Questions tagged [exponential-sum]

For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.

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Asymptotics for an exponential sum

I am trying to find an equivalent of : $\displaystyle{S_n=\sum_{k=1}^n}e^{i\sqrt k}$ I tried to elucidate the asymptotic behaviour of the subsequence : $$S_{n^2-1}=\sum_{k=1}^{n^2-1}e^{i\sqrt k}=\sum_{...
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Solution (recurrence relation) of non-linear DE using the method of power series

I have to solve this non-linear DE $y' -e^y -x^2 = 0 , y(0)=c$ using powerseries. $y(x) = \sum_{n=0}^\infty a_{n}x^n $ $y'(x) = \sum_{n=1}^\infty na_{n}x^{n-1} $ so we get $\sum_{n=1}^\infty na_{n}x^{...
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calculation / lower bound of $\sum _{k=0}^d \frac{e^{-\frac{k^2}{2}} \binom{d}{k} \binom{p-d}{d-k}}{\binom{p}{d}}$

Let $p>d>0$ be given integers. Is there any trick how I can analytically calculate / simplify the following term: $$\sum _{k=0}^d \frac{e^{-\frac{k^2}{2}} \binom{d}{k} \binom{p-d}{d-k}}{\binom{p}...
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2 votes
1 answer
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Evaluating an exponential sum along an arithmetic progression

I am familiar with the identity $$\sum_{1 \leq n \leq q}e^{2 \pi i n^2 / q} = \left( \frac{1+ (-i)^q}{1-i} \right)\sqrt{q}$$ I am wondering if there is a similar evaluation for the sum $$S(r):= \sum_{...
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Proving that periodic vector-valued function satisfies $f(\mathbf x)= f(\mathbf x')$ iff $\mathbf x' = \mathbf x + k + \mathbf am$

In order to create a layer for a neural network that uniquely encodes relative relationships between a set of angles, I'm looking for a differentiable function with some specific properties; I'm ...
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Exponential sums with the divisor function

Can anyone explain the first $\ll $ on line 8 on page 188 of "Jutila: On exponential sums involving the divisor function" for me? (https://eudml.org/doc/152693) Specifically, I think he is ...
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Number of zeros for a system of sum of exponential functions

We consider two integers $N,M>1$, and the system of sum of exponential functions $\displaystyle \sum_{i=1}^N a_{i,j}\,{\rm e}^{\sum_{k=1}^Mb_{i,k}\,x_k}=0,\quad j\in {1,\dots,M}.\qquad (*)$ In ...
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Calculate time until accrued compound interest reaches a fixed amount.

Suppose I start with an amount $a_i$ of each of several assets indexed $i\in \mathbb{I}$. The fixed interest rate associated with each asset is $r_i$. After some time $\Delta t$ the amount held of ...
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How to get closed form when $a(n)\alpha+b(n)\beta_0+c(n)\beta_1=3^m\alpha+\cdots$?

Let $n=(1b_{m-1}b_{m-2}\cdots b_1b_0)_2$, and we know $$\begin{aligned}g(n)&=a(n)\alpha+b(n)\beta_0+c(n)\beta_1\\ &=(\alpha\beta_{b_{m-1}}\beta_{b_{m-2}}\cdots\beta_{b_1}\beta_{b_0})_3\\ &=...
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2 answers
39 views

How to show $\displaystyle\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$

This result seems trivial, how would I show $$\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$$ Where $0 \le k,l \le n-1$ And $k \neq l \neq 0$. I tried to use the exponential definition ...
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Limit of summation of $\sum_{i=0}^{\infty} e^{-ik}$ series

I was wondering whether anyone was able to give some insight to determining the limit of the following summation $$ \sum_{i=0}^{\infty} e^{-ik} $$ If anyone has any idea or is able to point me in the ...
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Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time?

Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time? I'm trying to find the upper bound for the following series $$s(n) = 2^1 + 2^2 + 2^4 + 2^8 + 2^{16} + ...
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How to find real and imaginary parts of this complex exponential sum?

I understand how to find real and imaginary parts of regular complex exponentials using Euler's formulas, but this one stumps me: $$6e^{-0.3t+0.8 \pi t j}$$ By using rules of exponents, I can do the ...
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Series for the higher derivatives of the Zeta function around S=1 with Stieltjes constants?

The series for $\zeta'(s)$ about $s=1$ is: $-(\frac1{(s-1)^2}) -\gamma_1 +\gamma_2(s-1)-\frac12\gamma_3(s-1)^2$.... where $\gamma$ are the Stieltjes constants. I'm using this series to determine the ...
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2 votes
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72 views

Fourier series approximations for characteristic functions of intervals in $[0,1]$

The following technique for approximating the characteristic function of intervals in $[0,1]$ using Fourier series with coefficients having inverse quadratic decay is given in the book Diophantische ...
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exponential pattern question??

I guess this is a exponential pattern question? You have 365 individual numbers in groups of 7 The first group of 7 has an individual unit value of .01 With each successive group of 7, the individual ...
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Sums of powers of exponentials

Suppose that the identity $$e^{i \theta_1} + \ldots + e^{i \theta_n} = e^{i k \theta_1} + \ldots + e^{i k \theta_n}$$ holds true $\forall k \neq 0$. Is then true that we must have $\theta_i = 0$, $\...
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Product representation of some exponential functions?

I was looking at some old questions of mine and stumbled upon this quesiton, which I could not solve: Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The ...
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3 votes
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109 views

How to find the Gauss sum $\displaystyle \sum_{x=0}^{p-1} e^{2\pi i(ax^2+bx+c)/p}$?

How to find the sum: $\displaystyle \sum_{x=0}^{p-1} e^{2\pi i(ax^2+bx+c)/p}$ where p is prime ? I'm sure I need to use this: $\displaystyle \sum_{x=0}^{p-1} e^{2\pi i(x^2)/p}$ = $\sqrt{p}$ when $ p≡3(...
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Is there a closed form expression for the following exponential sum?

As part of a task given to me, I need to find a value/expression for the below sum: $$\sum_{a=0}^{2N} e^{\frac{i \pi a^2}{2N}}$$ I've tried plugging it into python but isn't the nicest thing and I've ...
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Do complex exponential equations have solutions in general?

Suppose we have an equation of the form $$ \sum_{j=1}^n w_j \exp (x_jt)=0 $$ for a set of (real?) coefficients $w_j$ and $x_j$ and for $n>1$. Is it in general true that there is a (complex) $t$ ...
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Sum of exponentials - derivation

In a couple of scientific papers, I've found this sum identity involving exponentials: $\sum_{n=0}^{\frac{E_{BD}}{\Delta E}} e^{\gamma(n \Delta E - E_{BD})} = \frac{1}{1 - e^{\gamma\Delta E}}$ ...
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3 votes
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Standard way to evaluate $\sum_{k=0}^\infty \frac{x^k}{(k+1/2)!}$?

Is there any standard way to evaluate the following summation? $$ S(x):=\sum_{k=0}^\infty \frac{x^k}{(k+1/2)!} $$ where $(k+1/2)! = (k+1/2)(k-1/2)\ldots (1/2)$. EDIT : As some of you asked for ...
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How to calculate sum of a sigma notation for sum expression with exponents?

So I have got this practice problem $$\sum_{k=1}^n \frac{(-1)^k\cdot2^{2k}}{3}$$ Now I am fairly new to the sigma notation for the sum of elements. I know the basic stuff like the sum of the ...
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Is there an expression for $\sum_{k=0}^{\infty} 2^{-(N^k)}$ in terms of an integer $N>1$? [closed]

I was trying to solve for the limit as $n\rightarrow\infty$: $\displaystyle A_n=\sum_{i=1}^{n} a_i; a_i=\frac{2^{-i}}{i}$ and I landed at the inequality $\frac{N-1}{N}\left(S_{n+1}^{(N)}-a_{1}\right)&...
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Why is $\frac{|1 - e^{2 \pi i \alpha N}|}{|1 - e^{2 \pi i \alpha}|} \leq \frac{\sin(\pi \alpha N)}{\sin(\pi \alpha)} \leq \frac{1}{||\alpha||}$?

I came across this chain of inequalities in notes I am reading. $|\sum_{1 \leq n \leq N}e^{2 \pi i \alpha n}| \leq \frac{|1 - e^{2 \pi i \alpha N}|}{|1 - e^{2 \pi i \alpha}|} \leq \frac{\sin(\pi \...
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1 vote
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Finding the initial payment that is exponentially increasing each month.

For example: After $12$ months, I want to have paid a total of $\$100,000$. Each month I want to increase my payment by $12\text{%}$ over the previous month. (1.12x) How would I go about finding the ...
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0 votes
1 answer
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Summation of Exponential

The equation below is a posteriori density under the assumption that the prior distribution for $\mu$ is normal. And I wonder why do the highlighted portions where $\mu$ and $x_{k}$ are swapped when ...
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-2 votes
1 answer
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Does this sum have a closed formula? [closed]

$$c^1 + c^2 + \cdots + c^n = c\frac{c^n - 1}{c-1}$$ Right? What about $c^\frac{1}{m} + c^\frac{2}{m}+ ... c^\frac{n}{m}$? Does it also have a closed formula? Thanks a lot!
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4 votes
1 answer
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Exponential sums and Partial fractions

Let $e(x):=e^{2 \pi i x}$. For integers $j$ and $k \geq 2$, is there a closed formula for $$ \sum_{\ell=1}^{k-1} \frac{e\left(\frac{j \ell}{k}\right)}{1-e\left(\frac{\ell}{k}\right)}? $$ For certain $...
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Invertibility of special Hankel matrix

At first the simple case. Let $h_k\in \mathbb C$ of the form $$h_k=\sum_{j=1}^Mc_j\;z_j^k\qquad\qquad \text{for }k\in\mathbb Z_{>0}.$$ Where $c_j\in \mathbb C$ are some weights and $z_j\in \mathbb ...
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Real numbers non linear equations system with exponentials

let $A_1 , A_2, \cdots A_p$ be any $p$ real numbers such that : $$A_1 + A_2 +\cdots +A_p =0$$ $$e^{A_1}+ e^{A_2}+ \cdots +e^{A_p}=p$$ Do we necessarily have $A_1 = A_2 = \cdots = A_p = 0$ ?
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1 vote
1 answer
173 views

What is $\sum_{i = 1}^n i^i$? How about $\sum_{i = 1} ^ n i ^ {1/i}$? [closed]

I was doing some sums when this idea popped into my head. What is the $$\sum_{i=1}^n i^i$$ I have been trying to find a relation using induction but hadn't had any succes. Any other ideas? What about ...
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7 votes
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For which $r \in \mathbb R$ is the series $S(r)$ finite?

For each $r \in \mathbb R$ we let $$L_r := \left\{ \begin{pmatrix} a+cr \\ b+dr \\ c \\d \end{pmatrix} : a,b,c,d \in \mathbb Z \right\}, \quad W:= \left\{ \begin{pmatrix} 0 \\ 0 \\ x_3 \\ x_4 \end{...
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1 vote
1 answer
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Modified exponential summation [duplicate]

How do we prove that $$1+\frac{2^3}{2!}+\frac{3^3}{3!}+\frac{4^3}{4!}+\cdots=5e$$ I'll post the answer to this question as a knowledge share.
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Absolute value of the theta function on the unit nome

I would like to evalue the absolute value of the theta function with unit nome, i.e. when $|q|=1$ or equivelantly when $\tau\in\mathbb R$. The theta function is expressed as $$\vartheta(z;q=e^{\pi i \...
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  • 411
3 votes
1 answer
85 views

Show that the Kloosterman Sum $S(a,b,p^k)$ vanishes under the given condition.

Let $a,b,k\in \mathbb{N}$, $k>1$. $p $ be an odd prime number and $p$ does not divide $a,b$. The Kloosterman sum is given by $$S(a,b,p^k)=\sum_{x\in (\mathbb{Z}/p^k \mathbb{Z})^\times}e\Big(\frac{a ...
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3 votes
1 answer
218 views

Is there a closed form for $ \sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y} $?

I would like to find a closed formula for this equation: $$ \sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y} $$ Both the denominator and also the exponent is changing in each step. How is it possible ...
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1 vote
1 answer
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Bounds on $\lim_{m\to\infty}\sum_{s=x}^m\frac{s!}{s^n(s-x)!}$?

Anyone familiar with this exotic infinite sum? I'm assuming $n$ and $x$ are positive integers as well as $n-x>1$ (for convergence). $$\lim_{m\to\infty}\sum_{s=x}^m\frac{s!}{s^n(s-x)!}$$ One can ...
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0 votes
0 answers
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Complex representation of a real wave

I have the following formula for a wave: $u(x)=a \cos(kx-\omega t + \phi)$. I am trying to recover the complex representation of it $u(x)=\Re(a e^{i\phi}e^{i(kx-wt)}$. I started by using Euler's ...
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  • 111
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Infinite sum over $e^{i \frac{\pi (n-x)^2}{a}}$

I would like to evaluate the infinite sum $$\sum_n \exp \left( \frac{i\pi}{a} (n-x)^2\right)$$ where $x,a\in \mathbb R$. For certain values of $a$ it becomes a Delta comb of various frequencies, ...
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  • 411
1 vote
1 answer
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Can I treat $\Delta t$ in different ways when $\lim_{\Delta t \to 0}$? i.e. $\Delta t = 0$ in one section and $\Delta t = d t$ in another?

The Problem Note: This is a possible solution to a question I asked a short time ago. I am trying to prove that this sum $$ \frac{1}{2 \sqrt{\pi}} \sum_{j=1}^{\infty} \sqrt{\frac{1}{j \Delta t}} (\cos{...
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1 answer
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Is this complex exponential conversion correct?

I am trying to convert $5\cos(100\pi t + 25)$ to a complex exponential. I get that Euler's formula is such that $\cos(nt) = \frac{1}{2}(e^{nit} + e^{-nit})$. I have my answer as $\frac{5}{2}(e^{(100\...
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0 votes
1 answer
62 views

how to show convergence and evaluate $n\sum_{k=1}^{2n} \frac{e^{-n/k}}{k^2}$

I tried this : $$\lim\limits_{n \rightarrow +\infty}n\sum_{k=1}^{2n} \frac{e^{-n/k}}{k^2} = \int_{0}^{1} \frac{e^{-1/x}}{x^2} dx$$ to evaluate, and to show convergence, I think $$n\sum_{k=1}^{2n} \...
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2 votes
3 answers
158 views

Delta function and $\sum_{t}\exp\{ i k t\} $

I am interested in evaluating the following, which appears in an integrand, $$ \sum_{t=0}^{\infty}e^{ikt} $$ where $k$ is real. Using the following relation $$ \sum_{t=-\infty}^{\infty}e^{ikt} = 2\pi\...
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0 votes
2 answers
54 views

Solving a Recurrence with a Summation of Exponents as a Term

This is totally a homework problem, but it's due in 27 minutes so I know I won't get anything in time for that. I don't want the answer, I just want to learn the technique. I've started a MS program ...
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0 answers
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Expansion of $b^{(x+a)^{-n}}$ where $0 \le b \le 1$

The expansion of the negative binomial series is given below. $$ (x+a)^{-n} = \sum^\infty_{k=0} (-1)^k \begin{pmatrix}n + k - 1\\k\end{pmatrix} x^ka^{-n-k} $$ when $|x| < a$. What would be the ...
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2 answers
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$|\sum\limits_{k\in \mathbb{Z}}P(X=k)e^{ikt}|=1$ implies $X$ is almost surely constant

Question : Let's suppose $P$ is a probability, $X$ a random variable defined on $\mathbb{Z}$. Let's suppose we have $|\sum\limits_{k\in \mathbb{Z}}P(X=k)e^{ikt}|=1$ for all $t\in \mathbb{R}$. Prove ...
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1 vote
1 answer
98 views

Closed formula for sum of exponentials

I'd like to have a closed formula for $\sum\limits_{i=0}^n e^{\sqrt{i}x}=1+e^{\sqrt{1}x}+...+e^{\sqrt{n}x}$, something similar to the formula $\sum\limits_{i=0}^ne^{ix}=\frac{e^{(n+1)x}-1}{e^x-1}$. ...
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0 answers
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summation methode of Fejér

let's define $f:\mathbb{R} \to \mathbb{C}$ a $2\pi$ periodic continuous function. Let's say $s_{n}(x)=\sum_{k=-n}^{n} f\hat (k) e^{ikx}$. Let's say $t_{n}(x)=\frac{1}{n+1}\sum_{k=0}^{n} s_{k}(x)$. ...
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