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Questions tagged [exponential-sum]

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

2
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1answer
40 views

Exponential Type Series [duplicate]

I'm looking for a closed expression (if it exists) of the following sum: $$\sum_{m=0}^{\infty} \frac{m^n}{m!}c^m$$ where $n \geq 1$ is a positive integer, and $c$ is a fixed constant. The series seems ...
0
votes
1answer
39 views

Log-Sum-Exp as an approximation of min function

I can prove that the function: $$f(\tau, x_1, x_2, ..., x_N) = -\tau \log \frac{1}{N} \sum_{i=1}^{N} \exp{\left(-\frac{x_i}{\tau}\right)} $$ converges to $\min(x_1, x_2, ..., x_N)$ for $x_i \geq 0$ as ...
4
votes
1answer
102 views

Evaluating $\sum_{n=0}^{\infty}ne^{1-n}$ using calculus

I'm trying to evaluate the following integral which popped up in MIT Integration Bee 2015 which involves the floor function. $$\int_{0}^{\infty}\left(xe^{1-x}-\lfloor x\rfloor e^{1-\lfloor x\rfloor}\...
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0answers
6 views

Heat kernel trace bound

Are there any ways to efficiently bound the sum of the following type: $$\varphi(t) := \sum_{k=1}^{\infty}{\exp(- k^{\frac{2}{d}} \cdot t)}$$ where $d > 0$, $d \in \mathbb{N}$. This sum ...
1
vote
0answers
22 views

Sum of exponential series of equal mean and variance

Assuming $A$ and $B$ are two non-negative real-valued random variables such that $\mathrm{E}(A)=\mathrm{E}(B)$ (equal means) $\mathrm{Var}(A)=\mathrm{Var}(B)<\epsilon$ (equal small variances) is ...
0
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0answers
12 views

Solve $\sum_nA_nr^{y_n} = V$ for $r$

I'm trying to evaluate the overall return on a sequence of share purchases over some time, where each purchase of value $A_n$ occurred at $y_n$ time ago. The total current value of the portfolio is $V$...
0
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2answers
26 views

Why is $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\delta(k-N+k_0)$?

EDIT: $\delta$ is the Dirac delta function and in the context it is defined as $\delta(0)=1$ and $0$ for all $n\neq 0$. I am having trouble concluding that $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\...
0
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2answers
28 views

Exponential double angle formula

My question is whether someone could provide a proof for the following identity: $$ \frac{1 - e^{int}}{1 - e^{it}} = e^{i(n-1)t/2} \frac{\sin(nt/2)}{\sin(t/2)} $$ Motivation: The left hand side is ...
0
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2answers
47 views

Find $P(X_1+X_2<X_3)$

Given that $X_i\sim Exp(\lambda_i),i\in\mathbb{N}$, find $P(X_1<X_3)$ $P(X_1+X_2<X_3)$ I know that for 1. $P(X_1<X_3)=P\big(X_1=\min\{X_1,X_3\}\big)=\frac{\lambda_1}{\lambda_1+\...
0
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1answer
26 views

Convergence value of series with exponentials

I was solving a problem and I came across the following expression, $$\sum_n^N {N \choose n}\exp[-\beta n\omega]$$ I was looking for the convergence of this series but I couldn't find any resources ...
0
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1answer
37 views

Infinite sum - rewriting $\exp{1/z}$

I'm having some trouble understanding a step. It's from this question on math stackexhange. It's about finding the laurent series of $\exp{(z + 1/z)} $ i understand we can do this: $\exp{(z + 1/z)} =...
0
votes
1answer
60 views

Finding value of $\int_{0}^{1} \int_{y}^{1} e^{x^2} dx dy$

I have an integral, $$\int_{0}^{1} \int_{y}^{1} e^{x^2} dx dy$$ I tried to apply integration by parts on the inner integral with respect to $x$ but it didn't seem to progress. Does anyone have a ...
3
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0answers
42 views

Infinite complex series of hyperbolic sine.

Can it be said that the following equivalence is true? $$ 0 = \sum_j^\infty (e^{\zeta_j} - e^{-\zeta_j}) \Leftrightarrow \sum_j^\infty e^{\zeta_j} = \sum_j^\infty e^{-\zeta_j} \quad , \quad \...
1
vote
1answer
45 views

fitting triple exponential term function to data

The function, I am trying to fit to data is: $$y(x) = −(A+B)e^{−x/a_1} + 𝐴e^{−x/a_2} + Be^{−x/a_3}$$ this function is a little bit different to Is it possible to find initial parameters when fitting ...
10
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2answers
224 views

Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$ .

Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$. It is clear that the equation can be rewritten as follows: $$ (3 \times 7)^a+(4 \times 7)^b=(5 \times 7)^c $$ If $a=b=c=2$ then ...
0
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1answer
49 views

Need help with Natural Logs

Firstly, I don't know much about Natural Logs to begin with. I am actually an SQL developer and I have a query where I am calculating a 'Return' metric as follows: ...
5
votes
2answers
96 views

Sum of exponential terms and binomial

I would like to calculate the following expression with large $m$: $$\sum^{m}_{q=1} \frac{(-1)^{q+1}}{q+1} {{m}\choose{q}} e^{-\frac{q}{q+1}\Gamma}.$$ But, due to the binomial, the computer gets ...
0
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1answer
21 views

quadratic gauss sum calculation in sage

I tried to calculate quadratic gauss sum in SAGE but it works just for primes 3 and 5 which are $i\sqrt{3}$ and $\sqrt{5}$ respectively. p=3 print sum((legendre_symbol(x,p))*(e^(2*piIx/p)) for x in ...
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2answers
33 views

Evaluate $\sum_{y=0}^{2^n-1}e^{2\pi iy(z-x)/2^n}$

I am trying to evaluate $$\sum_{y=0}^{2^n-1}e^{2\pi iy(z-x)/2^n}$$ where $n\in\mathbb{Z}$ and $x,z\in\mathbb{Z}_{2^n}$. Clearly, if $x=z$, then $\sum=2^n$. But I am unsure about when $x\neq z$. I ...
2
votes
1answer
23 views

Expressing $c = 1 - \exp\left(\lambda_1 p + \lambda_3 q\right)$ as a product of two terms

This question is motivated by the response provided in this question Considering the same equation which is shown below $$c = 1 - \exp\left(\lambda_1 R^2 \left(\frac{2\pi}{3}-\frac{\sqrt{3}}{2}\right)...
2
votes
1answer
48 views

Calculating moments of a function of independent Gaussians

Suppose we have $$f(g) = \log\left(\sum_{i = 1}^N \begin{cases} |a_i| e^{g_i}, & a_i \geq 0 \\ |a_i| (2 - e^{g_i}), & a_i < 0 \end{cases} \right) $$ where $g = (g_1, \ldots, g_N)$, each $...
0
votes
1answer
44 views

How to derive the discrete delta function from geometric sum of complex sinusoids?

Using this in the context of Fourier transforms. This should probably be an easy derivation for you guys, but I forget how to derive it. $$ \sum_{n=0}^{L-1}e^{-j\frac{2\pi k}{L}n} = \frac{1-e^{-j2\pi ...
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0answers
24 views

Finding the close form of the expression containing two summations

I am trying to deduce an equation and right now I am stuck at a point where there are two summations. Below is the equation where I am stuck. Kindly, help to deuce this further to obtain a closed form ...
1
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3answers
61 views

Prove a set has an area of $\;e^h - 1$

Prove through appropriate estimations with simple sets that the set $$E =\{(x, y) ∈ ℝ^2 ∣∣ 0 ≤ y < e^x \quad \& \quad 0 ≤ x < h\}$$ has an area of $$e^h - 1.$$ I believe I have to ...
0
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0answers
28 views

Roots of a real exponential sum

Suppose I had some exponential sum $\ f(x)\ $ of the form: $$f(x) = \sum_{i=1}^{N} \left( c_i \ e^{a_i x} \right)$$ where: $$c_i, a_i \in R$$ $$a_i \leq 0$$ Is there a quick way to find the roots, $\ \...
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0answers
23 views

Laplace Transform, inverse Laplace transform differential equation

how would you find the solution x(t) of $\frac{dx}{dt} = rx + 1000\sum_{i=1}^\infty \delta(t-i)$ for $x(0)=0$ using laplace transformation? Am I heading in the right direction as I am stucked? $sX(...
0
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0answers
36 views

How to take log of sum product of matrix and e^x?

I wonder how to take log of this sum function where it has $e$ inside $$f(x) = \sum_{ij}V_{ij}\exp\left((x-N_{ij})^2\right)$$ where $V_{ij}$ and $N_{ij}$ are matrix size $i \times j$
1
vote
2answers
18 views

Summation over multiple arguments

This might seem stupid, but I'm really stuck. I don't understand how to calculate the following explicitly: $$\sum_{s_1=\pm1} \sum_{s_2=\pm1} \sum_{s_3=\pm1} e^{-{s_1s_2}}e^{-{s_2s_3}}$$ (it's the ...
2
votes
2answers
77 views

Convergence of $\lim_{N\to\infty}\sum_{n=1}^N \exp(-N\sin^2(\frac{n\pi}{2N}))$ and $\lim_{N\to\infty}\sum_{n=1}^N \exp(-\sin^2(\frac{n\pi}{2N}))$

I can't find the right approach to tackle the question whether $$\lim_{N\to\infty} \sum_{n=1}^N \exp\Bigg(-N \sin^2\left(\frac{n\pi}{2N}\right)\Bigg)$$ and $$\lim_{N\to\infty} \sum_{n=1}^N \exp\Biggl(-...
3
votes
1answer
117 views

All real number for which $n$ in $5^n+7^n+11^n=6^n+8^n+9^n$ [duplicate]

Finding all real number $n$ in $$5^n+7^n+11^n=6^n+8^n+9^n$$ Try: From given equation $n=0,1$ are the solution But i did not understand any other solution exists or not Although i have ...
0
votes
1answer
31 views

Stuck at summation of exponential.

I want to find the coefficient but I am stuck. The formula is: $$\frac{1}{10}\sum_{n=0}^{9} e^{-jk\omega_0n}$$ In order to solve this I am using the following formula: $$\sum_{n_1}^{n_2} \alpha^{n} ...
0
votes
2answers
75 views

Is there a closed form for $\sum_{k=0}^n2^k\binom{2n+1}{2k}$, the sum of binomial coefficients times powers of two, for even indices?

I'm hoping for a nice and simple closed form for the sum $$\sum_{k=0}^{n}2^k\binom{2n+1}{2k}.$$ Searching this site I found many nondescript titles but no duplicates, though I wouldn't be surprised if ...
0
votes
2answers
63 views

How do I sum 1/6( e^t + e^2t + … + e^6t)? [duplicate]

I have a question about a sum when calculating moment generating function. The question is : "Find the moment generating function for each of these two random variables. (i) $X$ = outcome a die toss, ...
3
votes
3answers
116 views

re-calculating exponential sum when exponent changes

If $A_c$ can be calculated as follows: $A_c=\sum_{k=1}^{N} a_ke^{ck}$ Where c is a known real constant and $a_k$ is a known series comprising real numbers which cannot be described by a function $f(...
0
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0answers
32 views

Bounds for a character sum $\sum_{n } \dfrac{\eta(n/N)\chi(n)\sin(2\pi \delta n)}{n}$

Let $\chi$ be a primitive Dirichlet character of large modulus $q > 1$, and let $\delta \in \mathbb{R}$ be fixed. Assume $\eta : \mathbb{R} \to [0,1]$ is smooth compactly supported on $[1,2]$. For ...
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0answers
17 views

Weighted and exponential distribution

I have no advanced mathematics knowledge and I would like to come up with a formula that would express my thinking. I have a pot of money of $1,000 that I would like to distribute to 3 persons ...
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votes
3answers
30 views

Exponent Question

This is the equation I am trying to solve: $$4^{27} + 4^{27} + 4^{27} + 4^{27} = 4^{y-1}.$$ I understand that I should convert to base $4$ and solve but I am not sure about how to do that, are there ...
1
vote
1answer
62 views

How to represent a sum of exponentials as one exponential

Let us say, we have a sum of exponentials, such as: $$ a (e^{iwt}+e^{-iwt})+b (e^{i2wt}-e^{-i2wt})+c (e^{i3wt}+e^{-i3wt}). $$ Is there a way that represents the above sum as a sum of two exponentials ...
2
votes
1answer
93 views

Maximize the modulus of a sum of complex exponentials on a given interval

I am trying to find $$\hat{t}=\underset{t \in [a,b]}{\text{argmax}}\,\left|\sum_{j=1}^n e^{i x_j t}\right|$$ where $t$ is a real number, $[a,b]$ is a given interval, $i$ is $\sqrt{-1}$, and $x_j$ are ...
0
votes
1answer
27 views

Multi-term exponential curve fit

Good day everyone. On a simple 2D chart with the horizontal axis as the time t. Let's say you have a few data points available. A function F(t) must pass through each one of these data points. This ...
-2
votes
1answer
29 views

I need help simplifying a sum problem that involves a binomial raised to a power

I have come across a problem in my homework that describes the sum of a binomial squared, and I can't think of a way to simplify it. I have an idea that it would involve $\frac{\left(n\right)\left(n+1\...
0
votes
2answers
114 views

Summation of Double Exponential Series [closed]

Is there any known closed form or tight bound analysis (big-O or big-$\Theta$) for $\sum_{i = 0}^{n} 2^{2^i}$?
0
votes
1answer
39 views

Finding real and imaginary parts of $\frac{1-e^{2i \pi x}}{R \left(1-e^{\frac {2i \pi x}{R}} \right)}$

I have a function given by $$\frac{1-e^{2i \pi x}}{R \left(1-e^{\frac {2i \pi x}{R}} \right)}$$ Using Euler's formula, I expand into real and complex components: $$\frac{1-\cos 2 \pi x-i\sin2 \pi x}...
0
votes
1answer
124 views

Inequality for exponential sum

I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
1
vote
1answer
39 views

Reverse summation of a complex exponential

In the equation below: $$s_l(t)=\sum_{k=-\lfloor{N_{sc}^{RB}/2}\rfloor}^{\lceil{N_{sc}^{RB}/2}\rceil-1}a_{k^{(-)},l}^\,.e^{j2\pi(k+1/2)\Delta f(t-N_{CP,l}T_s)}$$ where: $0\le t\lt (N_{CP,l}+N)\times ...
0
votes
1answer
36 views

can we preserve correctness of inequality after adjusting all involved exponents?

Problem Lets assume we have an inequality such that it involves only positive, real values and all exponents within this inequality are integer multiples of $2n$ (even), we suppose this inequality is ...
0
votes
0answers
26 views

Add periodic functions to obtain constant

I have to fulfill the following equation $$0=K +\sum_{m=0}^{m'}C^m_+e^{i\sqrt{\omega^2-(\frac{\pi m}{a})^2}z}+\sum_{m=0}^{m'}C^m_-e^{-i\sqrt{\omega^2-(\frac{\pi m}{a})^2}z}$$ $m'$ is defined such ...
1
vote
0answers
51 views

How to make Sum of sinusoidal signals with different frequencies have different absolute values of positive peak and negative peak?

Let me define a signal $$f(t)=\sum_{i=1}^N a_i \cos(\omega_it+\theta_i),$$ where $\omega_i = 2\pi f_i$ with a frequency $f_i$. I want to make a specific $f(t)$ by determining $a_i$'s and $\theta_i$'s ...
1
vote
1answer
29 views

Optimization Problem for Exponential Polynomials

This question was asked on mathoverflow more than a year ago, with no answers Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity $$ \max_{f(n)\leq \...
3
votes
0answers
78 views

Probability that one M/M/1 queue empties before another queue with no external arrivals?

Suppose there are 2 independent queues A and B having m and n customers respectively. The service time of each of the queues are Exponentially distributed with service rate as $\mu_{A}$ and $\mu_{B}$. ...