Linked Questions
10 questions linked to/from What does Mellin inversion "really mean"?
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Examples of common false beliefs in mathematics
The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
74
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9
answers
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Motivating the Laplace transform definition
In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem.
The Laplace transform of a function $f(t)$, ...
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8
answers
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How does one motivate the analytic continuation of the Riemann zeta function?
I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
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answers
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Motivation and physical interpretation of the Laplace transform
Concerning the one-sided Laplace transform,
$$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$
what is a motivation to come up with that formula? I am particularly interested in "physical&...
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answers
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An "analytic continuation" of power series coefficients
Cauchy residue theorem tells us that for a function
$$f(z) = \sum_{k \in \mathbb{Z}} a(k) z^k,$$
the coefficient $a(k)$ can be extracted by an integral formula
$$a(k) = \frac{1}{2\pi i}\oint f(z) z^{-...
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3
answers
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Ramanujan's Master Formula: A proof and relation to umbral calculus
The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...
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votes
2
answers
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Newton series and Fourier transform - is there an analogy?
Fourier expansion for a function:
$$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$
Newton series expansion of a function:
$$f(x)...
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votes
2
answers
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What's the matrix of logarithm of derivative operator ($\ln D$)? What is the role of this operator in various math fields?
Babusci and Dattoli, On the logarithm of the derivative operator, arXiv:1105.5978, gives some great results:
\begin{align*}
(\ln D) 1 & {}= -\ln x -\gamma \\
(\ln D) x^n & {}= x^n (\psi (n+1)-\...
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votes
3
answers
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Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
I am considering a PhD research topic. I only have a math Bachelor's degree with working experience mostly in teaching and I have been working on a paper. I have deep interest in Fourier Series and ...
3
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answer
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Solution for Moment problem
I want to invert a sequence of moments and find a function f(x) satisfying:
$$m_r=\int x^{r}f(x) dx=\int x^{r} dF(x)$$
The sequence of moments is given by:
$m_{2s+1}=0$
$m_{2s}=\sum_{k=1}^{s}\binom{...
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