0
votes
0answers
15 views

Calculate $\lim_{z\rightarrow -n} \frac{\Gamma'(iz)}{\Gamma^2(iz)}$

We know that: \begin{equation} \lim_{z\rightarrow -n} \frac{\Gamma'(z)}{\Gamma^2(z)}=(-1)^{n+1} n! \end{equation} What if there is $iz$ instead of $z$? i.e. \begin{equation} \lim_{z\rightarrow -n} ...
6
votes
0answers
111 views
+250

$-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)=S$

Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5),\quad H_n:=\sum_{k=1}^n\frac{1}{k}\ \ ...
0
votes
2answers
42 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
0
votes
0answers
12 views

normality of set of analytic functions whose derivative is normal

I have this question from old preliminary exam problem set. (a) Show that if F⊂H(G) is normal then F′={f′:f∈F} is also normal. (b) Does F⊂H(G) normal imply F′={f: f′∈F} is normal? Otherwise give a ...
2
votes
1answer
30 views

Find the Laurent series of the function and give the residue

I have $f(z)={z^2\over {z^2-1}}$. I want to find the Laurent series of $f$ and the residue at the point $z_0=1$. Can I say that $f$ has a pole of order 2 at $z=1?$ Or is that only used when the ...
1
vote
0answers
33 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
0
votes
2answers
45 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
1
vote
1answer
44 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
2
votes
3answers
116 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.
1
vote
0answers
30 views

Conditional convergence of $\sum _{n=1}^{\infty} a_n$ and $\sum _{n=1}^{\infty} \log(1+a_n)$

In the Ahlfors Complex Analysis book section on infinite products, there is a result that the the series $\sum _{n=1}^{\infty} |a_n|$ converges exactly when $\sum _{n=1}^{\infty} | \log(1+a_n)|$ does. ...
11
votes
1answer
179 views

Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$

The so-called "two squares theorem" can be proven by establishing the following identity: $$\left(\sum_{n=-\infty}^\infty e^{\pi i \tau n^2}\right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi ...
4
votes
0answers
77 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
0
votes
1answer
33 views

Finding the radius of convergence of a series with only even powers.

I have the series $\displaystyle\sum_{n=0}^{\infty} \dfrac{z^{2n}}{(2+i)^n}$, and I need to find its radius of convergence. I cannot think how to arrange this so I can find $ R= \lim ...
1
vote
1answer
43 views

Which of the following set is countable?

Let $X$ be the set roots of unity in $\mathbb C$. Let $S(X)$ be the set of all sequences of elements in $X$. Which of the following subset of $S(X)$ is countable ? The set $ \ \ A \ \ $ of all ...
1
vote
2answers
63 views

Infinite sums and integrals using residues

I have no idea how to solve these two, any help? $\mathtt{i)}$ $$\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\frac{e^{tz}}{\sqrt{z+1}}dz$$ $$ a,t\gt0$$ $\mathtt{ii)}$ $$ \sum_{n=1}^\infty ...
0
votes
2answers
35 views

Radius of convergence of the given power series in powers $(z+2)^n-1$.

$$\sum_{n=0}^{\infty}\frac{(z+2)^{n-1}}{(n+1)^24^n}$$ Please help me solve this question on finding the radius of convergence..
1
vote
1answer
41 views

Proof complex series

I have to prove this: $\displaystyle\sum_{n=1}^\infty n\alpha^n = \displaystyle\frac{\alpha}{(1-\alpha)^2}$ if $|\alpha | < 1$ I think this is a geometric series, and i have to solve it with a ...
2
votes
2answers
92 views

Sum of $\sum\limits_{n=1}^\infty q^n \sin(nx)$

How to find $\sum\limits_{n=1}^\infty q^n \sin(nx)$, where $|q|<1$ and $x \in \mathbb{R}$? I was thinking about rewriting it as $\sum\limits_{n=1}^\infty (q(\Im(\cos x+i\sin x)))^n$. It is a ...
0
votes
1answer
156 views

Equality of two series.

EDIT: Consider $m,n\in\mathbb{Z}$ with $\frac{m}{n}\in \mathbb{R} -\mathbb{Z}$ and $n>1$. Given integers $m,n$ I want to prove that $$\bigg\lfloor \frac{m}{n} ...
6
votes
3answers
305 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
2
votes
1answer
26 views

Effect of sigmas inequality on sequences

We have two nets of complex numbers $\{z_\alpha\}_{\alpha\in I},\{w_\alpha\}_{\alpha\in I}$ for some set $I$ which might be uncountable, and we have $$\sum_{\alpha\in I}|w_\alpha|\leq ...
0
votes
1answer
24 views

Integrating Trigonometric Series

Let $f(x)=c_0+c_1e^{i\theta}+c_2e^{2i\theta}+...+c_ne^{ni\theta}$ where $c_k\in \mathbb R$. We need to show $$\int_{0}^{2\pi}f(e^{i\theta})\overline {f(e^{i\theta})} d\theta =2\pi\sum_{k=0}^{n} c_k$$ ...
1
vote
1answer
30 views

Is this complex function harmonic?

Let us consider the following convergente series in the set $0<x<1$ and all real $y$: $$h(x+iy)=∑_{n=2}^{∞}(-1)ⁿ⁻¹((n^{2x-1}-1)/n^{x})n^{iy}$$ My question is: Is this complex function ...
1
vote
1answer
39 views

Compact convergence of series

I am trying to show that $$f(z) = \sum_{k=1}^{\infty} \frac{(-1)^{k}}{z+k}$$ converges compactly over $\mathbb{C}$ and starting to think that this statement is false after several attempts. If I ...
0
votes
0answers
31 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...
0
votes
1answer
27 views

Multi part problem to prove functional relation of the exponential function

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that $\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = ...
0
votes
0answers
21 views

Show that the full series have no zeros in $A$

Let us consider the convergent series $h:Ω→ℂ$ given by $$h(α,β)=∑_{n=2}^{∞}(-1)ⁿ⁻¹a_{n}(α)n^{iβ}$$ where $α,β$ are reals in the domain of convergence and $a_{n}(α)$ is a real increasing sequence with ...
0
votes
1answer
44 views

Prove triangle inequality for complex series

First thing I notice is the strict inequality.. shouldn't it be $\leq$? In any case, how can I do this one?
1
vote
2answers
69 views

Evaluating $\lim_{n\to\infty}\left(\frac{1-i}{4}\right)^n$

It's been a while since I have taken Calculus II so my experiences on sequences and series has gone down the drain. I'm trying to find the limit of the sequence ...
0
votes
1answer
34 views

Conjugate of a complex series is series of the conjugates

I am talking about part (i). We know it's convergent so it converges to some $\large z_s \in \mathbb{C}$ , don't know how to show it's equivalent to the series of the conjugates though.
3
votes
4answers
159 views

$\forall t \in \mathbb{R}, e^{i \alpha_n t} \rightarrow 1 \implies \alpha_n \rightarrow 0$? [duplicate]

$\alpha_n \in \mathbb{R}$ is a fixed sequence of real numbers, for which the following holds: $$ \forall t \in \mathbb{R}, \lim_{n \rightarrow \infty}e^{i\alpha_nt} = 1 $$ Is it necessarily the case ...
0
votes
1answer
35 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
1
vote
1answer
45 views

Question on $2^N$th Roots of Unity within a function.

Prove that, if $w$ is a $(2^N)$th root of unity, where $N \in \mathbb N$, then: $$\lim_{r\to 1^-}|f'(rw)| = \infty$$ Where: $$f(z) = \sum\limits_{j = 1}^\infty 2^{-j}z^{2^j}$$ I haven't done left ...
0
votes
2answers
92 views

Prove that $f(z) = \sum\limits_{k = 1}^\infty \frac{z^{2^k}}{2^k}$ is continuous in the closed unit disc and holomorphic inside it.

I have started off by assuming that there is a disc of radius $r$ for which $|z|<r$ for $r \in (0,1)$ and $z \in D_r$. This implies that $|z|^{2^k} < r^{2^k}$ And after that, I don't know ...
1
vote
2answers
40 views

$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$

I'm reading the proof that $$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$$ There is a function $$h(z) =\pi \cot (\pi z) -[ \frac{1}{z} + \sum_{n ...
1
vote
1answer
64 views

Stirling approximation / Gamma function

Is it possible to obtain the Stirling approximation for the factorial by using the gamma function ? $$\Gamma(z) = \lim_{n \to +\infty} \frac{n! n^{z}}{z(z+1) \dots (z+n)}$$ Any hint ?
5
votes
2answers
175 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and maybe easy question, but I was not able to find an answer. If $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a linear change of coordinates it can be brought to the ...
1
vote
1answer
46 views

A rearrangement of an absolutely convergent complex series is also absolutely convergent

I just completed the following proof. Is it valid? Let $\sum_{k=1}^{\infty} a_k$ be an arbitrary convergent series that also converges absolutely. Then $\sum_{k=1}^{\infty} a_k \in \mathbb{C}$ and ...
1
vote
1answer
44 views

Proving convergence of real and imaginary parts

I am trying to prove that a complex sequence $(z_n)$ converges if and only if $(\Re(z_n))$ and $(\Im(z_n))$ converge. Now $\impliedby$ was straightforward, but I got a bit stuck with $\implies$: ...
3
votes
2answers
80 views

Show that $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$

Problem: I need to show that the power series $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$. I tried to prove it by contradiction by assuming that diverges for finitely ...
4
votes
2answers
158 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
0
votes
1answer
29 views

Is there a ratio (root) test for complex valued sequences?

I am reading a book about complex analysis (Complex Analysis by Ian Stewart/David Tall). Can we use the ratio test (or the n-th root test) for complex valued sequences as well? If so, how can this ...
6
votes
1answer
159 views

The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the ...
1
vote
2answers
96 views

Finding the $n$th term of some sequences given by its few first terms

Does anyone know how to find the $n$th term of the following sequences: $(1)\ $ $1,0,0,0,1,0,0,0,1,0,0,0,1,....\\$ $(2)\ $1,0,0,0,-1,0,0,0,1,0,0,0,-1,....\ $ $(3)\ $ $1,0,-1/2,0,1/3,0,-1,4,0,...\\$ ...
1
vote
1answer
52 views

Radius of Convergence of $\sum\ z^{n!}$

Does anyone know how to find the radius of convergence of the series $\sum\ z^{n!}$, where $z$ is a complex number? I tried to use the definition: $\frac{1}{R}=Limsup|\frac{a_n+1}{a_n}|$, but I ...
1
vote
0answers
21 views

Complex integral with branch cuts.

The problem is the following, $$ \ \int_{-\infty}^{\infty} du e^{-iu w }\bigg( \cos (\theta u) - i \frac{\xi }{\theta} \sin (\theta u)\bigg)^{-1/2} $$ When we go to complex $u$ plane there are branch ...
3
votes
1answer
64 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
3
votes
3answers
72 views

Radius of convergence of the series $\displaystyle\sum\limits_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n}$

I am doing the following problem and would like to know whether my answer is correct or not: Find the Radius of convergence for the complex series $\displaystyle\sum\limits_{n=1}^n ...
2
votes
1answer
45 views

$F$ entire with $\lim_{k \rightarrow \infty} F(z + N_{k}) = h(z)$ for every $h$ entire

Prove that there exists an entire function with the following "universal" property: given any entire function $h$, there is an increasing sequence $\{ N_{k}\}_{k=1}^{\infty}$ of positive integers ...
1
vote
0answers
40 views

Behaviour of $\sum_{n=1}^{\infty}\frac{z^{n}}{n}$ for $|z| = 1$ [duplicate]

How to show that the power series $$\sum_{n=1}^{\infty}\frac{z^{n}}{n}$$ ( which has radius of convergence $1$ ) converges in all points of $\partial D(0,1)$ except $z = 1$ ?