0
votes
0answers
12 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
2
votes
1answer
148 views
+50

Cauchy product $\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}a_{n-k}b_k$

I have been told that, if $\{a_n\}_{n\in\mathbb{N}}$, $\{a_{-n}\}_{n\in\mathbb{N}^+}$, $\{b_n\}_{n\in\mathbb{N}}$ and $\{b_{-n}\}_{n\in\mathbb{N}^+}$ are absolutely summable complex sequences, ...
0
votes
0answers
32 views

Function $f(z)=\frac{\sin \pi(z-\lambda_n)}{\pi(z-\lambda_n)}$ and infinite product.

What is the relationship between the infinite product $$\prod_n \left(1-\left(\frac{z}{\lambda_n}\right)^2\right), \ \ \ \ \ z\in \mathbb C, \lambda_n\in \mathbb R$$ and the function $$f(z)=\frac{\sin ...
7
votes
0answers
96 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
0
votes
0answers
42 views

complex limits, how to show they go to 0?

In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are: $e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number ...
1
vote
0answers
26 views

Boundedness of sequence

Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$ where the parameters $\alpha, ~\beta$ are complex numbers, and the ...
11
votes
5answers
175 views

Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$

Do you know any nice way of expressing $$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$ ? Some simple manipulations involving the integrals lead to an expression that also uses the hypergeometric series. ...
4
votes
3answers
172 views

How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?

In an older question here in MSE I've asked for the term for the "slicing" of a power series in partial series and have learned that it is "multisection". I' ve been looking at the behaviour of the ...
0
votes
2answers
57 views

How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
1
vote
1answer
27 views

Series of points in a bounded sector of a complex half-plane

The question is: consider an infinite sequence of points which lie in a bounded sector of the complex plane, whose angular width is strictly less than pi (that is, it's an open sector of a ...
0
votes
1answer
21 views

A question about the relation between divergence and absolute divergence.

Princeton Lectures in Complex Analysis by Stein and Shakarchi says the following: If $|z| > R$, then a similar argument proves that there exists a sequence of terms in the series whose ...
0
votes
2answers
64 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
1
vote
2answers
47 views

Sum of Complex series

Let $\theta\in\mathbb{R}$ and $\theta \neq k\pi$ for $k\in\Bbb Z$. By summing a geometric progression show that $$1 + e^{2i\theta} + e^{4i\theta}+e^{6i\theta} + e^{8i\theta}= ...
1
vote
2answers
67 views

When can an infinite sum and complex integral be interchanged?

Are there some conditions under which the following two are equal? $$\displaystyle \oint_C \sum f_n(z)= \sum \oint_C f_n(z)$$ In the case of real valued functions, the condition $f_n(z) \geq 0$ ...
0
votes
0answers
17 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
2
votes
1answer
41 views

Complex Analysis Weierstrass M-Test

Prove that each of the following series converges uniformly on the corresponding subset of $\mathbb C$: $$\begin{align*} \text{(a)} \; & \sum_{n=1}^\infty \frac{1}{n^2 z^{2n}}, & & ...
2
votes
0answers
49 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...
1
vote
1answer
30 views

Analytic Extension: Imaginary Stripe

I was always wondering the following: Given a real analytic function there exists a positive radius of convergence for every point. This won't be affected by allowing complex numbers so it extends ...
1
vote
1answer
22 views

Computing the radius of convergence of a given series

Could anyone help me with the following problem? I'm getting $1$ as the answer, but I found the solution (without justification) to this problem online and it says the answer is $1/2$. Determine ...
1
vote
1answer
31 views

Radius of convergence of series given radius of convergence of another series

I'm hoping someone might be able to verify my solution to the following problem: Suppose that the series $\sum c_n z^n$ has radius of convergence $R$. Find the radius of convergence of the $\sum ...
1
vote
1answer
70 views

Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
4
votes
1answer
95 views

What is $\zeta(n)$ as $n$ tends to $\infty$? How fast it goes to the limit?

What is $\zeta(n)$ as $n\to\infty$? How fast it goes to the limit?
2
votes
1answer
77 views

Power series difficulty

How would I find the region of convergence of the series of $\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting $\frac{z+1}{z-1}$ as $\frac{2}{z-1}+1$ but I don't think that helps. Thanks
0
votes
2answers
28 views

Radius of convergence query

Find the radius of convergence of the series of $\frac{2^n(4z-8)^n}{n}$ My answer: $(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let $c_{n}=\frac{2^{3n}}{n}$. Then $\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$ so ...
6
votes
2answers
166 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
2
votes
2answers
43 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
1
vote
0answers
12 views

Taylor's expansion of the singular part of an analytic function

Assume $f$ is analytic on the annulus $R_1<|z-a|<R_2$. Assume $R_1<r<|z-a|$. Define $f_2$ by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$ $f_2$ is analytic on $|z-a|>r$. ...
0
votes
2answers
38 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
2
votes
0answers
69 views

What is the limit of this sequence of complex numbers?

Let $z_1$ and $z_2$ be two complex numbers in the upper half-plane. Does the sequence $c_n = \exp^n\left(\sqrt{\log^n(z_1)*\log^n(z_2)}\right)$ converge to a fixed point as $n\to\infty$? If so, what ...
4
votes
5answers
239 views

How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$ \sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)} $$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
1
vote
0answers
28 views

Exchanging Limits in Series inversion

I have the Lagrange Bürmann formula as follows: $$\sum^{m-1}_{n=1}\frac{1}{n!}(w-b)^n\lim_{z \rightarrow a} \frac{d^{n-1}}{dz^{n-1}} \left(f^{'}(z)\frac{z-a}{g(z)-g(a)}\right)^n$$ Where g(z) is the ...
3
votes
1answer
79 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
1
vote
2answers
33 views

Limit of complex numbers' sequence (related to Möbius transformation)

Problem Let $T(z)=\dfrac{7z+15}{-2z-4}$. Let $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$ Find $\lim_{z_n \to \infty}z_n$ I am having a lot of difficulties trying to solve this. I've tried to find a ...
1
vote
0answers
59 views

how to prove $\sum_n |b_n|^2<\infty$

$\{b_n\}$ is a complex sequence, If for all $\ell^2$ sequences $a_n$, we have $\sum_n \bar{a}_nb_n$ converges . Prove that $\sum_n|b_n|^2<\infty$
1
vote
2answers
40 views

Equivalent metrics in $\mathbb C$ (via stereographic projection)

Problem statement Let $\overline{d}$ be the distance in $\bar{\mathbb C}$ induced by the distance of $\mathbb R^3$ through the stereographic projection, i.e., if $z,z' \in \bar{\mathbb C}$, we define ...
0
votes
0answers
35 views

series functions of complex variable $ z $ and alternating zeta function convergence

Let $f_n(z)$ and $p_n(z)$ two series functions of complex variable $z$ defined as the following: $f_n(z)$=$ \sum_{n}exp({(-1)}^{n-1}{n^{-z}})) $ $p_n(z)$=$exp(\sum_{n}({(-1)}^{n-1}{n^{-z}})$ ...
1
vote
0answers
21 views

How to characteristize or distinguish those lacunary series with same natural boundary from each other

There are lots of different lacunary series with integral coefficients and with same natural boundary. As we know,Some functions can be distinguished or characterized by their poles. Now,the ...
1
vote
1answer
36 views

Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
1
vote
1answer
33 views

Show that $\sum_{k=1}^N\frac{1}{(k+a)(k+b)}=\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k}-\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k+N}$

I am quite stuck on this problem and I don't know how to proceed. The question states: Let $a,b,N\in\mathbb{N}$, $b>a$, $N\geq b-a$. Show that ...
2
votes
1answer
38 views

If $\sum_{n=1}^\infty |a_n|^2<\infty$, Then : $\sum_{n=1}^\infty a_n$ Converges $\Leftrightarrow \prod_{n=1}^\infty(1+a_n)$Converges

Prove that for complex sequence $\{a_n\}_{n\in\mathbb{N}}$ : if $\displaystyle \sum_{n=1}^\infty |a_n|^2<\infty$, Then : $\displaystyle \sum_{n=1}^\infty a_n$ Converges $\Leftrightarrow ...
1
vote
1answer
37 views

Divergent Complex sequence with distinct points and without accumulation point tends to $\infty$?

Is it true that any Divergent complex sequence with distinct points and without accumulation point tends to $\infty$ ? (One can also replace distinctness condition by condition finitely repeating ...
3
votes
0answers
74 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
0
votes
0answers
41 views

Change of Double Sum to Single Sum for Large N

I am reading article on X-ray interference. In the article authors claim that it is possible to replace double sum $\frac{1}{N} \sum_{n=1}^{N} \sum_{k=1}^{N} e^{-i \phi_{n}} e^{i \phi_{k}}$ by ...
5
votes
4answers
165 views

How to find the following sum? $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - \frac{1}{{4n + 7}}} \right)} $

I want to calculate the sum with complex analysis (residue) $$ 1 - \frac{1}{7} + \frac{1}{9} - \frac{1}{{15}} + \frac{1}{{17}} - ... $$ $$ 1 + \sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - ...
1
vote
1answer
32 views

Is it correct to say that this series diverge in the given annulus?

Let's consider the annulus $1\le|z|\le2$ and $$f(z)=\frac{2z+3}{z^2+3z+2}$$ Find its Laurent series. First I simplify the expresion: ...
0
votes
1answer
26 views

Find Laurent's series of these two functions around $z_o$

Find the Laurent series of $f(z)=\frac{z}{(z+1)^2}$ around $z_o=-1$, and $g(z)=z\exp(\frac1{z+i})$ around $z_o=-i$. For $f$, what they're asking is to find the series in $0<|z+1|$. On the ...
1
vote
2answers
34 views

a series derived from a holomorphic function converges implies that the coefficients converge to $0$

Let $D=\{z\in\mathbb{C}\mid |z|<2\}$. Let $f:D\setminus\{\frac{i}{2}\}\longrightarrow \mathbb{C}$ be holomorphic with $f(z)=\sum_{n=0}^\infty a_nz^n$ for any $|z|<\frac{1}{2}$. Suppose $a_n\neq ...
2
votes
1answer
67 views

series of an arbitrary sequence multiplying 1/n

Let $(r_n)_{n=1}^\infty$ be an arbitrary sequence of numbers in $[0,1]$. The series $\sum_{n=1}^\infty\frac{1}{n^2\sqrt{|x-r_n|}}$ converges for almost all $x$ in $[0,1]$. Is it true or not true? I ...
1
vote
4answers
879 views

Lagrange's Trigonometric Identity

Lagrange's Trig identity is $$ 1+\cos\theta+\cos 2\theta +\cdots + \cos n \theta=\frac{1}{2}+\frac{\sin\frac{(2n+1)\theta}{2}}{2\sin \frac{\theta}{2}},\quad (0<\theta <2\pi). $$ How can we prove ...
1
vote
0answers
26 views

How do we show $\ln z=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$ for all $z\in\mathbb{C}$ with $|z-1|<1$?

Let $$g:B_1(1):=\left\{z\in\mathbb{C} :|z-1|<1\right\}\to\mathbb{C}\;,\;\;\;z\mapsto\ln z-\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$$ (1) In a first step, I'm asked to show, that $g$ is ...