Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [exponential-sum]

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

In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function $ \exp(2\pi ix).\,$ Therefore a typical exponential sum may take the form $$ \sum \exp(2\pi ix_n), $$ summed over a finite sequence of real numbers $x_n$.

477 questions
0
votes
1 answer

Understanding solution to: $\sum_{k=0}^{N-1}\sum_{m=0}^{N-1}x[m]e^{-\rm j\frac{2\pi}{N}mk}e^{-\rm j\frac{2\pi}{N}nk}$

I cannot understand the solution that is presented below: \begin{eqnarray} X[k]&=&\sum_{m=0}^{N-1}x[m]e^{-\rm j\frac{2\pi}{N}mk}.\nonumber\\ y[n]&=&\sum_{k=0}^{N-1}X[k]e^{-\rm j\frac{2\pi}{N}nk}\nonumber\\ …
user2051561
  • 103
0
votes
1 answer

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 gms of the item to the box, then on the 7th day…
user93353
  • 466
  • 2
  • 18
0
votes
1 answer

Discrete Gaussian density function sum drawn from Gaussian distribution

Doing some analysis of my problem, I have come up with the following equation (I tried to search this problem but no luck) $S_N = \sum \limits_{i=1}^{N} \exp{\left(-\frac{X_i^2}{2\sigma^2}\right)}$ where $X_i \ \ \text{~} \ \ {N(0, \sigma^2)}$ It…
ben Heo
  • 1
  • 3
0
votes
0 answers

Why are there multiple spirals in the Weyl-type sum?

This question comes from the Mathematica documentation. In the documentation of $e$ 1, it shows a neat example of Weyl-type sum: Letting $$ f(n;r,p) \triangleq \sum_{k=1}^{n} e^{i~r~k^p}, $$ then the sequence…
SY Z
  • 1
0
votes
1 answer

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 expressing the multiplication of exponential…
dkssud
  • 25
0
votes
0 answers

What is the sum of quadratic function?

I would like to know the result of the summation of $\sum\limits_{n=0}^{500}e^{i2{\pi}{an^2}/(b+cn)}$, in which a, b, and c are real constant value. No need for an exact solution, an approximation would be fine. Thanks in advance.
Peteryu010
  • 1
0
votes
1 answer

How to write out this result? 10^116.75

I entered the following into my calculator: 10^116.75 And from a Google Search, I landed on this: 5.623413e+116
John Smith
  • 193
0
votes
1 answer

Is there a closed-form solution for this summation?

Context: I want to find the amount of time it takes for the sum of exponential growth to reach a given value. $x_t=\sum\limits_{i=0}^{t}x_0(1+r)^i$ In other words, is there a formula I can use to solve for $t$? If there is a solution, can you please…
Cameron
  • 109
0
votes
1 answer

Explain on why the require time is not linear.

Here is the question: I got capital X, and I will compound interest (4%) to roll the capital, until the the interest grow > 3000. So, I start from 6000 (X), here is my example excel: I mark down the cell number on when it reaches more than 3,000,…
Ted Wong
  • 111
0
votes
1 answer

summation of an arbitrary function multiplied by exponential

I'm trying to find some function $g(k)$ such that $$\sum_{k=0}^{\infty} g(k) \frac{(n \lambda)^k}{k!} = 0 $$ The textbook says that there is only one solution, that is $g(k)=0$ for all $k$. But I cannot see why it is so. It is also constrained that…
Satish
  • 108
0
votes
1 answer

Convert 60 Linear steps to Exponential from (0 - 5)

Let's say I have 60 linear steps from 0 - 5. therefore 5/60 = 0.0833 (0, 0.0833, 0.2499,..4.91667,5) Thus creating a straight line. Now, what would I need to do if I wanted to convert these 60 steps from 0 - 5 into exponential? what equation could…
konobyBYnight
  • 1
0
votes
1 answer

Summation series for a $\times$ b^x

I know theres a summation series for basic $a^x$. Like $2^{x-1}$ is summed by $2^x-1$. Then how would you sum $5\times2^{x-1}$?
Gᴇᴏᴍᴇᴛᴇʀ
  • 915
  • 2
  • 14
  • 28
0
votes
1 answer

Calculating $\frac{e^{\frac{z_1}{T}}}{e^{\frac{z_1}{T}} + e^{\frac{z_2}{T}}}$ from $\frac{e^{z_1}}{e^{z_1} + e^{z_2}}$

Let's say I know the value of $\frac{e^{z_1}}{e^{z_1} + e^{z_2}} = x $ (say) Now I want to get the value of $\frac{e^{\frac{z_1}{T}}}{e^{\frac{z_1}{T}} + e^{\frac{z_2}{T}}} = y $ (say), where $T \epsilon \mathbb{Z^+}$, some integer constant. Now,…
Koustav
  • 105
0
votes
1 answer

Calculating Maximum Population Size After 106 Years, Starting With 3 Breeding Pairs

I have been looking all over for a formula or program to determine a maximum possible human population size within a $106$ year span. This specifically is in response the the claim by the Creationist organization "Answers in Genesis" that The Tower…
Greg Brahe
  • 1
0
votes
1 answer

binary search tree size calculation

how do I calculate the number of elements a binary search tree can hold if have the height? For example a tree of height 3 can have 7 elements (7=1+2+4), and a tree of height 4 can have 15 elements (15=1+2+4+8).
Kouros
  • 103
1
2
3