# Questions tagged [exponential-sum]

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

339 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
1answer
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### 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: ...
2answers
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### 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 ...
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### 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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### 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 ...
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### 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 ...
3answers
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### 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 ...
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### 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 $i \times j$
2answers
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### 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 ...
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### 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 ...
2answers
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### 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, ...
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### 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 ...
1answer
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### 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 ...
1answer
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### 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 ...
1answer
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### 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 ...
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