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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2
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1answer
18 views

Sketch $ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $

Sketch the following $$ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $$ I have considered this geometrically and ended up thinking that the complex numbers $z$ must satisfy $$0 < ...
0
votes
0answers
28 views

reference needed for a property of the Riemann sphere

I need reference citations for these fact, or the theorem's name for them The bijective conformal mappings from the Riemann sphere to itself are Möbius transformations. Another one is Every ...
0
votes
0answers
9 views

Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$

Let $z_1$ and $z_2$ be 2 elements of $C _∞$ . Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$ (the ...
3
votes
2answers
69 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
0
votes
1answer
34 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
2
votes
1answer
34 views

Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number.

Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number. Since $Im\frac{z-z_1}{z-z_2 }=0$, then there is no imagining part in ...
2
votes
3answers
29 views

Find taylor of $\psi (z)$ where $(e^z-1)^2=z^2 \psi (z)$ - first 3 terms

I was asked to find the first three terms in the taylor series of $\psi (z)$ around $z=0$ where $(e^z-1)^2=z^2 \psi(z)$ and I'm having a few difficulties. My original idea was to say $\psi ...
0
votes
1answer
17 views

Show that a complex map is onto

I consider $\mathbb{C}$ as a real vector space. For $(a,b) \in \mathbb{C}^{2}$, consider the map : $F_{a,b} \, ; \, \mathbb{C} \, \rightarrow \, \mathbb{C}^{\ast}$ such that : $$ \forall z \in ...
0
votes
0answers
18 views

Continuous complex function from Rudin's Real and complex analysis

Lemma 10.29 from Rudin's Real and Complex Analysis, p. 314 of the third edition states that "if $f \in H(\Omega)$, then $g:\Omega \times \Omega \to \mathbb{C}$ defined by \begin{equation} g(z, w) = ...
0
votes
0answers
20 views

Understanding Green's Theorem

When looking at Goursat's theorem in complex analysis, I came across the Wiki proof which involves beautiful application of Green's theorem. I saw Greens theorem simply as "connection between line ...
1
vote
1answer
41 views

Are complex numbers a trivial lie group of itself? [on hold]

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 $w$?
3
votes
1answer
40 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
72 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
votes
0answers
61 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
votes
0answers
35 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
votes
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
votes
1answer
21 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
votes
2answers
29 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 ...
0
votes
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 ...
0
votes
3answers
28 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 ...
1
vote
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
vote
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
votes
1answer
51 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
vote
1answer
20 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 ...
1
vote
2answers
23 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
votes
1answer
17 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
votes
1answer
28 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, $$\frac{-4 + 2i}{i-4}$$
-3
votes
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 ...
2
votes
3answers
62 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 ...
0
votes
1answer
22 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
vote
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: $$f'(c) = 3c^2$$ $$3c^2 = (f(i)-f(1))/(i-1) = (-i-1)/(i-1) = i$$ ...
1
vote
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
votes
3answers
58 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 ...
0
votes
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
vote
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.
0
votes
1answer
18 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
votes
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)} = ...
0
votes
0answers
13 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
vote
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
vote
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 ...
1
vote
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
vote
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 ...
1
vote
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 ...
1
vote
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 $\frac{a-b}{c-b} \in \mathbb{R}_{<0}$. ...
4
votes
1answer
40 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 ...
1
vote
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 ...
1
vote
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? ...
1
vote
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 ...
1
vote
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 ...
0
votes
0answers
17 views

Coefficient in Taylor Series expansion [duplicate]

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