Questions tagged [exponential-sum]

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

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If $\frac1{1-e^x}=\sum_{n=0}^\infty\left(\frac{\beta_n}{n!} x^n\right)$, then what is $\sum_{i=0}^{n-1}{\beta_i}(^nC_i)$ equal to?

I'm really confused and i have no idea how to proceed with this. If $$\frac{1}{1-e^x} = \sum_{n=0}^{\infty} (\frac{\beta_n} {n!} x^n)$$ then what is the following equal to? $$\sum_{i=0}^{n-1} {\...
Pratham Nayak's user avatar
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About $T(n,k)=\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+...+\frac{1}{(n+k)!}$ where $n,k\in \mathbb{N}$

For $n,k\in \mathbb{N}$ let $$T(n,k)=\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+...+\frac{1}{(n+k)!}$$ Which of the below statement(s) hold true (A)$T(n,k)\geq \frac{1}{2}$ (B)$T(n,k)> \...
Maverick's user avatar
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Closed form for $\sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$

I need a closed form for $$ \sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$$ where $x\in[1,\infty)$ For $x=1$ we have the sum as $$ \sum_{n=1}^\infty e^{-2\pi n}=\frac{1}{e^{2\pi}-1}$$ For $1<x&...
Max's user avatar
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1 answer
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Sum of powers of the sum of all the square numbers in a factorial

Here's the exact question: The sum of all positive integers $p$ such that $13! \div p$ is a perfect square, can be written as $2^a . 3^b . 5^c . 7^d . 11^e . 13^f$ where $a, b, c, d, e$ and $f$ are ...
Cuckoo Beats's user avatar
3 votes
1 answer
55 views

Special property of the normal distribution density concerning infinite sums?

It seems that equi-distant Gaussian bell curves almost add up to the constant function $1$ already for relatively small sigmas, i.e. $$ \lim_{s\rightarrow\infty} \sum_{n = -\infty}^{\infty} \frac{1}{\...
coproc's user avatar
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Proof that $\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$

I am looking for an analytical proof that : $$\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$$ Both CAS Wolfram alpha and sympy agree on the result : ...
mocquin's user avatar
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How to estimate the exponential sum $\sum_{|m|\le N} \frac{e^{im^{3/2}}}{m^{1/4}}$

I am interested in upper-bounding tightly the following sum $$\mathcal{S}(N)=\sum_{|m|\le N} \frac{e^{im^{3/2}}}{m^{1/4}}$$ better than the naive triangle inequality estimate $$\mathcal{S}(N)\lesssim ...
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Is there a general rule for addition or subtration of two numbers with the same base but different exponents?

I see the product and quotient rules in a quick online search, but I can't find amything for addition and subtraction. For example, a special case is: $$2^6 - 2^5 = 2^5$$ $$2^5 - 2^4 = 2^4$$ $$3^6 - 3^...
Andrew Singley's user avatar
3 votes
1 answer
177 views

Analytical evaluation of infinite series

I am trying to calculate the infinite series $$\sum _{n=-\infty }^{\infty } \frac{(-1)^{n+1} e^{-(n-1)^2\pi}}{1-e^{ (2 n-1)\pi}}\simeq -0.0903244354808$$ Are there any any analytical methods to ...
El Rafu's user avatar
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Exponential function to the power of a fraction

I have an equation such that $$1=e^{\frac{ylog(3+x)}{3+x}}-3 -x$$ Where e is the exponential function. How do I solve for x? According to the laws of exponents, $a^{m/n}=\sqrt[n]{a^m}$, Applying this ...
CorporateNationalism's user avatar
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2 answers
51 views

How to efficiently compute $\sum_{i=1}^n (x_i) ^ a$ for many different values of a and large n?

I need to compute $\sum_{i=1}^n (x_i) ^ a$ for many different values of a where n is large and x is fixed. I am in an environment where computing exponentials $(x_i) ^ a$ is effectively constant time. ...
Lalaland's user avatar
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Exponential sums reference

Looking for reference on exponential sums, in particular Jacobi, Gauss, Kloosterman and Ramanujan sums. The books mentioned in https://mathoverflow.net/questions/65429/exponential-sums-for-beginner ...
Ximenez's user avatar
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3 votes
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An infinite sum of products

I have to calculate this sum in closed form $$ \sum_{n=1}^\infty \prod_{k=1}^n \frac{x^{k-1}}{1 - x^k} $$ where $x < 1$. Numerical evaluation shows that this converges. The product can be performed ...
golfer's user avatar
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1 answer
33 views

Bounding exponential sums

There is classical inequality that seems to often appear: $$\sum_{i=1}^N \exp(2i\pi n \theta) \leq \|\theta\|^{-1}$$ where $\|\theta\|$ denotes the distance to the closest integer. I do understand ...
Amomentum's user avatar
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Double EMA into Single EMA

The exponential moving average (EMA) operator is defined as: $$y_t(x, \lambda) = (1-\lambda) \sum_{i=0}^\infty \lambda^i x_{t-i}$$ where $1-\lambda$ is the normalization factor, and the operator is a ...
joeP's user avatar
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Vanishing of a certain double sum of roots of unity

For $d \in 2\mathbb{N}$ and $0 \leq k,l \leq d-1$ the following double sum turned up in my computations: $$ f(k,l)=\sum_{x=0}^{d-1} \sum_{y=0}^{d-1} \varepsilon(x+y)\exp\left( \frac{2 \pi i\cdot(k x + ...
Bipolar Minds's user avatar
2 votes
1 answer
45 views

Deriving an upper bound for an exponential function

I am studying risk theory this semester and we are currently covering the concept of adjustment coefficients, $R$. Essentially, the result we derived in class was that, for a random claim of size $x$, ...
Ethan Mark's user avatar
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Most efficient way to compute $S(n, a, b) = \sum_{i=1}^n a^i i^b$?

Given $n$, $a$ and $b$, find most efficient way to compute $S(n, a, b) = \sum_{i=1}^n a^i i^b$ Most trivial way would be $O(n\cdot(n+b))$. If we use fast exponentiation then it would be $O(n\cdot(\log ...
ishandutta2007's user avatar
0 votes
1 answer
60 views

Numerical convergence of a sum

I want to study the convergence of the function given below. $$ \sum_{n = 1}^{\infty} e^{-\Gamma^2 n^2} $$ The $n$ are integers! Here, if we check numerically, the function converges based on $\Gamma^...
CfourPiO's user avatar
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1 answer
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How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$.

How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$? Here $\|\alpha x\|$ is the distance from $\alpha x$ to the nearest integer. My attempt was to rewrite ...
trynalearn's user avatar
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1 vote
1 answer
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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_{...
Adren's user avatar
  • 7,234
2 votes
2 answers
99 views

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^{...
DontWorry's user avatar
2 votes
1 answer
81 views

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_{...
Daniel's user avatar
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1 vote
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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 ...
monkeypuzzle's user avatar
0 votes
2 answers
48 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 ...
Governor's user avatar
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1 answer
182 views

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 ...
Christopher Briggs's user avatar
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0 answers
72 views

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} + ...
exponential's user avatar
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0 answers
128 views

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 ...
kik24's user avatar
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56 views

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 ...
FodderOverflow's user avatar
2 votes
0 answers
100 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 ...
Subin Pulari's user avatar
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1 answer
70 views

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 $\theta_i = 0$, $\...
zork's user avatar
  • 107
0 votes
0 answers
43 views

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 ...
mathoverflowUser's user avatar
3 votes
0 answers
151 views

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

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(...
Dariua's user avatar
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0 answers
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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 ...
Daniel T-K's user avatar
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0 answers
73 views

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) $t$ ...
k_moreno's user avatar
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0 votes
1 answer
36 views

Sum of exponentials - derivation

In a couple of scientific papers, I've found this sum identity involving exponentials: $\sum_{n=0}^{\frac{E_{BD}}{\Delta E}} e^{\gamma(n \Delta E - E_{BD})} = \frac{1}{1 - e^{\gamma\Delta E}}$ ...
Pablo's user avatar
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3 votes
1 answer
108 views

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 ...
Krishnarjun's user avatar
2 votes
1 answer
568 views

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 ...
Jasasul's user avatar
  • 53
1 vote
1 answer
134 views

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)&...
Ashwin B's user avatar
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4 votes
1 answer
94 views

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 \...
trynalearn's user avatar
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1 vote
1 answer
39 views

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 ...
Jean Von's user avatar
0 votes
1 answer
107 views

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 ...
dkssud's user avatar
  • 25
-2 votes
1 answer
49 views

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!
InSaNo's user avatar
  • 43
4 votes
1 answer
91 views

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 $...
Dzoooks's user avatar
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1 vote
0 answers
24 views

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 $A_1 = A_2 = \cdots = A_p = 0$ ?
Vincent NIEL's user avatar
1 vote
1 answer
243 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 ...
andu eu's user avatar
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7 votes
1 answer
274 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\}, \quad W:= \left\{ \begin{pmatrix} 0 \\ 0 \\ x_3 \\ x_4 \end{...
principal-ideal-domain's user avatar
1 vote
1 answer
39 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.
Abhishek A Udupa's user avatar
0 votes
0 answers
125 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 \...
Cameron's user avatar
  • 513
3 votes
1 answer
94 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 ...
Jishu Das's user avatar
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