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

learn more… | top users | synonyms

-4
votes
1answer
36 views

Help me solve this [closed]

$(1-(1-(1-n)^{-1})^{-1})^{-1}$ Please tell me how to solve the abv sum. The ans is $n$
0
votes
0answers
37 views

Computation with Exponential Generating Function

I am working a problem on generating functions but am stuck at a technical part which I think is actually supposed to be really easy, but I just don't have much experience with such things. We're ...
0
votes
0answers
34 views

Is there a way to solve this problem - sum of powers

$N_1^a + N_2^a + \ldots + N_n^a$ Where a is any number. Example $40^7+ 37^7 + 35^7 + 32^7 + ... 1^n=R$ If Iknow what $R$ is and that $ N_n$ is restricted to $0 \lt N_n \le 40$, Is there a way to ...
0
votes
1answer
30 views

Simplifying sums of exponentials

I am trying to solve a tricky little problem involving sums of exponentials. Consider the sequences $\{a_m\}$ and $\{b_m\}$. Both of these sequences are finite and $a_m>0 ~\forall m$ and ...
2
votes
1answer
28 views

Exponent Subtraction With Same Base

Can someone explain how this is valid? Thanks so much! $3\cdot 2^{2n+2} - 3\cdot2^{2n} = 3\cdot2^{2n}$
0
votes
0answers
39 views

How to find last 9 digits of a large exponent?

I was doing some programming problem where I had to find the sum of last 9 digits of very large exponents. I can't even think of calculating that big powers, but I have to find last 9 digits of the ...
1
vote
0answers
57 views

When does sum of exponentially many exponential functions go to zero?

I have $K=e^{n\beta}$ positive numbers $D_1\leq D_2\leq ...\leq D_K$. I want to find the maximum value of $\beta$ for which $e^{-nD_1}+e^{-nD_2}+...+e^{-nD_K}$ goes to zero as $n$ goes to infinity. ...
4
votes
1answer
61 views

Infinite sum involving powers and factorials

I am interested in evaluating the following infinite sum \begin{equation} \sum_{n=0}^{\infty} \frac{\alpha^{n}}{n!}n^{\beta} \end{equation} where both $\alpha$ and $\beta$ are real numbers. However, ...
0
votes
1answer
61 views

Maximizing the sum of exponentials whose exponents sum to $N$

Let $N \geq 1$ be a sufficiently large integer, let $a > 1$ be a real number, and let $n_1, \dots, n_t$ be integers between $0$ and $K$, where $K$ divides $N$. I want to determine the following: ...
0
votes
0answers
53 views

Lower bound on sum of exponentials

Let $\mathbf{K}_{1:n}$ be a vector of known positive constants and $f(\mathbf{\theta}_{1:n})=\sum_{i=1}^n K_i \exp(\theta_i)$, for $\theta\in \mathbb{R}^n$. Is it possible to find a 'tight' lower ...
2
votes
1answer
59 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
1
vote
3answers
90 views

Solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$

How can I solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$, where $N\geq 1, k_1,\ldots,k_N \in \mathbb{R}, k_1,\ldots,k_N < 0, x\in \mathbb{R}$ and $x >0$. I looked at the basic rules of ...
4
votes
0answers
190 views

Is Every transcendental entire function $f(z) = C + \exp a + \exp b + \exp c $?

Let $z$ be a complex number and let $f(z)$ be any transcendental entire function. Is it true that $f(z) = C + \exp a + \exp b + \exp c $ where $ a,b,c $ are entire functions of $z$ and $C$ is a ...
4
votes
1answer
64 views

question on identity of sums with exponent. [duplicate]

I want to show that for $x>0$: $$\sum_{n=-\infty}^\infty e^{-n^2\pi x}= \frac{1}{\sqrt{x}}\sum_{n=-\infty}^\infty e^{-n^2\pi / x}$$ It doesn't seem that a simple change of variables will do, like ...
4
votes
1answer
75 views

Discrete Fourier Transform of $\omega^{n(n-1)/2}$

For the sequence $x_0$, $x_1$, $\ldots$,$x_{N-1}$, let $\omega=e^{2\pi i/N}$ and define the discrete Fourier transform as $$X_k = \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x_n\omega^{nk}\,.$$ I'm interested ...
0
votes
1answer
40 views

Formula for $\sum_{i = 1}^n k^n$ [duplicate]

I know from my calculator the answer is $\sum_{i = 1}^n k^n$ = $\frac{k^{n+1}-k}{k - 1}$. I'd just like help understanding why.
3
votes
1answer
41 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 ...
0
votes
0answers
16 views

How to find a minima for a function

I'm trying to find a minimum value for the function $[(4N)^{r-q-2} r(r-1)...(r-q-1)]^\frac{1}{2^{q+2}-2}$ where $N$ is an integer, $r$ may or may not be an integer and $q$ is an positive integer ...
0
votes
3answers
86 views

$\sum\limits_{n=4}^{\infty } \frac{2^n + 8^n}{10^n} = ?$

im looking for hints on how to do: $\sum\limits_{n=4}^{n= \infty } \frac{2^n + 8^n}{10^n} = ?$ I thought this may have had something to do with geometric series but nothing obvious comes up ...
3
votes
0answers
59 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 ...
0
votes
2answers
73 views

How to compute $\sum_{n=1}^N e^{-( n-c)^2}$

I have to compute or at least find good upper and lower bounds on \begin{align*} \sum_{n=1}^N e^{-( n-c)^2/b} \end{align*} and \begin{align*} \sum_{n=1}^N ne^{-( n-c)^2/b} \end{align*} where $c$ ...
3
votes
1answer
76 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 ...
1
vote
1answer
33 views

Let $X~Bin(n, \lambda/n)$, $\lambda >0$. Show that for fixed $k \geq 0$, $P(X=k)\equiv \frac{e^-\lambda)\lambda^k}{k!}$

Let $X - Bin(n, \lambda/n)$, $\lambda >0$. By using approximation $(1-\frac{x}{n})^n\approx e^{-x}$. Show that for fixed $k\geq 0$, $P(X=k)\approx \frac{e^{-\lambda}\lambda^k}{k!}$ ...
1
vote
2answers
67 views

Limit log-sum of exponentials

I'm trying to compute the following limit: $$\lim_{\lambda \rightarrow \infty} \frac{1}{\lambda}\log\sum_{i=1}^n \exp[\lambda a_i]$$ I tried to solve with L'hoptials: $$= \lim_{\lambda \rightarrow ...
1
vote
2answers
53 views

Question on power, If 2x^2x^2x^2x… =4 Solve for x

I've seen this random example, in which can anyone give me clue how to solve for $ x $ here?
6
votes
2answers
210 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?
2
votes
2answers
166 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 ...
1
vote
1answer
76 views

Is there a closed form for $\sum_{n=1}^\infty \frac{q^{-n^2}}{n}$?

I found that $\sum_{n=1}^\infty q^{-n^2}$ can be expressed using a theta function. Is there also a closed form for the following series? $$\sum_{n=1}^\infty \frac{q^{-n^2}}{n}$$
0
votes
1answer
37 views

Progression with variable in exponent

what is the solution of this kind of progressions: $1 + \delta^{T} + \delta^{2T} + \delta^{3T} + \dots $ I have tried hard and look through the website but still I don't know hot to solve the issue ...
1
vote
0answers
343 views

sum of exponential series with power increasing by geometric series

Is there any way to reduce the following summation... $2^{ar^0}+2^{ar^1}....+ 2^{ar^n} $ to a simple equation? I feel like I can pull out a $2^a$ somehow and then treat it as a normal series but I ...
3
votes
2answers
26 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
274 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 $\alpha \in (0,1)$? ...
2
votes
1answer
173 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 ...
0
votes
2answers
90 views

quadratic Gauss sum over a power of 2

Is there a general formula for the generalized quadratic Gauss sum defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=0}^{c-1}e\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$ and $c$ is a power of ...
0
votes
1answer
59 views

Algorithm for smooth exponential curves

I want to plot an exponential curve between 256 and 0. Using the following equasion, I get the resulting data set. (Please note that I am rounding any decimals down to nearest whole number throughout ...
2
votes
1answer
102 views

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

How can I get the result of this sum: $$ x + x^2 +...+ x^n $$
1
vote
0answers
37 views

Plotting exponential partial sums in the complex plane

I was plotting the following sequence of points $(a_n)_{n = 0}^\infty$ in the complex plane for various reals $\alpha > 1$: $$a_n = \sum_{k = 0}^n e^{i k^\alpha}$$ I found that for many values of ...
1
vote
1answer
28 views

Trigonometical sum from Fourier analysis

(Edit) Note: $a\in \mathbb{R},0<a\leq\pi$. Also, the sum skips $n=0$ (that's where the other term comes from). Working through a Fourier analysis exercise I've got stuck in a clearly ...
1
vote
0answers
22 views

Solve $b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x$

Suppose, \begin{align*} b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x \end{align*} Assume $a_1,a_2,a_3, b_1,b_2, b_3>0$ What are the possible values of $a_1,a_2,a_3, b_1,b_2, ...
2
votes
1answer
38 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
1answer
176 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 = ...
7
votes
2answers
335 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 ...
3
votes
2answers
103 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
1answer
42 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 ...
2
votes
0answers
47 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 $a\neq 0$?
1
vote
1answer
146 views

solve two term equation with different fractional exponents

Suppose: $$a = bw^f + cw^g $$ where $a,b$ and $c$ are known, and $f$ and $g$ are known fractional exponents Ex. $50000 = 200w^{0.72} + 4000w^{0.19}$ How can one solve for the value of w?
5
votes
0answers
193 views

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

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

Get $a_1$, $a_2$, $b_1$, $b_2$ from $a_1 \times \exp{b_1 \times x} - a_2 \times \exp{b_2 \times x}$

I have experimental data which follow the function below. $$f(x) = a_1 e^{-b_1 x} - a_2 e^{-b_2 x} + \epsilon$$ ($a_1$, $b_1$, $a_2$, $b_1$ are all positive real numbers. $\epsilon$ represents ...
3
votes
1answer
100 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 ...
2
votes
1answer
170 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 ...