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

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1answer
26 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
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0answers
15 views

Multiple Integral Approximation

Does anybody know how to simplify following multiple integration ? ...
0
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0answers
28 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
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2answers
50 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
27 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 ...
1
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0answers
34 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.
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2answers
26 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
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1answer
29 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
39 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
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1answer
40 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
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3answers
86 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
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 ...
2
votes
1answer
34 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
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0answers
16 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
123 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
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2answers
44 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
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2answers
37 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
188 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 ...
0
votes
2answers
39 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
1
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1answer
58 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
2
votes
4answers
49 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 ...
1
vote
4answers
79 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
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votes
2answers
75 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}}- ...
0
votes
0answers
93 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 ...
2
votes
1answer
44 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 ...
0
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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
1answer
52 views

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 ...
0
votes
1answer
44 views

How can I derive this summation?

I have the following equation, $$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] ...
0
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0answers
33 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
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
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0answers
41 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. ...
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 ...
0
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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 ...
4
votes
2answers
1k 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] + ...
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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 ...
1
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3answers
119 views

Solve for Exponent in a summation

For a simple summation: $$ Z = \sum_{x=1}^X x^n $$ Solve for n. I have googled unsuccessfully and I do not remember this.
2
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0answers
86 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 ...
3
votes
2answers
107 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} ...
2
votes
1answer
104 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, ...
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 ...
3
votes
3answers
70 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
3answers
252 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 ...
1
vote
2answers
54 views

What is the distribution of the average of IID exponential RVs? (have to use MGFs)

Just a quick question as I cannot figure this out. Here is the problem: Q: What is the distribution of the following? ($Y_1 + Y_2 + Y_3 +\cdots+ Y_n$) $ /n$ if each $Y$ is independent and identically ...
3
votes
2answers
156 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 $\sum_{x=0}^{\infty} xe^{-x/\theta}$?
0
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3answers
72 views

How would you solve for $x$ in $7^{x}=5^{x-4}$?

How would you be able to make these the same base? I tried to put it into a base $10$ formula, but it ended up making $x=0$.
0
votes
1answer
34 views

Question about $\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Let $z$ be a complex number and $n$ a positive integer. Let $a_n$ be a sequence of $n$ real numbers such that $a_n > 1$ for every $n$. Define $f_n(z;a_1,a_2,...,a_n)=\Sigma_{i=1}^n ...
2
votes
1answer
65 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 ...
1
vote
2answers
50 views

Summation simplification on exponentiaion

I would like to ask help for the following question: Simplify $$\sum_{x=0}^{+\infty}{e^{-x/\mu}}$$ Will integration work here? Meaning, can I do it this way? ...
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 ...
2
votes
1answer
59 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 ...