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
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How can I resolve $\sum_{x=0}^{\infty} xe^{-x/\theta}$?

I stumble on this summation during an exercise. How can I resolve $\sum_{x=0}^{\infty} xe^{-x/\theta}$?
srodriguex
  • 801
3
votes
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Plotting exponential partial sums in the complex plane

I was plotting the following sequence of points $(a_n)_{n = 0}^\infty$ in the complex plane for various reals $\alpha > 1$: $$a_n = \sum_{k = 0}^n e^{i k^\alpha}$$ I found that for many values of $\alpha > 1$ the path traced by $(a_n)$ displayed…
Thomas
  • 932
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Exponential equation in x,y,z

Find all positive integers x, y, z satisfying $$x^{y^{z}} \cdot y^{z^{x}} \cdot z^{x^{y}}=5xyz$$ I took log on both sides, which led me to $$(\log x)(y^{z}-1)+(\log y)(z^{x}-1)+(\log z)(x^{y}-1)=\log 5$$ However, further simplification is not clear.…
Tejas
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The identity of a polynomial sum

I am wondering if there is a recursive formula to calculate $$S=1^{k}+2^{k}+3^{k}+\dots+n^{k}$$ Where $k$ and $n$ are natural numbers.
Denis
  • 187
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How many times do you need to double previous result to get at least $10^{82}$?

This is pretty straightforward, but I'd like to study, how find out, how many times do you need to double previous result of calculation to get some sum, for example: $10^{82}$ $1\times 2 = 2$ $2\times 2 = 4$ $4\times 2 = 8$ $8\times 2 = 16$ n.…
MarkokraM
  • 671
2
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1 answer

Simple expression:$ a - a^{-1}$ = ...

I got stuck with one simple expression, I hope get some help with it: If $a-\frac{1}{a}=\frac{3\sqrt7}{7}$, so $a^4+\frac{1}{a^4}=$
gintko
  • 169
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Express sum of complex exponentials as 1 + sum of cosines

I'm told that I can express the following sum of complex exponentials: $$\sum_{n=0}^6 e^{-j\Omega n}$$ As 1 plus 3 cosine terms. I'm having a really hard time arriving at this, I see where the 1 comes from, when n = 0 and I…
AlwaysLearning
  • 23
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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 formular to be right two conditions I thought that the left…
user2874612
  • 109
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vote
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Math Quiz question $(5b^4)^2$

So in this question, I got $25b^8$, though my teacher marked it wrong and said it was $5b^8$, he said don't multiply the bases. But $5$ is not the base, $b$ is the base and $5$ is the coefficient. I am confused, I checked my word in a calculator and…
Jason
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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. $$ \sum_{n=0}^{L-1}e^{-j\frac{2\pi k}{L}n} = \frac{1-e^{-j2\pi k}}{1-e^{\frac{-j2\pi k}{L}}} $$ which somehow…
user50420
  • 55
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Figure out all primes p and q such that

Figure out all primes p and q such that $p^3$ + 19$q^3$ + 2018 is the cube of a prime.
허인행
  • 63
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summation of product of exponential fuction

I'am trying to prove the following equation. $$\sum_{n=1}^{N}f(n-x_1)f(n-x_2)=\delta(x_1-x_2)$$ where $f(x):=\frac{sin(\pi x)}{Nsin(\frac{\pi}{N}x)}$ , and $x_1,x_2$ are integers. First, I have used the Euler theorem for sine functions. $$…
Hoil Kim
  • 31
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Indices Question grade A*

I am a student and I am having difficulty with answering this question. I keep getting the answer wrong. Please may I have a step by step solution to this question so that I won't have difficulties with answering these type of questions in the…
OliviaAages
  • 183
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solve two term equation with different fractional exponents

Suppose: $$a = bw^f + cw^g $$ where $a,b$ and $c$ are known, and $f$ and $g$ are known fractional exponents Ex. $50000 = 200w^{0.72} + 4000w^{0.19}$ How can one solve for the value of w?
Harry Marx
  • 11
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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 period, that's why i mentioned half life in the…
Max Williams
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