Questions tagged [exponential-sum]
For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.
426
questions
0
votes
0answers
13 views
For which $r \in \mathbb R$ is the series $S(r)$ finite?
For each $r \in \mathbb R$ we let
$$L_r := \left\{ \begin{pmatrix} a+cr \\ b+dr \\ c \\d \end{pmatrix} : a,b,c,d \in \mathbb Z \right\}$$
and
$$ L^*_r := L_r \cap \{ x \in \mathbb R^4 : x_1 \ne 0 \...
1
vote
1answer
23 views
Modified exponential summation [duplicate]
How do we prove that
$$1+\frac{2^3}{2!}+\frac{3^3}{3!}+\frac{4^3}{4!}+\cdots=5e$$
I'll post the answer to this question as a knowledge share.
0
votes
0answers
30 views
Absolute value of the theta function on the unit nome
I would like to evalue the absolute value of the theta function with unit nome, i.e. when $|q|=1$ or equivelantly when $\tau\in\mathbb R$. The theta function is expressed as
$$\vartheta(z;q=e^{\pi i \...
3
votes
1answer
61 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
$$S(a,b,p^k)=\sum_{x\in (\mathbb{Z}/p^k \mathbb{Z})^\times}e\Big(\frac{a ...
3
votes
1answer
210 views
Is there a closed form for $ \sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y} $?
I would like to find a closed formula for this equation:
$$
\sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y}
$$
Both the denominator and also the exponent is changing in each step. How is it possible ...
1
vote
1answer
35 views
Bounds on $\lim_{m\to\infty}\sum_{s=x}^m\frac{s!}{s^n(s-x)!}$?
Anyone familiar with this exotic infinite sum? I'm assuming $n$ and $x$ are positive integers as well as $n-x>1$ (for convergence).
$$\lim_{m\to\infty}\sum_{s=x}^m\frac{s!}{s^n(s-x)!}$$
One can ...
0
votes
0answers
38 views
Complex representation of a real wave
I have the following formula for a wave: $u(x)=a \cos(kx-\omega t + \phi)$. I am trying to recover the complex representation of it $u(x)=\Re(a e^{i\phi}e^{i(kx-wt)}$. I started by using Euler's ...
0
votes
0answers
35 views
Infinite sum over $e^{i \frac{\pi (n-x)^2}{a}}$
I would like to evaluate the infinite sum
$$\sum_n \exp \left( \frac{i\pi}{a} (n-x)^2\right)$$
where $x,a\in \mathbb R$.
For certain values of $a$ it becomes a Delta comb of various frequencies, ...
1
vote
1answer
77 views
Can I treat $\Delta t$ in different ways when $\lim_{\Delta t \to 0}$? i.e. $\Delta t = 0$ in one section and $\Delta t = d t$ in another?
The Problem
Note: This is a possible solution to a question I asked a short time ago.
I am trying to prove that this sum
$$
\frac{1}{2 \sqrt{\pi}} \sum_{j=1}^{\infty} \sqrt{\frac{1}{j \Delta t}} (\cos{...
0
votes
1answer
22 views
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\...
0
votes
0answers
23 views
Fitting a sum of exponentials to data
How do I find $N_1$ and $N_2$, given $(t,N)$ in the model $$N=N_1 e^{-\lambda_1 t} +N_2 e^{-\lambda_2 t}+B$$ and the coefficients $\lambda_1 ,\lambda_2$?
Each exponential function is supposed to ...
0
votes
1answer
58 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 $$n\sum_{k=1}^{2n} \...
2
votes
3answers
111 views
Delta function and $\sum_{t}\exp\{ i k t\} $
I am interested in evaluating the following, which appears in an integrand,
$$
\sum_{t=0}^{\infty}e^{ikt}
$$
where $k$ is real. Using the following relation
$$
\sum_{t=-\infty}^{\infty}e^{ikt} = 2\pi\...
0
votes
0answers
42 views
Close form/Simplification of this sum?
Does anybody know the close form of this sum $S$ or a way of simplifying it?
$$
S = 1+\sum_n^N e^{-a_n}+\sum_{n_1>n_2}^N e^{-a_{n_1}}e^{-a_{n_2}}+\sum_{n_1>n_2>n_3}^N e^{-a_{n_1}}e^{-a_{n_2}}...
0
votes
2answers
43 views
Solving a Recurrence with a Summation of Exponents as a Term
This is totally a homework problem, but it's due in 27 minutes so I know I won't get anything in time for that. I don't want the answer, I just want to learn the technique. I've started a MS program ...
1
vote
0answers
24 views
Expansion of $b^{(x+a)^{-n}}$ where $0 \le b \le 1$
The expansion of the negative binomial series is given below.
$$
(x+a)^{-n} = \sum^\infty_{k=0} (-1)^k \begin{pmatrix}n + k - 1\\k\end{pmatrix} x^ka^{-n-k}
$$
when $|x| < a$.
What would be the ...
0
votes
2answers
49 views
$|\sum\limits_{k\in \mathbb{Z}}P(X=k)e^{ikt}|=1$ implies $X$ is almost surely constant
Question :
Let's suppose $P$ is a probability, $X$ a random variable defined on $\mathbb{Z}$.
Let's suppose we have $|\sum\limits_{k\in \mathbb{Z}}P(X=k)e^{ikt}|=1$ for all $t\in \mathbb{R}$.
Prove ...
1
vote
1answer
82 views
Closed formula for sum of exponentials
I'd like to have a closed formula for $\sum\limits_{i=0}^n e^{\sqrt{i}x}=1+e^{\sqrt{1}x}+...+e^{\sqrt{n}x}$, something similar to the formula $\sum\limits_{i=0}^ne^{ix}=\frac{e^{(n+1)x}-1}{e^x-1}$. ...
0
votes
0answers
17 views
Log-sum-exp as inverse temperature approaches zero.
I have a question similar to this one regarding the Log-sum-exponential function.
I want to show the continuity of the function with respect to the inverse temperature $\beta$ around $\beta = 0$.
...
0
votes
0answers
9 views
Lower bound for the convergence radius of the Poincaré serie
This is a technical follow up to this question of which I repeat the definition for clarity.
Let $(X,g)$ be a Riemannian manifold on which a group $G$ acts properly by isometries $G\curvearrowright X$....
0
votes
0answers
14 views
How can I make this given condition only dependent on the norms of the vectors, so thereby radial.
The following condition is given. Here $u, x_k\in \mathbb{R}^n \ \ \forall k \in \{1,...,m\}$,
$$\sum_{k=1}^{m}(vol(K_{k}))^{\frac{1}{2}}(vol(K_{j}))^{\frac{1}{2}}f_{k,j}(u)e^{-2\pi i
<u,x_k>}=...
0
votes
0answers
37 views
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 $t_{n}(x)=\frac{1}{n+1}\sum_{k=0}^{n} s_{k}(x)$.
...
0
votes
1answer
38 views
Determinant of matrix with power series terms
I'm trying to find an alternative to the Taylor series which represents some function $f(x)$ as a sum of exponentials.
$$f(x) = a_0 e^{0x} + a_1 e^{1x} + a_2 e^{2x} + a_3 e^{3x} + ...$$
The ...
0
votes
0answers
24 views
Sum of double exponentials with different patterns.
Is there a formula for this expression without using sum? $$f(x) = \sum^x_{i=0}{2^i\cdot3^{i(i+1)/2}} = 2^0\cdot3^0 + 2^1\cdot3^1 + 2^2\cdot3^3+ 2^3\cdot3^6+ 2^4\cdot3^{10} + . . . + 2^x\cdot3^{x(x+1)/...
2
votes
0answers
34 views
Sum of powers of cosines over dyadic rationals
Let $n$ and $k$ be positive integers. When $n$ is odd, one can show that: $$\sum_{j=0}^{2^{k-1}-1}\cos^{n}\left(\pi\frac{2j+1}{2^{k-1}}\right)=\begin{cases}
-1 & \textrm{if }k=1\\
0 & \textrm{...
0
votes
1answer
33 views
Simplifying exponential terms
How would I simplify the following equation to solve for $x$?
$$
\frac{\exp(\frac{1-x}{20}) + \exp(\frac{10-x}{20})}
{\exp(\frac{1-x}{5}) + \exp(\frac{10-x}{5})} = 4
$$
What about if there were ...
0
votes
0answers
14 views
Equation with factorial, exponential and Sum [duplicate]
I need to prove that :
$\mathrm{\sum_{n=0}^{\infty} {p^n\over n!} |p-n| = 2{p^p\over(p-1)!}}$
Knowing that $\mathrm{\sum_{n=0}^{\infty} {p^n\over n!} = exp(p)} $
I've tried many ways :
$\mathrm{\sum_{...
0
votes
1answer
110 views
How to prove that the softmax function is not invariant under scalar multiplication.
I want to prove that the softmax function is not invariant under scalar multiplication.
How to continue from there to prove that S(x)i is not equal to S(xc)i ?
1
vote
1answer
47 views
Need help in finding the sum of the following infinite series.
I need to find the sum of the infinite series with the following general term
$\frac{2^{nlogn}}{n!}$
I initially started off by getting rid of the log in the the numerator, so that later became $\frac{...
0
votes
0answers
19 views
How do you solve $\sum _i\:a^{\:c_ix}\:\:=\:b$
I was wondering how can we approach the sum of several powers with the same basis.
$$
\sum _i\:a^{\:c_ix}\:\:=\:b
$$
$a,b,c_i$ being known constants and solving for $x$.
I'm also more interested in ...
1
vote
1answer
56 views
Math Quiz question $(5b^4)^2$
So in this question, I got $25b^8$, though my teacher marked it wrong and said it was $5b^8$, he said don't multiply the bases. But $5$ is not the base, $b$ is the base and $5$ is the coefficient. I ...
4
votes
0answers
53 views
Invariant for interesting set of functions generalizing $\sin$ and $\cos$ and other properties
In an attempt to generalize $\sin(x)$ and $\cos(x)$—but just a curiosity—I found some functions with fairly interesting properties. The idea was to extent the definition of $\sin$ and $\cos$ as unique ...
2
votes
2answers
93 views
Finding the Closed Form of a Multivariable Exponential Summation
Here was a problem I thought of after seeing the 2017 HMMT #5:
For all positive integers $n$, what is the closed form of the summation of $\sum_{a+b+c+d=n}(3^a)(9^b)(27^c)(81^d)$, where $a, b, c,$ and ...
0
votes
0answers
99 views
Optimizing sum of two exponential functions
I want to find the optimal solution to the following seemingly sum of two exponential functions:
$$\max_{x \in [0, 1]}~f(x) = \underbrace{x\exp\left(-a\left(bx^t+1\right)^\frac{1}{1-t}\right)}_{f_{1}(...
1
vote
0answers
41 views
If the Infinite sum of a series is known, what is the sum of element wise product with another series?
Suppose we know the summation of some series $G(n)$ such that, $$\sum_{n=1}^{\infty}G(n)=S_1.$$
Now assume another summation $S_2$ is expressed as,
$$S_2=\sum_{n=1}^\infty G(n) e^{i\frac{2\pi}{m}n}; \...
0
votes
1answer
37 views
solving a function with series
Is it possible to find a solution for the following equation with respect to the parameter $\gamma_k$ where the equation is $$\alpha\gamma_k\beta_k-\alpha\ln\sum_{m=1}^K\exp(\gamma_m\beta_m)-\theta\...
0
votes
0answers
26 views
What is the sum of quadratic function?
I would like to know the result of the summation of $\sum\limits_{n=0}^{500}e^{i2{\pi}{an^2}/(b+cn)}$, in which a, b, and c are real constant value.
No need for an exact solution, an approximation ...
0
votes
1answer
65 views
How to get the bound of a summation changed from $e^x$?
I meet a problem with calculating
$$
\frac{e^{-|\alpha|^2}}{\sum A_k^2}\left|\sum_{k=0}^{\infty} A_k \frac{(-\alpha)^k}{\sqrt{k!}}\right|^2\,.
$$
Only I know is the constrain that $\sum_{k=0}^{\infty} ...
2
votes
0answers
36 views
Bounding the character sum $\sum e((ax^k+bx)/p)$
The following question is from Carlos Moreno's "Algebraic Curves over Finite Fields":
I am trying to solve part (i), but I am keep getting stuck at roughly the same place. Write $C: y=x^d$ ...
2
votes
0answers
44 views
Can we exactly calculate the sum $S=\sum\limits_{n=1}^{N} e^{2ib\cos n\theta} e^{-in\theta}$?
Consider the sum $$S=\sum\limits_{n=1}^{N} e^{2ib\cos n\theta} e^{-in\theta}$$ where $\theta=\frac{2\pi}{N}$, $b$ is any real number (not small), $N$ is an integer. Here, $n$ is also an integer which ...
1
vote
2answers
64 views
The closed form solution to the equation with sum of exponential functions
I have a function
$\begin{equation}f(x)=\sum_{k=1}^{n}\left(\frac{1}{n}+\frac{ax}{k}\right)e^{-a(\frac{nx}{k}-b)}\end{equation}$,
where $a,b$ are both positive constants, $n$ is a positive integer.
$x^...
1
vote
0answers
54 views
Almost-geometric series whose coefficients doubly exponentially decay up to $1$
Is there a nice formula for this function of $|x|<1$?
$$f(x)=\sum_{n=1}^\infty x^n\cos\left(\frac{3 \pi}{2\cdot5^n}\right)=\Re \sum_{n=1}^\infty x^ne^{\frac{3\pi i}{2}\cdot5^{-n}}$$
Ideally the ...
35
votes
0answers
1k views
A Nice Problem In Additive Number Theory
$\color{red}{\mathrm{Problem:}}$
$n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
7
votes
1answer
102 views
A curious property of exponential sums for rational polynomials?
An article led me to generate some graphs of exponential sums of the form $S(N)=\sum_{n=0}^Ne^{2\pi i f(n)}$, where $f(n)= {n\over a}+{n^2\over b}+{n^3\over c}$ with $a,b,c\in\mathbb{N}_{>0},\,$ ...
9
votes
0answers
289 views
Why should $\sum_{m=1}^N e(\alpha m^3)$ be big for some $\alpha?$
I'm going through a "circle method" proof of the fact that every large enough natural number $n$ is the sum of nine cubes. At some point a lot of control over the function
$$f(\alpha)=\sum_{...
1
vote
1answer
62 views
Equidistribution of powers of primitive roots modulo $p$
Let me start with a nice experimental observation. Fix a large prime, say $p = 5003$. It turns out that $g = 2$ is a primitive root mod $p$. If we plot the powers of $g \in \Bbb F_p^{\times}$ (...
6
votes
1answer
122 views
Deciding convergence/divergence of $\sum_{m \geq 1} \frac{1}{m^3} \sum\limits_{\substack{k=1\\(m,k) = 1}}^m k \sin\left(\frac{2\pi k n}{m}\right)$
Let $n$ be a positive integer. I am attempting to determine whether the series
$$
\sum_{m \geq 1} \frac{1}{m^3} \sum_{\substack{k=1\\(m,k) = 1}}^m k \sin\left(\frac{2\pi k n}{m}\right)
$$
converges or ...
0
votes
1answer
56 views
Exponential Sum Approximation
Is it possible to show mathematically that, for a short portion, sum of two decaying exponential can be approximated by a single decaying exponential? i.e.
$Ae^{-ax}+Be^{-bx}\approx Ce^{-cx}$?
0
votes
1answer
183 views
Sum of discrete Gaussian series
Is there a sum formula for the following sequence?
It is a Gaussian with $N$ terms, $x_{0}$ and $\sigma$ are real numbers, with $0<x_{0}<N$
$$\sum_{x=0}^{N-1} e^{\frac{-1}{\sigma} \:(x-x_{0})^{...
12
votes
1answer
233 views
Number of zeros in difference of exponential sums: $\sum\limits_{i=1}^n a_i^x - \sum\limits_{i=1}^n b_i^x$
Let $$f(x) = \sum\limits_{i=1}^n a_i^x - \sum\limits_{i=1}^n b_i^x$$ where the $a_i$ and $b_i$ are positive reals such that $f(x)$ is not a constant zero for all real $x$.
Is it possible to find a ...
