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 ...
0
votes
1answer
70 views

Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
2
votes
1answer
82 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
2
votes
1answer
81 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
1
vote
0answers
39 views

Extracting coefficients from a transformed generating function

Let $G(z)=\sum_{k\geq 0} a_kz^k$ be a generating function such that $z^aG(1-z)=P(z)$, where $P(z)$ is a polynomial and $a$ is a positive integer. I'm interested in $P(z)[z^n]$, the coefficient in ...
3
votes
2answers
62 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
1answer
83 views

closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+…+\binom{n}{n}$ [duplicate]

closed form for $$\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$$ I tried to solve it by : $$\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}=\sum_{k=0}^{n/3}\binom{n}{3k}$$ ...
0
votes
1answer
41 views

Close form of a power series starting at $n=2$

This is the power series I am looking at $\sum_{n=2}^{\infty}{n(n-1)z^n}$. I want to find the closed form of this power series. This is my approach, if I divide the power series by $z^2$, then I ...
0
votes
1answer
58 views

Question on the Koebe-Bieberbach Theorem

Assume $f$ is injective and that $f(0) = 0$ and $f'(0) = 1$. The theorem states that $\exists r >0$ such that $D_r(0) \subset f(\mathbb D)$ and, at best, $r=1/4$ ($\mathbb D$ is the unit disc). ...
9
votes
1answer
428 views

Summation over Weierstrass $\wp$ functions

I've been trying to prove the following closed expression for a summation over Weierstrass $\wp$-functions: \begin{equation} \sum_{k=1}^{N-1} \wp_N(k) = ...
0
votes
1answer
33 views

About complex sum

Let $\left(c_{n}\right)_{n},\,\left(d_{n}\right)_{n}$ two successions of complex numbers and let $N$ a large natural number.Is it true that ...
0
votes
1answer
43 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 ...
0
votes
0answers
28 views

About complex exponential summation

Let $f:\,\mathbb{R}^{+}\rightarrow\mathbb{R},\, f\in C^{\infty}\left(\mathbb{R}^{+}\right)$ and such that $f\left(n\right)>0\,\forall n\in\mathbb{N}$. Let $c>0$ a real number, $N>0$ a large ...
2
votes
2answers
37 views

About complex sum and modulus

Let $\left(a_{n}\right)_{n},\,\left(b_{n}\right)_{n}$ two succession of non negative real numbers, $\left(c_{n}\right)_{n}$ a succession of complex numbers and $N$ a large natural number. Suppose that ...
0
votes
0answers
39 views

Determining the disc of convergence in two series and determining at which points on the boundary of the disc the series converges.

The two series are as follows: $f(z) = \sum\limits_{n = 1}^\infty n(z+1-i)^{2n}$ and $f(z) = \sum\limits_{n = 1}^\infty n^{-1}z^{n}$ I have worked out that the discs of convergence are, ...
0
votes
2answers
104 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 ...
0
votes
0answers
22 views

Is $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ a general analytic function?

I have an expression $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ where $c(k)\in\boldsymbol{C}$ and $k\in\boldsymbol{R}$. $f(z)$ is an analytic function, since it contains only non-negative ...
1
vote
0answers
53 views

Weierstrass and Borel summation

In the Wikipedia article on Borel summation, there is the following quote attributed to Gösta Mittag-Leffler: Borel, then an unknown young man, discovered that his summation method gave the ...
2
votes
1answer
62 views

Sum of sines $\sum_{k=0}^{n} \sin(\phi +k\alpha)$

I've got the following problem. I'd like to prove that $$\sum_{k=0}^{n} \sin(\phi +k\alpha) = \frac{\sin\left(\frac{n+1}{2}\right)\alpha + \sin\left(\phi + ...
3
votes
1answer
78 views

How to evaluate the summation $S_b$

This question is from my notebook, not hw or else, only exercise to understand better. I tried by myself. However, since my trail are too trivial, I dont need to write here. i am confused a bit. I ...
0
votes
2answers
127 views

Sum $\sum_{n=0}^{N} z^n=\frac{1-z^{N+1}}{1-z}$

My professor just gave this as given in class today and went onto something else. Show that $$\sum_{n=0}^{N} z^n=\frac{1-z^{N+1}}{1-z}.$$ I would like to know how he obtained this neat expression ...
4
votes
0answers
94 views

Simplifying a sum?

Define polynomials $P_{j,s}^{(r)}$ via the generating series $$\left(\frac{d^s}{dz^s}f(z)\right)^r=\sum_{j=0}^{\infty} P_{j,s}^{(r)}z^j,$$ where $r\geq 1$. Here, $f(z)=z+a_2z^2+a_3z^3+\cdots.$ I was ...
12
votes
2answers
258 views

An identity about the Pi and Riemann's zeta function

How to prove the following identity? $$\sum_{n=1}^{\infty}\frac{(m-1)^n-1}{m^n}\zeta(n+1)=\pi\cot\left(\frac{\pi}{m}\right)$$
25
votes
1answer
571 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
1
vote
1answer
31 views

Prove that for n~=n' sum is much smaller than the case with n=n'

Hi I want to prove that this summation is much smaller for $n\neq n'$ than for the case where $n=n'$. I have seen this fact with simulation results. But I don't know how to prove it in mathematics. ...
7
votes
2answers
2k views

Value of Summation of $\log(n)$

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
1
vote
1answer
106 views

How to calculate $\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$?

I was given an exercise: Calculate 1+$\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$ I recognize $$\sin(kx)=Im(cis(kx))=Im(cis^{k}(x))$$ and $$\sin^{k}(x)=(Im(cis(x)))^{k}$$ but I do not know ...
5
votes
3answers
2k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
0
votes
1answer
55 views

Complex analysis equality in a limited sum

Let $z=e^{i\theta}$ with $\theta \in [0,2\pi[$ . Consider the sum $$ \sum_{n=1}^{N} (e^{i\theta})^n. $$ How could this be equal to $$ \frac{1-e^{iN+T\theta}}{1-e^{i\theta}} \quad ? $$ I tried to ...
11
votes
3answers
3k views

How to prove Lagrange trigonometric identity [duplicate]

I would to prove that $$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$ given that ...
14
votes
5answers
918 views

Proving the identity $\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$

Can you help prove the functional equation: $$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$$ Specifically, I am looking for a solution using complex ...