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

learn more… | top users | synonyms

9
votes
0answers
230 views

Equidistribution of roots of prime cyclotomic polynomials to prime moduli

Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
7
votes
2answers
301 views

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
6
votes
2answers
196 views

Closed form of the sum $\sum\limits_{n=0}^\infty \exp(-n^3)$

I am trying to calculate the sum of the series $$\sum_{n=0}^\infty \exp(-n^3)$$ Can it be expressed in terms of known mathematical functions?
6
votes
1answer
429 views

Integrating $\sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$

Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$ Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
5
votes
1answer
225 views

Simplify exponential sum over $\mathbb{F}_p$ to prove identity

I have a sum involving $p$-th roots of unity (where $\frac{1}{t}$ is to be understood as the field inverse $t^{-1} \bmod p$ etc.) of the form $\begin{align*} &d_{j,k}=\sum_{a,b,c \in ...
4
votes
3answers
126 views

Sum of the sequence

What is the sum of the following sequence $$\begin{align*} (2^1 - 1) &+ \Big((2^1 - 1) + (2^2 - 1)\Big)\\ &+ \Big((2^1 - 1) + (2^2 - 1) + (2^3 - 1) \Big)+\ldots\\ &+\Big( (2^1 - 1)+(2^2 - ...
4
votes
1answer
237 views

Determining the Value of a Gauss Sum.

Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right) $$ for general $k$ and $n$? For $k=1$, on MathWorld we have that ...
4
votes
2answers
3k views

Baker-Hausdorff Lemma from Sakurai's book

I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + ...
4
votes
2answers
2k views

upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
4
votes
1answer
427 views

Proof for generalized sum of powers

Bernouli's Formula for sum of kth powers of first n natural numbers is given by: $$f_k(n)=\frac{1}{k+1}\sum_{j=0}^k{k+1\choose j}B_j(n+1)^{k+1-j}$$ where $Bj$ is the $j^{th}$ Bernoulli Number and is ...
4
votes
1answer
69 views

Character Sums, Weights, and Cohomology

I know, vaguely, that certain bounds for character sums over finite fields can be determined by looking at weights and the dimensions of certain compactly supported cohomology groups. However, I've ...
4
votes
0answers
104 views

Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for $$ \sum_{-\infty}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)} $$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + ...
4
votes
0answers
103 views

Multiplicative subgroup of a finite field with prescribed trace.

Any suggestions/methods/estimates for the following problem would be very appreciated. $l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q ...
3
votes
2answers
213 views

A “generalized” exponential power series

I'm wondering if $$ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} $$ what would this be $$ \sum_{k=0}^\infty \frac{x^{k+\alpha}}{\Gamma(k+\alpha)} = \large{?}_{\alpha}(x) $$ for $\alpha \in (0,1)$? ...
3
votes
3answers
75 views

How does $\displaystyle \frac{e^{-x}}{1-e^{-x}}$ become $\sum_{k=1}^{\infty}e^{-kx}$?

How does $\displaystyle \frac{e^{-x}}{1-e^{-x}}$ become $\sum_{k=1}^{\infty}e^{-kx}$? I know $\displaystyle \frac{e^{-x}}{1-e^{-x}}=\frac{1}{e^x-1}=\left(\sum_{k=0}^{\infty} ...
3
votes
2answers
330 views

How can I resolve $\sum_{x=0}^{\infty} xe^{-x/\theta}$? [duplicate]

I stumble on this summation during an exercise. How can I resolve $\sum_{x=0}^{\infty} xe^{-x/\theta}$?
3
votes
1answer
570 views

How to find a summation of a logarithmic function?

Suppose that I had to find $\log_{10}(8952!)$. Now, since $\log(a) + \log(b) = \log(ab)$, this can be rewritten to the following summation: $$\sum_{x=1}^{8952}{\log_{10}(x)}$$ Would there be a ...
3
votes
2answers
63 views

An upper bound for an average exponentiated weighted sum of a vector from $\{-1,1\}^n$

Suppose $\mathcal{S}=\{\mathbf{x}:\mathbf{x}\in\{-1,1\}^n\}$, that is, $\mathcal{S}$ contains all $2^n$ vectors of length $n$ containing -1 and 1. I am interested in the following average: ...
3
votes
2answers
22 views

Prove that $S_{n}(x) = \sum_{k=-n}^{n}c_{k}e^{ikx}$

The problem is to prove that every trigonometric sum of the form $$S_{n}(x) := \frac{1}{2}a_{0} + \sum_{k=1}^{n}(a_{k}\cos kx + b_{k} \sin kx)$$ can be expressed as $$S_{n}(x) = ...
3
votes
2answers
157 views

Sum of finite series of $\exp(an^2 + bn + c))$

Is there a way to simplify that sum to an expression without actual performing the summation, similar to the formula for calculating the sum of a (finite) geometric series? $\sum_{n=0}^{N-1} ...
3
votes
1answer
38 views

Showing $\sum_{n=-\infty}^{\infty}\exp\left(-\pi an^2+2\pi ibn\right)=a^{-\frac{1}{2}}\sum_{m=-\infty}^{\infty}\exp\left(-\frac{\pi(m-b)^2}{a}\right)$

How do I show that \begin{align} \sum_{n=-\infty}^{\infty} \exp\left(-\pi a n^2 + 2 \pi i bn\right) = a^{-\frac{1}{2}} \sum_{m=-\infty}^{\infty} \exp\left(-\frac{\pi(m-b)^2}{a}\right) \end{align} is ...
3
votes
1answer
59 views

Bounds for general character sums over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $\chi$ be the canonical additive character of $\mathbb{F}_q$, let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$, and ...
3
votes
3answers
266 views

Summation evaluation of $\sum_{j=0}^k n^{1/2^j}$?

How do I go about solving this: $\sum_{j=0}^k n^{1/2^j}$ So, the terms of this series are $n , n^{1/2},n^{1/4},n^{1/8},.......n^{1/2^k}$ Any insights on what the thought process should be, to ...
3
votes
1answer
66 views

Bounding $\sum \exp(x_1\pm \cdots\pm x_n)^2$ in terms of $x_1^2+\cdots +x_n^2$.

Suppose that $x_1,\ldots,x_n$ are positive real numbers such that $$ x_1^2+\cdots +x_n^2 < \epsilon. $$ Can we bound the quantity $$ 2^{-n}\sum_{b_1,\ldots,b_n\in\{\pm1\}}e^{\left(\sum_i ...
3
votes
1answer
86 views

Exponential series is cosh(x), how to show using summation?

I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} ‎\frac{(x)^{2n}‎}{(2n)!}‎ $$ I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that ...
3
votes
2answers
99 views

Inequality with a sum

I am reading Remarks on a Ramsey theory for trees: Janos Pach, Jozsef Solymosi, Gabor Tardos http://arxiv.org/abs/1107.5301 I am stuck at inequality in proof of Lemma 6. $n\geq 8$, $k=2\lfloor ...
3
votes
1answer
23 views

How do I estimate the error term when computing the number of integers which have the fractional part of their square roots in a given interval?

I'm trying to find the number of integers $n \leq N$ such that fractional part of $\sqrt n\in (\alpha,\beta]$ where $(\alpha,\beta]\subseteq(0,1]$. The approximate number is of course ...
2
votes
1answer
81 views

How to calculate the sum $ x + x^2 +…+ x^n$ [closed]

How can I get the result of this sum: $$ x + x^2 +...+ x^n $$
2
votes
1answer
34 views

Simple expression:$ a - a^{-1}$ = …

I got stuck with one simple expression, I hope get some help with it: If $a-\frac{1}{a}=\frac{3\sqrt7}{7}$, so $a^4+\frac{1}{a^4}=$
2
votes
3answers
86 views

Exponential equation in x,y,z

Find all positive integers x, y, z satisfying $$x^{y^{z}} \cdot y^{z^{x}} \cdot z^{x^{y}}=5xyz$$ I took log on both sides, which led me to $$(\log x)(y^{z}-1)+(\log y)(z^{x}-1)+(\log z)(x^{y}-1)=\log ...
2
votes
1answer
78 views

Two questions on trigonommetric sums and integrals

Is it true that $\int_{0}^{\infty}\sin(mx)\sin(nx) \, dx = \delta (m-n) $ although using Euler formula I get a linear combination of $ \delta(m-n) $ and $ \delta (m+n)$? What is the sum ...
2
votes
1answer
70 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
2
votes
1answer
57 views

Exponential function as a sum

I have an exercise that asks me to write $e^{2x}$ using a power series of $x+1$. I know that $$e^{2x}=\sum_{n=0}^{\infty}\frac{(2x)^{n}}{n!}$$ Then, I tried something like this $$x=y+1\Rightarrow ...
2
votes
2answers
183 views

Number of zeros in polynomial-exponential sums

Is there some bound (or even an exact solution) on number of real roots of polynomial-exponential sum of type $$f(x) = a_1b_1^x+a_2b_2^x+\cdots=\sum_{i=1}^N a_i b_i^x = 0$$ where $b_i>0, ...
2
votes
1answer
67 views

Convergence of $\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$

We know that $\sum_{ k\in\mathbb{N} } \frac{\lambda^k}{k!} = e^\lambda$. I'm interested in the convergence of $$S^{(n)}=\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n $$ for some value ...
2
votes
1answer
99 views

Standard deviation of phase for a random phasor sum

I have a phasor sum $a e^{j \theta} = \frac{1}{\sqrt{N}} \sum_{k=1}^{N} \alpha_k e^{j \phi_k }$ where $\phi_k = [-\pi, \pi]$, the standard deviation $\sigma_{\phi}$ of the phase is known and the ...
2
votes
1answer
107 views

Sum of exponential functions involving powers of two

I came across a weird series with exponential functions and powers of two: $$\sum_{k=0}^{\infty} \left(1 - e^{-2^{-k}z} \right), z \in \mathbb R_+$$ and have no idea how to solve this (if there even ...
2
votes
1answer
70 views

Equality of exponential functions from geometric series

I'm currently trying to understand why the first and second line of this equation are in fact equal. This is taken from "Introduction to the Physics of Waves" by Tim Freegarde from a chapter about ...
2
votes
1answer
141 views

What is the infinite sum of $a^{b^x}$?

What would $$\sum^{\infty}_{n=0}(1/2)^{4^n}$$ be and how to determine it? EDIT: Apologies. I can see this converges by the ratio test. My issue is working out its sum, more for fun really. It ...
2
votes
4answers
56 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
2
votes
1answer
107 views

Median and Mean of Sum of Two Exponentials

I have a cumulative distribution function: $$G(x) = -ae^{-xb} - ce^{-xd}+h$$ The associated probability density function is: $$g(x) = abe^{-xb} + cde^{-xd}$$ My problem concerns $x\ge 0, X \in R$. I ...
2
votes
1answer
165 views

Integrating exponential of multiple exponentials

I have a integral term that looks similar to $\int_0^\infty\exp(-u-ae^{-c_1u}-be^{-c_2u})\,du$ where the constants $a,b,c_1,c_2>0$. For the case where $b=0$ I can use the answer from: Integrating ...
2
votes
0answers
30 views

Determination of quartic Gauss sums

Typically, the Gauss sum over $\mathbb{F}_p$ of order $k$ means the quantity $$\sum_{n=0}^{p-1} e^{2\pi i n^k/p}.$$ In the book Gauss and Jacobi Sums of Berndt, Evans and Williams, a more general sum ...
2
votes
2answers
94 views

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where ...
2
votes
0answers
39 views

an exponential sum involving quadratics

Let $q$ be a prime and $a,b\in \mathbb{Z}$. Can we compute $$\sum_{k=1}^{q}e^{2\pi i\frac{ak^2+bk}{q}}$$ in terms of $a,b,q$? The sum is easy when $a=0$, but what about the case $a\neq 0$?
2
votes
1answer
81 views

Bounds on sum of cosines

Find bound on the following sums for $k\in \{1,\dots p-1\}$ where $p$ is a prime (even assumed to be $3\mod 4$, I used this to get to the current sum) find good upper and lower bounds on ...
2
votes
1answer
70 views

Express sum of complex exponentials as 1 + sum of cosines

I'm told that I can express the following sum of complex exponentials: $$\sum_{n=0}^6 e^{-j\Omega n}$$ As 1 plus 3 cosine terms. I'm having a really hard time arriving at this, I see where the 1 ...
2
votes
0answers
100 views

The sum of finite exponential series with a quadratic phase

How can I prove that: $$ \sqrt \frac K2 + i \sqrt \frac K2=\sum^K_{m=1}\exp\left(i\frac \pi Km^2\right) $$ When $K$ is even.
2
votes
2answers
41 views

Can the following equations be solved without the need of numerical methods?

I'm taking advanced algebra in school. I have been asked to solve two equations: $\log_{6}(1-x) + \log(x^{2}-9) = 2 \\$ $ 3^{x+2} + 2^x = 5 $ The teacher said this equations can be solved ...
2
votes
1answer
39 views

Evaluation of infinite sum related to problems in elasticity

I'm working on some problems with relation to elasticity (plate mechanics in specific) and while I've made some progress the following sum is giving me a hard time ...