,\infty)$ to $(w_1,w_2,w_3)=(2,1-i,0)$, the transformation should be $w(z)=\frac{2}{z+1}$ My calculations: $$\frac{w-2}{w-1+i}=k\frac{z-0}{z-i}$$ By ...
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3answers
46 views

An inequality about a bounded holomorphic function on unit disk

Suppose $f$ is a bounded holomorphic function on the open unit disk $D$. Show that $$\left(1-\left|z\right|\right)\left|f'\left(z\right)\right|\leq\sup_{w\in D}\left|f\left(w\right)\right|$$ for all $ ...
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1answer
32 views

Cauchy riemann and analytic functions

At what points, if any, is $f(z)$ analytic? $f(z)=(2x+y−x^2y)+i(3+2y−xy^2).$ Please help, very confused.. I know how to compare the C-R equations and know how to find them. One set yields the unit ...
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1answer
35 views

Show that there is a single real number θ such that $h(z)=e^{iz}k(z)$ for any $z$

Let $h$ and $k$ be complex valued functions each analytic on an open set containing the closed unit disk $B =\lbrace z \in \mathbb{C} \hspace{5pt} |\hspace{5pt} |z| \leq 1 \rbrace$, and suppose that ...
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0answers
43 views

Describe all real-valued functions which are analytic on $\mathbb{C}$

This is a homework question, so if I am wrong please do not explicitly give me the answer. Question: Describe all real-valued functions which are analytic on $\mathbb{C}$. My Answer: Given that we ...
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1answer
19 views

Prove that certain family is normal. Is the domain hypothesis not relevant?

Let $\mathbb{H} = \{ Im(z) > 0 \}$, and $H(\mathbb{H}) = \{ f: \mathbb{H} \rightarrow \mathbb{C} : f$ is holmorphic $\}$. Prove that $\mathbb{F} \subset H(\mathbb{H})$ defined as $\mathbb{F} = \{ ...
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1answer
46 views

Is zero part of this complex set?

I want to draw the complex set $\{z\in\mathbb{C}:0\leq \arg(z)\leq \pi/4\}.$ This set contains the area in the $1^{st}$ quadrant between the radial lines drawn at the two angels $0$ and $\pi/4,$ with ...
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2answers
42 views

Why does this inequality hold for $|y|\geq 1$?

My lecture notes use this inequality for a complex $z=x+iy$ with $|y|\geq 1$ $$|\cot(z)| \leq \frac{1+\exp(-2|y|)}{1-\exp(-2|y|)}.$$ How can I show it? My attempt: \begin{align*} |\cot(z)| &= \...
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0answers
41 views

On a property of a holomorphic function on a unit disk [duplicate]

Let $f$ be a holomorhic function on the open unit disk $D$. Suppose that there is an $r\in[0,1)$ such that restriction of $f$ to the annulus $$U=\left\{z\in C:r<\left|z\right|<1\right\}$$ is one-...
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0answers
40 views

theorem 2 of perfect powers with all equal digits but one

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets $a$ and $c$ not equal to ...
3
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2answers
64 views

radius of convergence of $1/(1+z^2)$ about $z=2$ using geometric series approach

I would like to calculate the radius of convergence of $f(z)= 1/(1+z^2)$ about $z=2$ using the geometric series approach. Let me first state that according to a theorem, the radius of convergence ...
0
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3answers
49 views

If $f : B(0,1) \rightarrow \mathbb{C}$ is analytic, and $\forall\ |z_n|\rightarrow 1$, {$f(z_n)$} is not bounded, then $f$ has at least one root

Title says all. I submitted my attempt at answering it. It would be interesting to see if anywone had a different idea though!
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2answers
47 views

Pointwise convergence of holomorphic functions

Let $(g_n)_n$ be a sequence of holomorphic functions on $U$, where $U$ is the open unit disk. Suppose the first $k$ derivatives of $g_k$ at zero all vanish, $g_k(0) = 0$, and finally that $g_n$ ...
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1answer
35 views

Complex analysis analytic functions

I need to tell where $f(z)$ is analytic.$f(z)=(2x+y-x^2y)+i(3+2y-xy^2)$. I found that the cauchy riemann eqns for $u_x=v_y$ but when comparing $-u_y$ and $v_x$ I get $-1+x^2=-y^2$ What does this ...
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2answers
61 views

If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$

Let , $\Delta = B(0,1)$. If $f:\Delta \rightarrow \Delta$ is holomorphic, and $f(0)=0$, prove that $|f(z)+f(-z)| \leq 2|z|^2$. My attempt so far: $\displaystyle f(z) = \sum_{n=0}^{+\infty} a_n z^n =...
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2answers
57 views

solutions of $\bar z = |z-2\Im(z)|^2$.

I need to find all the solutions of $\bar z = |z-2\Im(z)|^2$. I know that $z=x+iy$ and $\bar z=x-iy$ and then $2\Im(z)=2y$. But can someone show the algebra for what I do next?
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1answer
25 views

Schwartz lemma and higher derivatives

Let $F= \{f|f:D(0,1) \to D(0,1)$, holomorphic and $f(0)=a \}$ Using $φ_a(z)= {z-a \over 1- \bar az}$ and Schwartz lemma I can find an upper bound for $f'(0), f \in F$ How can I get a similar result ...
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1answer
71 views

Primitive of $\frac{1}{z-z_1}-\frac{1}{z-z_2}$ on an open set

Let $\Omega\subset\mathbb{C}$ be open and assume $z_1$ and $z_2$ belong to the same connected component of the complement of $\Omega$. First, prove that there exists a holomorphic function $f$ on $\...
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0answers
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Why does $\lim_{z \to c} f(z)(z-c)=0$ imply that $\int\limits_{\partial B}f(\zeta)d\zeta=0 $

I am using the book Theory of Complex Functions by Remmert. In chapter 7 there is a corollary that says for a function $f:D\to\mathbb{C}$, $D$ domain: If $f$ is bounded around $z$, then $\int\limits_{...
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2answers
73 views

A generalization of holomorphic functions

Let's fix a matrix $A\in M_{2}(\mathbb{R})$. Assume that the following vector space of smooth functions is closed under complex multiplication: $$\mathcal{S}_{A}=\{f:\mathbb{C}\to \mathbb{C}\...
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0answers
36 views

I want to take the Fourier transform of Schrodinger equation of hydrogen atom.

I have the Schrodinger equation of hydrogen atom as $$ i\hbar \frac{\partial \varphi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\varphi+V\varphi$$ where $\varphi$ is a function of $r, \theta, \phi,t$....
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0answers
43 views

Why isn't this a [PARTIAL] solution to Lewy's Example

I was considering the example of the equation from: https://people.maths.ox.ac.uk/trefethen/pdectb/lewy2.pdf $$ \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} - 2i(x + iy) \frac{\...
3
votes
1answer
63 views

A Contour Integral

I'm interested in computing the integral: $$ - \frac{1}{2 \pi} \int_{- \infty}^{\infty} dE \; \frac{e^{-iEt}}{E^2 - \omega^2 + i\epsilon}. $$ I have two small queries: How does one choose the ...
0
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1answer
39 views

Proof for continuous function on an open subset of $\mathbb{C}$ which take on a specific form.

This is a homework question from Complex Variables by Joseph L. Taylor. The questions reads as follows: Prove that if $f$ is a continuous function defined on an open subset $U$ of $\mathbb{C}$, ...
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1answer
31 views

alternative definition of analytic functions

Definition: A map $f : U → V$ between open subsets $U ⊂ \mathbb C^n, V ⊂\mathbb C^m$ is called complex differentiable if it is (totally) differentiable in the sense of real analysis and if the Jacobi ...