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

learn more… | top users | synonyms

2
votes
2answers
31 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 ...
0
votes
2answers
39 views

Inequation of an sum smaller than 1

I'm trying to figure out the following $$ \sum^{\infty}_{n=3} \dfrac{q!^2}{n!^2} < 1 $$ How I can show it if $q \geq 2$? Maybe with telescoping sums? Thanks, Landau
-1
votes
2answers
79 views

How to approximate this large sum of exponential terms

Is there any way to approximate the following sum: $$ \sum\limits_{i_1=1}^N\sum\limits_{i_1=2}^N \cdots \sum\limits_{i_k=1}^N \cdots\sum\limits_{i_N=1}^N \exp(-r_{i_1}-r_{i_{k+1}}-r_{i_{2k+1}}- ...
2
votes
1answer
67 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 ...
1
vote
1answer
52 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} ...
0
votes
1answer
42 views

Calculate remaining quantity based on half-life with everyday supplementation

Half life of a thing is 'h' days. I have a box where everyday I put 'n' quantity into the box. How do I calculate the quantity remaining after 'd' days. i.e. if half life is 7 days. Each day I add 10 ...
0
votes
1answer
88 views

exponential upper bound on sum of exponentials

For what value of $c_3$ can I guarantee that $(a+b)\exp(-c_3\theta)>a\exp(-c_1\theta)+b\exp(-c_2\theta)$ where $a,b,c_i>0$
9
votes
0answers
219 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 $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
4
votes
0answers
100 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 ...
2
votes
0answers
87 views

Descartes Rule of Sign for exponential sums

I have the following exponential sums ($x\in\mathbb{R}$) $$f(x)=\sum_{i=1}^Na_iP_i(x)b_i^x$$ where $P(x)$ is some monomial, e.g., $x^2, x^3,\dots$, so $f(x)$ looks like ...
2
votes
0answers
79 views

Upper Bound on $\frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$

Is there any procedure to find an upper bound of the following expression? $$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$ Here $u,\beta\in\mathbb{R},\ u>1,\ ...
2
votes
0answers
45 views

a functional space for the study of exponential sum

I would like to study the exponential sum in an appropriate functional space. In particular: $f(x): \mathbb{R} \to \mathbb{R} $ $f(x) = \chi_K \sum_{i=1}^{M} R_i \exp{(s_i x)}$ where $R_i, s_i \in ...
1
vote
0answers
40 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
vote
0answers
17 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 ...
1
vote
0answers
42 views

please evaluate this sum

$$\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{3^{a+n}}{a+n}-\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{1}{a+n}$$ I need help evaluating the above, step by step would help. Thanks for a>0 or ...
1
vote
0answers
29 views

Math equations of electron scattering

I'm trying to figure out the missing step here, in a problem about X-ray crystallography. I am referring to the attached image: In the image, A= electron density, Z= distance traveled, λ= X-ray ...
1
vote
0answers
138 views

sum of $p$th roots of unity

Let $p$ be an odd prime. Let $\mathbb Z_p$ be the ring of integers from $0$ to $p-1$, i.e. $\mathbb Z_p=\left\{0,1,\dots,p-1\right\}$. Let $r$ be a positive integer. Then, for all values of $p$, is ...
1
vote
0answers
185 views

Approximation of an exponential sum

Consider the flowing exponential sum with $x,\,y \ge 0$ and $c_i,\, d_i$ is a real number for $i=1,..,N$ $$ E=\sum\limits_{i = 1}^N \exp \left(- \left( x - c_i \right)^2 - \left( y - d_i \right)^2 ...
1
vote
0answers
77 views

Sum of power and Bernoulli intuitive discover

I really looked up to the 10th page of google, and the PDF's I find aren't complete, and definetively have NOT intuitive explanation about Bernoulli's discover. I know that he observed the formulas of ...
1
vote
0answers
189 views

Non-trivial upper bound for binomial sum

I'm trying to upper bound the following sum: $$ \sum\limits_{k=0}^n \begin{pmatrix} n \\ k \end{pmatrix} e^{-\frac{m}{2^k} } $$ where $n>0$ is fixed and $0\leq m \leq 2^n$. A trivial upper ...
0
votes
0answers
15 views

Multiple Integral Approximation

Does anybody know how to simplify following multiple integration ? ...
0
votes
0answers
29 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, ...
0
votes
0answers
105 views

Derivative of Log of Summation of exponential function (base e)

A financial formula that I am implementing requires that I find the first derivative of a function to find a local maxima, from scratch. Can someone please help me with finding the first derivative of ...
0
votes
0answers
29 views

Maths for a half-life style calculation

Struggling to see a very simple way to do the following calculation. It strikes me that it must be similar to working out what percentage of uranium atoms in a sample of uranium will decay in a given ...
0
votes
0answers
35 views

Simplified formula for the following sum

I have the following sum $$\sum_{n\ge 0}\sum_{0\le \alpha_i\le n+1,\\ \sum_{k=1}^N \alpha_i=n+1} \frac{e^{-(\rho+s)\left(\sum_{k=1}^N \alpha_kT_k\right)}\left(\rho\sum_{k=1}^N ...
0
votes
0answers
43 views

Removing backups in an exponential fashion

Background: I want to create a backup system that utilizes the full space of a hard-disk. Given that all backups are approximately equal in size this means that I can save a fixed amount of backups. ...
0
votes
0answers
13 views

Find the inverse function of a function relating to limited exponential sum

The function is given out as: $$y = 4x + {x^m} + {x^{ - m}},where{\text{ 0 < m}} \leqslant {\text{1, 0 < }}y < 6;$$ Closed form will be highly appreciate,but approximate results is also ...
0
votes
0answers
11 views

Is there a 'mild' product function?

I'm simulating an economy, each person has a list of integers representing the quantity of each resource they possess (for example: 5 water, 6 food, 2 education). From this I want to calculate ...
0
votes
0answers
29 views

About complex exponential summation

Let $f:\,\mathbb{R}^{+}\rightarrow\mathbb{R},\, f\in C^{\infty}\left(\mathbb{R}^{+}\right)$ and such that $f\left(n\right)>0\,\forall n\in\mathbb{N}$. Let $c>0$ a real number, $N>0$ a large ...