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

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1
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2answers
18 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
42 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
181 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?
1
vote
1answer
52 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 ...
0
votes
2answers
63 views

Exponential sum [duplicate]

$$ \sum_{k=1}^\infty \frac{k^2}{2^k} = 6$$ But how? What formula should we use? It is not geometric series. Thanks Ann
1
vote
1answer
66 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
21 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
44 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
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
198 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
79 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
1answer
57 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
13 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
74 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
24 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
23 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
18 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
33 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
53 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
287 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
52 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
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 ...
2
votes
0answers
34 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 $a\neq 0$?
0
votes
0answers
29 views

Solving a sum-exponential equation

I was wondering if someone could point me to the right resource towards numerically solving an equation of the form: $c = \dfrac{\sum_i a_i^{2x}}{\left( \sum_i a_i^x \right)^2}$ $c$ and $a_i$ are ...
0
votes
1answer
35 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?
4
votes
0answers
84 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} + ...
0
votes
2answers
42 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
49 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
90 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 ...
3
votes
1answer
56 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 ...
2
votes
1answer
77 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
64 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 ...
1
vote
1answer
47 views

Solve equation with unknown in exponents

This is in continuation of this but not related to it completely. I am interested in finding a solution to the equation: $m' = m - \sum \limits_{j=1}^{m} (1 - d_{O_j}/n)^k$. where $m,m',n$ and ...
0
votes
0answers
20 views

Multiple Integral Approximation

Does anybody know how to simplify following multiple integration ? ...
0
votes
0answers
41 views

Approximate Sum of exponentials

I want to closely approximate the following sum into possibly a single term (not as infinite summation) $SUM= \exp^{(-a_1x_1-a_2x_2-a_3x_3)} - \exp^{(-v)}\exp^{(-b_1x_1-b_2x_2-b_3x_3)}$ here $a_i, ...
1
vote
1answer
56 views

Help in simplifying this nasty expression obtained after binomial expnasion

I have arrived to the following expression and was wondering if anyone can help me further simplify to something nicer, $$F= 1- [1-\text{exp} (- \alpha(N) ) ]^N= 1- \sum_{k=0}^{N} \binom{N}{k} ...
2
votes
1answer
56 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
0answers
76 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.
1
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2answers
27 views

How do I solve for $\delta$ in this question

$316.45 = 100e^{\delta(10)} + 100e^{\delta(5)}$ I don't know why I can't do this. I thought of using $\ln$ but I don't think $\ln(A+B) = \ln(A) + \ln(B)$ or does it?
1
vote
1answer
32 views

Proof that $ y(n) = ∑_{k=-∞}^{∞}\ {a}^{-k}u(n-k)u(-k) = \frac{1}{1-a }$ if $n>0$

Can someone explain the steps and how the boundaries for the summation change to result in the answer (And possible for the case where $n\leq 0$. I am not really a mathematician, don't know if the ...
-1
votes
3answers
40 views

Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
0
votes
1answer
162 views

Stuck on derivative of logarithm of sum of exponentials

let's say that I need to calculate the following expression: $$ \frac{\partial\mathrm{log}(\mathrm{exp}(w_1 * x_1 + b_1) + \mathrm{exp}(w_2 * x_2 + b_2))}{\partial w_1} $$ How do I start? The rules ...
0
votes
3answers
89 views

Can I know the value of an infinite serie?

$$\sum\limits_{n=0}^{\infty}\frac{n}{e^n}$$ I have found through a software that the value is $\dfrac{e}{(e-1)^2}$. I've been trying to do it manually but I am getting $\dfrac{\infty}{\infty}$, ...
2
votes
2answers
33 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
38 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 ...
1
vote
0answers
21 views

Help in deriving proof with exponential and little-oh

I'm working through a proof of the Kelly Criterion found at http://www.elem.com/~btilly/kelly-criterion/. In checking up on some of the derivations I can't follow the last step in this sequence ...
2
votes
1answer
133 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 ...
1
vote
2answers
54 views

The result of exponential sum formula

I am awakard to deal with math problem I am trying to understand the first condition that is (k-r)=mN I can understand when (k-r) is mN, the left formula is 1 But I don't know how to left ...
1
vote
2answers
40 views

Identity involving trigonometric sum

I have to prove that $$\overset{N}{\underset{n=-N}{\sum}} \left(N-\left|n\right|\right)e^{2\pi inx}=\left|\overset{N}{\underset{n=1}{\sum}} e^{2\pi inx}\right|^{2}=\left(\frac{\sin\left(N\pi ...
5
votes
1answer
222 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 ...