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, ...

learn more… | top users | synonyms (2)

0
votes
1answer
54 views

How to find a conformal mapping that transfers the first quadrant onto the upper half plane?

How to find a conformal mapping that transfers the first quadrant $D_0 = \{(r, \theta) \mid 0 \leq r < \infty\}$ onto the upper half plane $D_1 = \{w \mid \operatorname{Im}(w) \geq 0\}$ such that $...
0
votes
1answer
36 views

About the convergence of the complex series $\sum_{k=1}^{\infty}\frac{1}{k(k+z)}$

I'm struggling to show that the series $$\sum_{k=1}^{\infty}\frac{1}{k(k+z)}$$ converges for all $z\in \mathbb{C}\setminus \{-1,-2,\cdots\}$. WolframAlpha says it indeed does converge, but I have no ...
-1
votes
1answer
32 views

$r(\theta) e^{i\theta}$ - Parametrization of the square $\gamma$

In the question Contour integral - Circle instead of a square achille hui explains that $r(\theta) e^{i\theta}$ with $r(\theta)\min\left(\frac{1}{|\cos\theta|}, \frac{1}{|\sin\theta|}\right)$ is a ...
4
votes
2answers
136 views

Analytic continuation for $\zeta(s)$ using finite sums?

$\zeta(s)$ converges for $\sigma >1$ but not for $\sigma =1/2.$ But for some reason for $s = 1/2 + i t $ and fixed finite $N,~$ $\zeta_N(s) =\sum_{n=1}^N\frac{1}{n^s}$ is very close to $\zeta(s)$ ...
2
votes
2answers
73 views

Proof that a Function is Entire

For every $z \in \mathbb{C}$, let us define \begin{equation} H(z) = \int_{0}^{\infty} t^{-t} e^{tz} dt. \end{equation} I have tried to prove that $H$ is holomorphic in the whole plane and that \begin{...
0
votes
1answer
60 views

Contour integral - Circle instead of a square

I would like to solve the integral $\int_{\gamma} \frac{1}{z \bar{z}} dz$. Here $\gamma \subset \mathbb{C}$ is a square centered at the origin and where his vertices are parallel to the axes. Could I ...
1
vote
1answer
54 views

Milne-Thompson Theorem with a Vortex

I'm doing a problem related with Milne-Thompson theorem which tells that: "A cylinder of radius $a$ is immersed in a counter-clockwise whirlpool, which we model here as a potential vortex of intensity ...
0
votes
0answers
11 views

Proving a series is holomorphic

Let $\chi$ be a Dirichlet-character modulo $N$. I would like to prove that $$g(s) := \sum_{p} \sum_{n \geq 2} \frac{\chi(p)p^{-ns}}{n}$$ is holomorphic for $\mathrm{Re}(s) > \frac{1}{2}$. This from ...
0
votes
1answer
23 views

$\int_{\gamma} \frac{dz}{z}$ - Circle immersed the point $z=0$

I would like someone explain to me why in the resolution of $\int_{\gamma} \frac{dz}{z}$ where $\gamma$ is the square of vertex $-1-i$, $-1+i$, $1-i$ and $1+i$, it is possible to take a unit circle $\...
2
votes
1answer
103 views

$\int_C \frac{\log z}{z-z_0} dz$ - Cauchy theorem with $z_0$ outside the interior of $\gamma$

Let the domain $O=\mathbb{C}-(-\infty,0)$, the point $z_0 \in O$ and the circle $\gamma=C(0,r<|z_0|)$ in the positive direction. Compute $\int_C \frac{\log z}{z-z_0} dz$. How do I solve the ...
0
votes
1answer
14 views

Developing complex product to derive identity

I'm referring to this question : Finding a trigonometric polynomial The OP says : On the unit circle $$f(\theta) = F(e^{i\theta}) = c \prod_{j=1}^n\frac{(e^{i\theta}-\beta_j)(1-\overline\beta_j e^...
1
vote
1answer
35 views

In using $\int_C(z+\frac{1}{z})^{2n}\frac{1}{z}dz=\binom{2n}{n}2\pi i$, how does it possible to compute $\int^{\pi}_{-\pi} \cos^{2n} t dt$?

In using the result $\int_C(z+\frac{1}{z})^{2n}\frac{1}{z}dz=\binom{2n}{n}2\pi i$, how does it possible to compute $\int^{\pi}_{-\pi} \cos^{2n} t dt$. A hint given by the textbook is to compute $\int^...
2
votes
1answer
61 views

Determine all the biholomorphic functions $\mathbb{C}\rightarrow \mathbb{C}$.

Determine all the biholomorphic functions $\mathbb{C}\rightarrow \mathbb{C}$. My attempt: First, we show that $z_{0}=0$ is not essential singularity of $g(z)=f\left(\frac{1}{z}\right)$, indeed, if $...
1
vote
1answer
55 views

Solutions to $z^5+2+e^z=0$

I am trying to show that this equation has exactly three solutions in the half plane $\{z:\,\Re (z)<0\}$. Ideas My first thought was to use Rouche's theorem. Since it applies to closed contours, ...
0
votes
1answer
22 views

$\sum^{2n}_{k=0} \binom{2n}{k} \int_C \frac{z^{2k-2n}}{z} dz$ - Cauchy theorem

I have to compute $\int_C(z+\frac{1}{z})^{2n}\frac{1}{z}dz$, where $n \in \mathbb{N}$, and $C$ is the unit circle with positive orientation. How could I compute $\sum^{2n}_{k=0} \binom{2n}{k} \int_C \...
0
votes
1answer
21 views

Complex conjugate of an analytic function $\phi$ is constant where $\bar{\phi}(x)=\overline{\phi(x)}$

Let $\phi:U\rightarrow \mathbb{C}$, where $U\subseteq\mathbb{C}$ is open and connected, and $\phi$ is analytic in $U$. Assume that $\bar{\phi}:U\rightarrow \mathbb{C}$ is analytic in $U$ where $\bar{\...
0
votes
0answers
33 views

the set of Dirichlet series converging for $Re(s) > \sigma$

$F(s) = \sum_{n=1}^\infty a(n) n^{-s}$ converging for $Re(s)$ large enough. let : $$A_1(x) = \sum_{n \le x} a(n), \qquad \qquad A_{k+1}(x) = \sum_{n \le x} A_k(x)$$ is it true that $\scriptstyle \...
0
votes
1answer
36 views

If $\partial A \subseteq L = \bigcup\limits_{i = 1}^n {{L_i}} \subseteq A$ then $\partial A = \bigcup\limits_{i = 1}^k {{L_i}} $?

Let $A,L_i\subset \mathbb{C}$ and $L = \bigcup\limits_{i = 1}^n {{L_i}} $ $\partial A \subseteq L \subseteq A$ $\partial A$ is boundary of $A$. $A,L_i$ have Euclidean topological. $L_i$ are ...
1
vote
2answers
47 views

Finding the power series of $\frac{1}{z^2}$

I need help finding the power series for this function around $z=1$: $$f(z)=\frac{1}{z^2}$$ My attempt to solve this: We know: $\frac{1}{1-x}=\sum_{n=0}^\infty x^n$, so I tried substituting $z=w-1$.$$...
0
votes
2answers
56 views

Solve $\cos z=2$ in $\mathbb{C}$

Solve $$\cos z=2 \qquad z \in \mathbb{C}$$ I consider $z=x+i y$, so: $$\cos(x+iy)=\cos(x) \ \cos(iy)-\sin(iy) \ \sin(x)=\cos(x) \ \cosh(y)-i \ \sinh(y) \ \sin(x)$$ I have to satisfy this ...
0
votes
0answers
30 views

proving a function has a primitive

If a; b are distinct complex numbers let $z(t) = (1-t)a + tb$; $0 <= t<= 1 $, be the segment joining a to b. (a) Prove that, outside this segment in the complex plane, the ratio z-a/z-b is ...
2
votes
0answers
20 views

How can I justify the swap of the limit with the contour integral?

I want to show that if $f$ is analytic then $$f^{''}(w) = \frac{1}{\pi i}\oint_C \frac{f(z)}{(z-w)^3}\,dz.$$ Consider the following: \begin{align} f''(w)&=\lim_{h\rightarrow 0} \frac{f'(w+h)-f'(...
4
votes
1answer
70 views

How to prove this complex inequality elegantly?

Question Let $z_{1,2}\in U(0,1)\subset \Bbb C$, prove that $$\frac{|z_1|-|z_2|}{1-|z_1||z_2|}\le\left|\frac{z_1+z_2}{1+\overline{z_1}z_2}\right|\le\frac{|z_1|+|z_2|}{1+|z_1||z_2|}$$ Actually I haven'...
0
votes
1answer
25 views

Fundamental theorem of calculus for analytic functions.

I have to solve the following problem but I have no idea where to start. Any hint or suggestions could be really helpful, thanks! Suppose the continuous functions $f(e^{i \theta})$ on the unit circle ...
3
votes
1answer
72 views

How to prove $|y|\le |\sin(z)|\le e^{|y|}$ in complex analysis where $z=x+iy$?

How to prove $$|y|\le |\sin(z)|\le e^{|y|}$$ in complex analysis where $z=x+iy$. I don't think I need the entire solution. May be just a set up or approach. I started with sin(z) formula and reached ...
1
vote
3answers
52 views

Find the laurent series for $\frac{1}{z(z-2)^2}$ centered at z=2 and specify the region in which it converges.

My attempt: $$\frac{1}{z(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{A}{z}+\frac{B}{z-2}+\frac{C}{(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{(1/4)}{z}+\frac{(-1/4)}{z-2}+\frac{(1/2)}{(z-2)^2}$$ This is ...
1
vote
2answers
31 views

Inverse of the Cross Ratio for Mobius Transformation from Circle to Circle

I'm reading Conway's complex functions of one variable, and in chapter 3 he goes over Cross-Ratios. He defines the cross ratio to be $(z,z_1,z_2,z_3)=\frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, where ...
3
votes
1answer
38 views

Cauchy's theorem for contour integration

I have to compute $\int_C(z+\frac{1}{z})^{2n}\frac{1}{z}dz$, where $n \in \mathbb{N}$, and $C$ is the unit circle with positive orientation. So let $z(t)=\cos (t) + i \sin (t)$, with $-\pi \leq t <...
1
vote
1answer
23 views

Solving for $z$ given 3 constraints: $\DeclareMathOperator{\Re}{Re}\Re[z^4]=1/2$ , $z\bar{z}+2|z|-3=0$, $\arg z \leq \frac{\pi}{4}.$

Let $z$ be a complex number satisfying $$\DeclareMathOperator{\Re}{Re}\Re[z^4]=\frac{1}{2}$$ $$z\bar{z}+2|z|-3=0$$ $$\arg z \leq \frac{\pi}{4}.$$ Find $z$ As a side note this ...
1
vote
1answer
103 views

The simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial? [closed]

Why the following is true? «The simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial?»
0
votes
0answers
41 views

Integral over a square

I want to integrate $f(z)= 1/ (z \bar{z})$ over a square centered at the origin oriented counter clockwise. I know this function is holomorphic except at $(0,0)$ My teacher says this integral is ...
0
votes
1answer
22 views

How to show: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$

I am struggling to show below in a big question: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$ Tried to analyse with geometric ...
2
votes
0answers
16 views

Values of Eisenstein Series

I'm trying to prove the algebraic independence of $\pi, e^\pi$ and $ \Gamma(1/4)$ while using Nesterenko's Theorem ($\{q, P(q), Q(q), R(q)\}$ contains at least three algebraically independent numbers ...
1
vote
1answer
47 views

sum of roots of unity multiplied by k+1

I'm considering the following sum: $$\sum\limits_{k=0}^n (k+1)\epsilon^k,$$ where $\epsilon=e^{\frac{2\pi i}{n}}$. I write the sum as $$\frac{\rm d}{{\rm d}\epsilon}\sum\limits_{k=0}^n \epsilon^{k+1}=\...
1
vote
1answer
43 views

Number of solutions to a polynomial within unit disk

Consider the polynomial: $$f(z)=z(z^4+1)$$ I was asked to find all $z$ on the unit circle such that $\Im[ f(z)]=0$, and to find the correspond values for $\Re[f(z)]$ - this I have done. I then had to ...
2
votes
2answers
32 views

Cauchy integral formula and contour

I was doing fourier transform of lorentzian function $$ B(k)=\frac{1}{1+\left(\frac{k-k0}{\Delta k}\right)^2} =\frac{\Delta k^2}{\Delta k^2+\left(k-k_0\right)^2} $$ $B(k)$ is neither even nor odd ...
1
vote
1answer
53 views

Complex analysis limit problem [closed]

I've recently started complex analysis and I'm having a bit of trouble with these questions: Find the limit as $n$ approaches infinity of $z^n$ where: (a) $|z|>1$ (b) $|z|\le1$ where $...
1
vote
1answer
51 views

Complex Numbers trig equation

I am struggling with proving $|\sin(z)|\geq|\sin (x)|$ in complex numbers where $z=x+iy$. I kinda know that I have to use trig identities and what not, but cannot find the approach. I posted a picture ...
2
votes
1answer
43 views

Why does a circle in the complex plane get “deformed” under a complex map? [closed]

So I am currently learning complex analysis. I am told that small circles centred at $\alpha$, when mapped under $f$, are scaled by $\lvert f'(\alpha)\rvert$ and rotated anticlockwise by $\...
0
votes
0answers
24 views

Convexity of a complex function

I have a problem in which I need to find the minimum of the function $f:\mathbf{C}\rightarrow\mathbf{R}$ given by $$f(s) = v^*M(s)^*M(s)v$$ with $v \in \mathbf{C}^{n+1}$ and so that $|v_i|$ ...
1
vote
2answers
16 views

Is infinity allowed in the definition of a fractional linear map?

I need to create a fractional linear map of the form $$ F(z) = \frac {az+b}{cz+d}, $$ where a,b,c, and d are complex numbers such that $ad-bc \neq 0$ and $F(0) = 1$, $F(1)=\infty$, and $F(\infty)=0.$ ...
2
votes
3answers
39 views

Radius of convergence involving $z^{n^2}$

Consider the following complex power series: $\displaystyle\sum_{n=1}^\infty a_nz^{n^2}$ where , $\displaystyle a_n = \frac{1}{n!}$. My approach is that by ratio test $$\lim\limits_{n\to\infty} \...
1
vote
2answers
41 views

Complex singularities of $f$ - Radius of convergence

A example in my textbook explain that $x^2-2x+2=0$ has the solutions $x = 1± i$. The distance between $x=0$ and $x = 1± i$ is $\sqrt{2}$ in the complex plane. So the radius of convergence of the ...
1
vote
2answers
34 views

Radius of convergence - Complex plane

A example in my textbook explain that $x^2-2x+2=0$ has the solutions $x = 1± i$. The distance between $x=0$ and $x = 1± i$ is $\sqrt{2}$ in the complex plane. So the radius of convergence of the ...
2
votes
0answers
72 views

Optimal distibution of a roots of a polynomial

Let $p(x)$ be a monic polynomial with $n$ complex roots: $$ p(x)=(x-\lambda_1)(x-\lambda_2)\cdots(x-\lambda_n), \qquad \lambda_i \in \mathbb{C}, $$ Define $$ f(x)=\frac{1}{\big(1+|x|^{2}\big)^n}|p(x)...
2
votes
1answer
23 views

Prove: Show $\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$ for $f(z)=z^n$ and any $n \in N$

Prove that if $f:\mathbb{D} \to \mathbb{D}$ (where $\mathbb{D}$ is the unit disk) is given by $f(z)=z^2$, the for all $z \in \mathbb{D}$, we have $$\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$$ ...
3
votes
0answers
36 views

Degree of the Divisor of a Theta Function

Let $(\gamma_1, \gamma_2)$ be a base for a lattice $\Gamma$ in $\mathbb C$, and $\theta$ a theta function, ie an holomorfic function such that $\theta(z+ \gamma) = \theta(z)e^{2i\pi(a_\gamma z + b_{\...
3
votes
0answers
54 views

Modular Discriminant and Pentagonal Numbers

I am asked to show $$(2\pi)^{-12}\Delta(\tau) = q \cdot \Big (\sum_{n\in \mathbb{Z}} (-1)^n \cdot q^{(3n^2+n)/2} \Big)^{24}$$ where $\Delta:\mathbb{H} \to \mathbb{C}$ is the modular discriminant, $q=e^...
1
vote
1answer
25 views

On an integral inequality involving the characteristic function (probabilistic setting).

I am following Probability and Measure by Billingsley Could someone give me a couple of steps on how the above inequality was found?
1
vote
1answer
28 views

Criticize my math at finding radius of convergence of $\sum_{n=0}^{\infty} a^{n^2}z^{n}$

$\sum_{n=0}^{\infty} a^{n^2}z^{n}$ Let R be the radius of convergence. Then we have $\frac{1}{R} = \lim \sup |a^{n^2}|^{1/n} = \lim \sup |a|^{n} = \lim_{n \rightarrow \infty} [ \sup \{|a^{n}|,|a^{n+1}|...