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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6
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2answers
137 views

What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$?

This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer. I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + ...
3
votes
1answer
53 views

Almost everywhere convergence of some series

Let $\{r_n\}$ be an arbitrary numerical sequence. Prove that $\sum_{n=1}^\infty\frac{1}{2^n\sqrt{|x-r_n|}}$. Prove that it converges almost everywhere on $\Bbb R$.
0
votes
2answers
119 views

Calculating residue of $z\sin{ \frac {z+1}{z-1}}$

Let $f=z\sin{ \frac {z+1}{z-1} }$. Calculate the residue of $f$ in $z=1$. I think $f$ has an essential singularity at $z=1$ so the only way I can proceed is with Laurent series. I've defined ...
1
vote
3answers
106 views

Prove that $f$ is constant if $f'=0$

Suppose that $f$ is holomorphic on a domain $D$ and $f'=0$ on $D$. Prove that $f$ is constant on $D$.
1
vote
1answer
34 views

complex residue involving exponent of quotient of polynomials

I was trying to work out an integral and came to trying to find the complex residue of $$R = \exp\left(\frac{Ax^2 + Bix}{Dx + 1}\right)$$ at $x = -D^{-1}$. I used partial fractions to get: $$ = ...
1
vote
2answers
45 views

Limit concept under Complex analysis

Prove that $$ \lim_{z\to i} \dfrac{3z^4-2z^3+8z^2-2z+5}{z-i} = 4+4i $$
1
vote
1answer
134 views

One-one analytic functions on unit disc

Is the following statement true? Suppose, $ f:D\to \mathbb C $ is an analytic function where $ D $ is the unit disc of radius $ 1 $ around $0 $. Suppose, $ f $ is analytic on the boundary of $ D $ as ...
0
votes
0answers
27 views

Sketching sets in $\mathbb{C}$

This might seem too elementary but I'm teaching myself complex analysis and have very minor doubts about the following sets. I'm asked to sketch: A. $\{z \in \mathbb{C}: |e^{z^2}| \le e \}$ B. $\{z ...
0
votes
0answers
35 views

Is there a general notation for the Mobius group?

Let $a,b,c,d$ be complex numbers such that $ad-bc\neq 0$. Define $f(z)=\frac{az+b}{cz+d}$. Such function $f:\overline{\mathbb{C}}\rightarrow \overline{\mathbb{C}}$ is called a Mobius transformation. ...
2
votes
1answer
36 views

Prove that $\sum_{n=1}^\infty \frac{z^n}{z^{2n}+1}$ converges when $\|z\|\neq 1$

I considered the partial fraction decomposition, $$ \sum_{n=1}^\infty \frac{z^n}{z^{2n}+1}=\frac{1}{2}\sum_{n=1}^{\infty}\bigg(\frac{1}{z^n-i}+\frac{1}{z^n+i}\bigg) $$ is it sufficient to say that ...
4
votes
1answer
108 views

What's the intuition for the fact that $\mathscr{O}(-k)$ and $\mathscr{O}(k)$ are so different?

maybe this question makes no sense and I just cannot accept the fact that dual the line bundle is different from the respective line bundle itself. Since it looks like that manifolds are more ...
0
votes
1answer
42 views

About $e^{i \theta}$ and mapping complex number.

So, $e^{i \theta}$ is a unit circle centered at the origin, and $e^{i\theta}-1$ is a unit circle centered at $-1$. If we have a complex function $M_a(z) = \frac{z-a}{\overline{a}z-1},$ I am required ...
1
vote
1answer
27 views

Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
0
votes
0answers
217 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
2
votes
1answer
63 views

A Question about Non-Conservative Vector Fields

In my multivariable calculus class, we spent some time discussing the vector field that was the gradient of arctan(y/x). This field was shown to be non-conservative in closed regions which enclosed ...
3
votes
1answer
77 views

Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)

When I was reading "Advances in Moduli Theory" by Shimizu Yuji, I´ve found a weird way of writing the Kodaira-Spencer map $\rho$. For a given analytic family of complex compact manifolds $\pi ...
1
vote
1answer
25 views

How do i prove that Mobius transformation is continuous at its pole?

Let $a,b,c,d$ be complex numbers such that $c\neq 0$ and $ad-bc\neq 0$. Define $T(z)=\dfrac{az+b}{cz+d}, \forall z\in\mathbb{C}\setminus\left\{-\dfrac{d}{c}\right\}$. Define $T(-\frac{d}{c})=\infty$ ...
0
votes
2answers
21 views

Parametric equations in complex analysis

I am trying to find $\int_C (1+i-2z')dz$ where$z'$ is the conjugate of $z$ and where C is the parabola $y=x^2$ from $z_1=0$ to $z_2=1+i$. How do I write the parametric equations for this?
0
votes
1answer
21 views

Is there a general formula for range of complex inversion map?

The map $z\mapsto 1/z$ has a nice mapping property. That is, it takes a circle $|z-\alpha|=|\alpha|$ to a straight line in a complex plane. Is there a formula for this line? Moreover, when ...
0
votes
1answer
44 views

Maximizing the product of projections of a vector on another vectors

I want to get the $N\times1$ complex vector $\mathbf{x}$ which maximizes this real valued function $f=\mathbf{x}^{H}\left (\mathbf{a}_{1} \mathbf{a}_{1}^{H}\mathbf{x}\mathbf{x}^{H}\mathbf{a}_{2} ...
1
vote
1answer
50 views

Showing $f$ must be constant on $\mathbb{C}$ given 3 conditions

Suppose we know the following about a function $f(z)$. i. $f(z+1)=f(z)$ and $f(z+i)=f(z)$ for all $z$ in $\mathbb{C}$. ii. $f$ has only isolated singularities (if any) in $\mathbb{C}$ iii. $f$ has ...
2
votes
3answers
290 views

how to show that this complex series converge?

If $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}$$ Converges( s is real) and $\operatorname{Re}(z)>s$. Then $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{z}}$$ also converges. $a_n$ is complex sequence.
3
votes
3answers
82 views

Bound on $|e^z - 1|$

I'm trying to show that $|e^z - 1| < e - 1$ when $|z| < 1$. The hint is to consider $e^z - 1 = \int_0^z e^w dw$. Thank you!
0
votes
0answers
22 views

Showing $\int_{\gamma}f(z)dz = \int_{\gamma_1}f(z)dz + \int_{\gamma_2}f(z)dz$ with non analytic points.

Suppose $f$ is analytic on the complex plane except at $z_1,z_2$, that $\gamma_1$ and $\gamma_2$ are simple closed curves with $z_1,z_2$ in their interiors and $\gamma_1$ and $\gamma_2$ are in the ...
0
votes
4answers
63 views

Integral of a function $f:\mathbb{R}\rightarrow \mathbb{C}$

My real analysis book defines derivatives and integrals only for a function $f:A\rightarrow \mathbb{R}$, where $A\subset \mathbb{R}$. But, when talking about Fourier series, it comes out an integral ...
1
vote
1answer
28 views

A mapping that maps circles centered at origin to lines parallel the real axis

I need a mapping that maps a circle (with the center at the origin) to lines parallel to real axis:
1
vote
2answers
59 views

Conformal Mapping Between Two Domains (log)

Does anyone have a recommendation as how to go about solving this problem? I want a conformal from G to H where $$ G = \{ z \in \Bbb C \ | \ |z|<1, |z+i|>\sqrt{2} \}, S = \{ z \in \Bbb C \ | \ ...
3
votes
2answers
61 views

Non-constructive proof that $\sum_{j=1}^n j^k$ is a polynomial $p(n)$ of degree $k+1$

So it can be shown that there are special polynomials (I forget their name) $p_k$ of degree $k$ that satisfy $\sum_{j=1}^n p_k(j) = n^{k+1}$, and that these polynomials are linearly independent so ...
1
vote
0answers
18 views

doubt about definition of holomorphic polynomials

In a topic of several complex variable theory (in particular functions on $\mathbb{C}^2$), I came across a term homogeneous holomorphoic polynomial. By the word, I think it is a polynomial in complex ...
0
votes
0answers
39 views

Is there a good introductory complex-analysis text in general setting, namely Riemann sphere?

I have studied first 1~3 chapters of some complex analysis texts (Ahlfors, Conway, Silverman) Well, i specially like Ahlfors in many ways but this text doesn't seem to develop a theory in a general ...
0
votes
2answers
61 views

What is the definition for holomorphic functions on the Riemann sphere?

I'm trying to study complex analysis in general setting, but i have troubles with defining things in this general setting. I have skimmed definitions in Ahlfors, Conway and wikipedia, but these all ...
2
votes
1answer
48 views

Expand $\frac{z}{(z^2 + 1)(z^2 + 4)}$ for $1 < | z | < 2$ in Laurent series

Please help. Expand $\frac{z}{(z^2 + 1)(z^2 + 4)}$ for $1 < | z | < 2$ in Laurent series.
1
vote
1answer
37 views

Winding number of a loop

I am reading a paper and unfortunately I have found a problem. I have a continuos map $[0,1]\times [0,1]\to U_q(\mathbb{C})$, where $U_q(\mathbb{C})$ are unitary matrices. We can consider the map ...
0
votes
0answers
20 views

What is the domain of complex tangent function?

What is the domain of $\tan z =\frac{\sin z}{\cos z}$ ? Is the domain $D=\{z\in\mathbb{C} : \cos z \neq 0\}$? Or considering the Riemann sphere, is $\tan z$ defined on $D$ as $\infty$?
0
votes
0answers
36 views

Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
3
votes
1answer
91 views

Showing that $\int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} = \pi\sqrt{2}$

Show that $$ \int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} = \pi\sqrt{2} $$
1
vote
1answer
22 views

How to do Laurent Series Expansion

$f(z) = \frac{3}{(z-1)(z-7)}$ in the domain $\{z: 1<|z|<7\}$ I remember the first step is to put $f(z)$ in partial fractions.This gives me $ \frac{3}{8(x-7)}$ $- \frac{3}{8(x+1)}$ What the ...
1
vote
0answers
33 views

homeomorphism from $\mathbb R^2 \cup \infty$ to $S^2$

Check that the map $$\phi(z)=\left\{ \begin{array}{l l} \left(\frac{2rcos \theta}{r^2+1},\frac{2rsin \theta}{r^2+1}, \frac{r^2-1}{r^2+1} \right)& \quad \text{if $z=re^{i\theta} \in ...
1
vote
1answer
67 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
3
votes
0answers
81 views

the partial fraction of ${\pi}\cot{\pi}z$ from the partial fraction of $\frac{\pi^{2}}{\sin^{2}{\pi}z}$

I want to deduce the equation $${\pi}\cot{\pi}z=\frac{1}{z}+ \sum_{n=1}^{\infty} \frac{2z}{z^{2}-n^{2}}$$ where the convergence is uniform on compact subsets of $\mathbb{C}-\mathbb{Z}$ from the ...
1
vote
0answers
29 views

Limit of sequence of arguments

In an example in Brown and Churchill's complex analysis book they take a sequence $z_n=-2+i\frac{(-1)^n}{n^2}$ and show that $z_n \to -2$. However when using polar coordinates letting $r_n=|z_n|$ and ...
2
votes
1answer
24 views

Complex functions with certain properties.

I'm trying to prove that on $U=\mathbb{C}- \{0\}$, there exists no non-constant holomorphic function such that $f(z)=f(2z)\ \forall z\in U$, but on the other hand, that there does exist a nonzero ...
1
vote
1answer
84 views

Laurent series expansion, principal part

I need to find the principal part of the Laurent series for $f(z) = \frac{e^{2z}}{1-\cos(z)}$, around $z = 0$. Also, I have to use the undetermined coefficient method. I don't know how to proceed. ...
4
votes
1answer
145 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
0
votes
1answer
53 views

Prove that a pseudo-hyperbolic ball is a Euclidean ball. Find the radius and center of the Euclidean ball.

We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by $$ \rho(z,w) = |\phi_w(z)|, \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$ where $z,w \in \mathbb D.$ ...
1
vote
0answers
54 views

Constructing Riemann surfaces

At the risk of asking a question that has been already answered, I have been trying to figure out how to construct the Riemann surface of slightly more complicated examples, but after reading examples ...
1
vote
1answer
34 views

Finding a function $g$ which is analytic on the region $E=\{ z \in \mathbb{C}: |z| >1 \}$

Find a function $g$ which is analytic on the region $E=\{ z \in \mathbb{C}: |z| >1 \}$ and maps $E$ one-to-one onto $H= \{ w \in \mathbb{C}: Re w<0 \}$. My approach: The region $E=\{ z \in ...
0
votes
0answers
33 views

Question concerning the gamma function in relation to other holomorphic functions when $Re(\xi) > 0$

Let $f$ be indefinitely differentiable on $\mathbb R$ that has compact support. $\implies f$ belongs to the Schwartz space. Consider: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
0
votes
1answer
81 views

A problem about a sequence of polynomials

Let $\Omega = \{z: |z|<1 $and $|2z-1|>1\}$, and suppose $f \in H(\Omega)$. a) Must there exist a sequence of polynomials $P_n$ such that $P_n \rightarrow f$ uniformly on compact subsets of ...
0
votes
1answer
22 views

How do i analyze this complex diagram?

I'm asking how to analyze diagrams like this : http://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Complex_LogGamma.jpg/600px-Complex_LogGamma.jpg What do distinct colors here mean? What do the ...