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
24 views

Are complex numbers a trivial lie group of itself?

Let $z$ be a complex number, then let's define a map $e^{T(*)}$. Let $w = e^{T(z)}$, where $T$ is some real number. Then is $z$ a lie group of $w$?
2
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1answer
33 views

How to indentify the power series I need to use?

Let $$ f(z) = \frac{1}{(z - 4)(z + 8i)} $$ a) Find the domains where f(z) is valid b) Find its power series at such domains Considering three singularities, I believe the domains are: $$ D_{1} = ...
2
votes
1answer
55 views

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$ such that $$\limsup_{z\rightarrow 0}\left|\frac{f(z)}{\sin z}\right|<\infty$$ and $$\limsup_{z\rightarrow ...
4
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0answers
53 views

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ I tried with polar coordinate. So let $z=r(\cos \theta +i \sin \theta)$. the LHS $=|2z|=2|z|=2r$ The RHS, I do some ...
2
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0answers
32 views

Show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$

Let $a\in\mathbb{C}, |a|>e, n\in\mathbb{N_1}$ I have to show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$ First, $f(z)=e^z$ and $g(z)=-az^n$ are entire. On $\partial B(0,1)$ we have ...
0
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1answer
20 views

Mapping of region into a line

Define a function $f:\mathbb{C}\rightarrow\mathbb{C}$, and let $z=x+iy$, then $$f(z)=u(x,y)+iv(x,y)$$ Given that $f$ maps a region into a line, what is the relation between $u(x,y)$ and $v(x,y)$? I'm ...
2
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1answer
20 views

Evaluation of an integral using nonrigorous methods

I was trying to solve the following integral $$ G(\alpha,m,n)=\int_0^{\infty}\cos(2nx)e^{-\alpha x}x^{m-1}dx;n\in N,\alpha>0,m\ge1. $$ By doing a change of variable I brought it to the integral $$ ...
3
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2answers
28 views

Complex Analysis holomorphic function question

I have a Complex Analysis assessment question about holomorphic functions: Let f be a function on a plane and satisfies $f'(z) = f(z)$ and $f(0) = 1$ i) Give an example of a function with this ...
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0answers
15 views

Coupling complex functions

After several calculations I end up with two complex functions: $$g(z)=zA(z)+\overline{z}A(\overline{z})+z^{-1}B(z)+\overline{z^{-1}}B(\bar{z})$$ and ...
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2answers
19 views

Finding $f(i)$ for an Entire Function $f$ that Maps a Line to a Subset of Itself and Sends $1$ to $0$.

Consider the line $L=\lbrace x+iy: x=y\rbrace\subset \mathbb{C}$, suppose that $f:\mathbb{C}\to\mathbb{C}$ is entire and satisfies $f(L)\subset L$. Given $f(1)=0$, find $f(i).$ (Ans: $f(i)=0)$ I am ...
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0answers
10 views

Parametrizing regions of complex plane

Let $\Omega=\mathbb{C}\setminus \lbrace t e^{it} \ \vert t \in \mathbb{R}_{\geq0} \rbrace$ I need to write $\Omega= \coprod_{i=0}^{\infty} R_i$ where each $R_i$ is the region bounded by from $t=2k ...
1
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1answer
26 views

Complex gamma function

Let $z$ be a complex number with positive real part. By induction on $n$, show that $$ \int_{t=0}^1{t^{z-1}(t-1)^n}dt = \frac{n!}{z(z+1)...(z+n)}. $$ Substitute $t = \frac{u}{n}$ and let $n → ∞$ to ...
4
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1answer
47 views

Find all holomorphic functions $f$ satisfying $f(1-f(z))=f(z)$?

Find all holomorphic functions $f$ (on $\mathbb{C}$) satisfying $f(1-f(z))=f(z)$? First, every constant function $f(z)=w$ is holomorphic and satsfies our condition. Now assume $f$ is not constant. ...
1
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1answer
16 views

Find all differentiable equations using Cauchy-Riemann equations

Let $z=x+iy$ and $f(z)=u(x,y)+iv(x,y)$. I want to use the Cauchy Riemann equations to find all differentiable functions of the form $$Re( h(z))=2x^2+2x+1-2y^2$$ So I used the C-R equations with ...
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2answers
20 views

Show this is complex differentiable

Prove or disprove $f$ defined on a complex neighborhood of $0$, $$f(x)=\begin{cases} \frac{1}{1+\frac{1}{x}} &\text{ if } x\not= 0\\0 &\text{ if } x=0.\end{cases}$$ is complex differentiable. ...
0
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1answer
16 views

Convergency of the power series at two points

Consider the power series $$\sum_{n=0}^{\infty}a_{n}(z+3-i)^{n}.$$ The series converges at $5i$ & diverges at $-3i$. Then which is correct ? (a) convergent at $-2+5i$ & divergent at ...
2
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1answer
27 views

Complex Numbers - Finding Limits

$$\lim_{z\to 1+i}\frac{z^4 + 2i}{iz-3}$$ Attempt: I substituted $z = 1+i$ in the numerator and denominator: Since $i^2 = -1$ I got $(1+i)^4 = -4$ So, $$\frac{-4 + 2i}{i-4}$$
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1answer
50 views

All solutions to $ z^{4} = -4 - i16 \sqrt{5} $ [on hold]

I am working on some exercises for my introduction to complex variables class and I have no idea how to solve this question. Given that $ (\sqrt{5} - i)^{4} = z^{4} = -4 - i16 \sqrt{5} $ Find ...
1
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2answers
46 views

Principal value of Fourier Integral

I have tried to find the principal value of $$\int_{-\infty}^\infty {\sin(2x)\over x^3}\,dx.$$ As $ {\sin(2x)\over x^3}$ is an even function, its integral may not be zero in the given limits. I ...
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1answer
20 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
1
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1answer
19 views

complex analysis differentiation and existence of a point?

If $f(z) = z^3$ prove that there is no point $c$ on line segment $[1,i]$ s.t. $(f(i)-f(1)) / (i-1) = f'(c)$. So differentiating: $$f'(c) = 3c^2$$ $$3c^2 = (f(i)-f(1))/(i-1) = (-i-1)/(i-1) = i$$ ...
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1answer
18 views

Points of a connected open subset joined by a curve

Let $\Omega$ be a connected open subset of $\Bbb C$. Is it necessarily true that any two points of $\Omega$ can be joined by a non-selfintersecting curve, that is, an injective continuous map ...
3
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3answers
48 views

Determine the region of convergence of series of complex functions

I have this problem. Find the region of convergence of the following series of complex functions $$ \sum_{n=1}^\infty \frac{2^n}{z^{2n}+1} $$ The progress I have made so far is that when n goes to ...
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0answers
23 views

Determine the set of points that satisfy $Re\left(\frac{z-z_1}{z-z_2}\right) =0$ for $z_1,z_2$ are fixed

Determine the set of points that satisfy $$Re\left(\frac{z-z_1}{z-z_2}\right) =0$$ for $z_1,z_2$ are fixed here is what I got so far Let $z=x+iy$ $z_1=a+ib$ $z_2=c+id$ where $x,y,a,b,c,d \in R$. ...
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1answer
20 views

Writing same equation in different forms

I am working with a unit circle with imaginary integration. I know from experience that this can be written as $f(\theta)=\cos t+ i \sin t$ or $e^{i \theta } $ My question would be if i have a circle ...
1
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1answer
36 views

Why $f(z)=z^2$ is single valued?

Why $f(z)=z^2$ is single valued where $z\in\mathbb{C}$? From definition we have $$z^2=e^{2 \log z}=e^{2(\ln|z|+i(2k\pi+Arg(z)))}$$ I dont get it ;/ Maybe it's getting late.
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1answer
16 views

Proper holomorphic maps and the degree of the map

Suppose f is holomorphic and maps U onto V, both being disks. If f is proper, does this induce a well defined degree for f? And does the converse hold? What are some tools that can help me see if ...
2
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0answers
13 views

Generalization of the Jacobi-Anger expansion to higher harmonics

I know the Jacobi-Anger expansion relation which gives the Fourier development of $e^{i z \cos(\theta)}$ and ${ e^{i z \sin(\theta)} }$, such that $$ \begin{cases} e^{i z \cos(\theta)} = ...
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0answers
12 views

Show that there is no branch of $z^{\frac{1}{n}}$ for $n \gt 1$ in the domain $B(0,1)-\{0\}$

Show that there is no branch of $z^{\frac{1}{n}}$ for $n \gt 1$ in the domain $B(0,1)-\{0\}$ Suppose there is one. Then Since $z^{\frac{1}{n}}=e^{\frac{1}{n}\log z}$, it is equivalent to saying that ...
1
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0answers
9 views

$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

Let $f \in M_k(\Gamma)$ not null in $\mathbb{H}$. I want to show that there exists a $C \in \mathbb{C}^{\times}$ and $n\in\mathbb{N}$ with $f = C\Delta^n$. I think one can show that for a $k>0$ ...
1
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1answer
23 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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0answers
27 views

Cuscs of a subgroup of $\Gamma$

I'm going to be completely honest about this: I need the solution of this to get permitted to the exam in complex analysis. The topic is not even relevant for the exam and I am absolutely not able to ...
1
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2answers
25 views

Find all singularities of a function and determine its types

Find all singularities of a function and determine its types $$f(z)=\frac{e^{iz}-1}{\sin{z}}e^{\frac{1}{z}}$$ I already showed, that $f$ has poles at points $z=\pi n$ where ...
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0answers
26 views

Transfomation for evaluation of real integrals using complex analysis [on hold]

In questions regarding calculation of improper integrals e.g $\int_{-\infty}^{\infty}\frac{x^2+1}{x^4+1} dx$, you consider the function $f(z) = \frac{z^2 + 1}{z^4+1}$ and apply the Cauchy's residue ...
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2answers
27 views

Points on a straight line (Complex Analysis)

I encouter a problem in complex analysis course : Let $a, b, $ and $c$ be three distinct points on a straight line with $b$ between $a$ and $c$. Show that $\frac{a-b}{c-b} \in \mathbb{R}_{<0}$. ...
2
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0answers
28 views

Cauchy-Riemann Equations Written as Complex Conjugate

Apparently, it can be shown that the Cauchy-Riemann equations can be written simply as, $df/dz^*=0$. I do not understand how it does not immediately follow from this that $df/dz=0$. When we proved ...
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0answers
13 views

Continuity of a function on the ring of formal power series, with a metric defined.

Let $E$ be the ring of formal power series over a field $K$. Consider $S,\ T \in E$. Define a metric $d$ on $E$ by $d(S,T)=0\ $ if $S=T$ and $a^{(-k)}$ for $k=\mathrm{order}(S-T)$, where $a>1$ is ...
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2answers
24 views

An explanation of how the range of a complex function works? Specifically $f(z)=z^2$ for$Re(z)>0$, $Im(z)>0$ and in the first quadrant..

I'm going through this complex analysis textbook, and it tells me that the range of the aforementioned function is $Im(w) \geq 0$. To me, that makes no sense. Could someone explain that, by chance? ...
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1answer
29 views

sketching lines and curves in the complex plane.

Well, my question is I a have the line equation x=1 and I need to know wich is thw image under $w=z^{2}$, then I parametrize it like $\theta=\pi /2$, the next step was to squred it $(\theta-\pi ...
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1answer
15 views

Series Expanding a Function: Complex Answer?

I have $$ f(x)=2\arccos\left(\frac{x}{2}\right)-x\sqrt{1-\frac{x^2}{4}} $$ and my friend says that the series of this function about $x=2$ (truncated to the first term) given $x\leq2$ is ...
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0answers
17 views

Coefficient in Taylor Series expansion [duplicate]

Find the coefficient of $(z-\pi)^2$ in the Taylor series expansion around $\pi$ if $$f(z) = \begin{cases} \frac{\sin z}{z-\pi} & \quad, z \neq \pi \\ -1 & \quad, z=\pi \end{cases}$$
2
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1answer
21 views

Power series of dependent and independent variables

Let $f(z,w)$ be an analytic function in two variables where $w=w(z)$ is dependent on $z$ ($z$ is the independent variable). Then $f(z,w)$ has a power series expansion centered at $(z_0,w(z_0))$ ...
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1answer
64 views

How to define Square Root

I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
3
votes
3answers
51 views

Is the function complex differentiable at (0,0)?

(in Complex) $$ g(z) = \begin{cases} \frac{z^5}{|z|^4} & \text{if $z \neq 0$} \\ 0, & \text{if $z = 0$ } \end{cases} $$ For the function above, is it differentiable at $z=0$? I am trying to ...
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0answers
21 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
3
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1answer
41 views

Sketching curves in the complex plane

Well I really need your help here because I need to sketch the curve $|z-1|=1$ in the z-plane and then its image under the $w=z^{2}$ but the thing is that I dont know how to sketch that function. In ...
2
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0answers
20 views

Integrating around simple pole and semicircle

Let $f$ be a holomorphic function on $\mathbb{C}$ with simple pole at $z_0$. Then if $\Gamma$ is a circle around $z_0$ oriented counter-clockwise with radius $r$ and $r\rightarrow 0$, then ...
2
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1answer
39 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
1
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0answers
21 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
2
votes
2answers
36 views

Reflections of circles through a circle are circles

To make things easier, we will try to reflect some general circle through the unit circle. We can use the inverse of the Cayley transform to map our analytic arc in the $z$-plane to the real line in ...