Questions tagged [exponential-sum]
For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.
463
questions
-4
votes
0
answers
66
views
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?
$$\sum_{i=0}^{n-1} {\...
0
votes
0
answers
44
views
About $T(n,k)=\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+...+\frac{1}{(n+k)!}$ where $n,k\in \mathbb{N}$
For $n,k\in \mathbb{N}$ let $$T(n,k)=\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+...+\frac{1}{(n+k)!}$$
Which of the below statement(s) hold true
(A)$T(n,k)\geq \frac{1}{2}$
(B)$T(n,k)> \...
- 8,105
5
votes
0
answers
90
views
Closed form for $\sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$
I need a closed form for $$ \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}$$
For $1<x&...
- 107
1
vote
1
answer
57
views
Sum of powers of the sum of all the square numbers in a factorial
Here's the exact question:
The sum of all positive integers $p$ such that $13! \div p$ is a perfect square, can be written as $2^a . 3^b . 5^c . 7^d . 11^e . 13^f$ where $a, b, c, d, e$ and $f$ are ...
- 87
3
votes
1
answer
55
views
Special property of the normal distribution density concerning infinite sums?
It seems that equi-distant Gaussian bell curves almost add up to the constant function $1$ already for relatively small sigmas, i.e.
$$ \lim_{s\rightarrow\infty} \sum_{n = -\infty}^{\infty} \frac{1}{\...
- 1,478
1
vote
1
answer
63
views
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 :
...
- 223
1
vote
0
answers
45
views
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
$$\mathcal{S}(N)\lesssim ...
- 5,032
0
votes
0
answers
24
views
Is there a general rule for addition or subtration of two numbers with the same base but different exponents?
I see the product and quotient rules in a quick online search, but I can't find amything for addition and subtraction.
For example, a special case is:
$$2^6 - 2^5 = 2^5$$
$$2^5 - 2^4 = 2^4$$
$$3^6 - 3^...
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 ...
- 590
0
votes
0
answers
23
views
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 ...
0
votes
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.
...
- 175
0
votes
0
answers
39
views
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 ...
- 97
3
votes
1
answer
129
views
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 ...
- 86
1
vote
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 ...
- 345
0
votes
1
answer
20
views
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 ...
- 101
4
votes
0
answers
78
views
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:
$$
f(k,l)=\sum_{x=0}^{d-1} \sum_{y=0}^{d-1} \varepsilon(x+y)\exp\left( \frac{2 \pi i\cdot(k x + ...
- 1,389
2
votes
1
answer
45
views
Deriving an upper bound for an exponential function
I am studying risk theory this semester and we are currently covering the concept of adjustment coefficients, $R$. Essentially, the result we derived in class was that, for a random claim of size $x$, ...
- 2,097
0
votes
1
answer
93
views
Most efficient way to compute $S(n, a, b) = \sum_{i=1}^n a^i i^b$?
Given $n$, $a$ and $b$, find most efficient way to compute
$S(n, a, b) = \sum_{i=1}^n a^i i^b$
Most trivial way would be $O(n\cdot(n+b))$.
If we use fast exponentiation then it would be $O(n\cdot(\log ...
0
votes
1
answer
60
views
Numerical convergence of a sum
I want to study the convergence of the function given below.
$$ \sum_{n = 1}^{\infty} e^{-\Gamma^2 n^2} $$
The $n$ are integers!
Here, if we check numerically, the function converges based on $\Gamma^...
- 113
1
vote
1
answer
36
views
How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$.
How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$?
Here $\|\alpha x\|$ is the distance from $\alpha x$ to the nearest integer.
My attempt was to rewrite ...
- 1,621
1
vote
1
answer
94
views
Asymptotics for an exponential sum
I am trying to find an equivalent of : $\displaystyle{S_n=\sum_{k=1}^n}e^{i\sqrt k}$
I tried to elucidate the asymptotic behaviour of the subsequence :
$$S_{n^2-1}=\sum_{k=1}^{n^2-1}e^{i\sqrt k}=\sum_{...
- 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^{...
- 69
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
$$S(r):= \sum_{...
- 364
1
vote
0
answers
29
views
Proving that periodic vector-valued function satisfies $f(\mathbf x)= f(\mathbf x')$ iff $\mathbf x' = \mathbf x + k + \mathbf am$
In order to create a layer for a neural network that uniquely encodes relative relationships between a set of angles, I'm looking for a differentiable function with some specific properties; I'm ...
- 11
0
votes
2
answers
48
views
How to show $\displaystyle\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$
This result seems trivial, how would I show $$\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$$
Where $0 \le k,l \le n-1$
And $k \neq l \neq 0$.
I tried to use the exponential definition ...
- 447
0
votes
1
answer
182
views
Limit of summation of $\sum_{i=0}^{\infty} e^{-ik}$ series
I was wondering whether anyone was able to give some insight to determining the limit of the following summation
$$
\sum_{i=0}^{\infty} e^{-ik}
$$
If anyone has any idea or is able to point me in the ...
0
votes
0
answers
72
views
Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time?
Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time?
I'm trying to find the upper bound for the following series
$$s(n) = 2^1 + 2^2 + 2^4 + 2^8 + 2^{16} + ...
0
votes
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 ...
- 1
0
votes
0
answers
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 ...
- 175
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 ...
- 188
0
votes
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 $\theta_i = 0$, $\...
- 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 ...
- 555
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(...
- 31
0
votes
0
answers
57
views
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 ...
- 21
0
votes
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) $t$ ...
- 161
0
votes
1
answer
36
views
Sum of exponentials - derivation
In a couple of scientific papers, I've found this sum identity involving exponentials:
$\sum_{n=0}^{\frac{E_{BD}}{\Delta E}} e^{\gamma(n \Delta E
- E_{BD})} = \frac{1}{1 - e^{\gamma\Delta E}}$
...
- 1
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 ...
- 951
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 ...
- 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)&...
- 159
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 \...
- 1,621
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 ...
- 13
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 ...
- 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!
- 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 $...
- 2,095
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 $A_1 = A_2 = \cdots = A_p = 0$ ?
- 91
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 ...
- 951
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
$$L_r := \left\{ \begin{pmatrix} a+cr \\ b+dr \\ c \\d \end{pmatrix} : a,b,c,d \in \mathbb Z \right\},
\quad W:= \left\{ \begin{pmatrix} 0 \\ 0 \\ x_3 \\ x_4 \end{...
- 5,946
1
vote
1
answer
39
views
Modified exponential summation [duplicate]
How do we prove that
$$1+\frac{2^3}{2!}+\frac{3^3}{3!}+\frac{4^3}{4!}+\cdots=5e$$
I'll post the answer to this question as a knowledge share.
- 1,136
0
votes
0
answers
125
views
Absolute value of the theta function on the unit nome
I would like to evalue the absolute value of the theta function with unit nome, i.e. when $|q|=1$ or equivelantly when $\tau\in\mathbb R$. The theta function is expressed as
$$\vartheta(z;q=e^{\pi i \...
- 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 ...
- 390