Questions tagged [special-functions]
This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
4,888 questions
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Two Term Dilogarithmic Identity Simplification
I was able to reach a non-trivial two term golden ratio ($\varphi$) Dilogarithmic identity from independent results found in recent literature and was left stuck trying to simplify it further, but was ...
- 31
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Reduction formula or Whipple/Bailey-type transformation for a double terminating, double balanced Kampé de Fériet function $F_{1:2;2}^{1:3;3}(1,1)$
Context:
I have a double terminating, double balanced Kampé de Fériet function:
$$F_{1:2;2}^{1:3;3}\left[\left.\begin{matrix}
M:\:& A,\;B,\;C\; &;& F,\;G,\;H\\
N:\:& D,\;E\; &;&...
- 11
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Good known bounds for cumulative distribution functions
Since I have to teach a course involving some statistical hypothesis tests, while not being a statistician myself, I am looking for some simple, but possibly accurate, bounds for cumulative ...
- 219
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The solution of ODE using Frobenius' Method, how to explain its convergence?
How to state the convergence of the solution of this ODE and what is the solution? Also, Is there online site to check its solution so that I could analyse it whether my answer is correct, MatLab, ...
- 2,557
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Regularizing an exponential integral
Is there any way of regularizing the following integral and showing equality?
$$\int \int \int_{\mathbb{R}^3} \exp \left (- i (k_1+k_2)(k_2+k_3)(k_3+k_1) \right ) \, dk_1 dk_2 dk_3 = \frac{2 \pi \ln ...
user1295974
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Prove that $2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x$
I'd like to prove that
$$2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x.$$
Ok, someone said that this holds, but I tried really ...
- 307
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Maximisation of functions of the form $f(x) = \sqrt{1 - x^2} + (ax+b)x$
I am studying a function arising in the analysis of robust aggregation rules in distributed learning, but the question is purely analytical. The function I am facing depends on parameters $a, b > 0$...
- 948
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Prove $B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$
Question. Is there a simpler way to prove $$B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$$ where $B_n(x)$ is the $n$-th Bernoulli polynomial and $E_n$ is the $n$-th Euler number?
I have verified ...
user1295974
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Inverse Fourier transform for the square root of the Ohmic bath spectral function
I think this is a bit hopeless but let me ask just in case. Consider the real and positive function:
$$
\hat{f}(\omega) = \sqrt{\frac{\omega}{1-e^{-\frac{\omega}{T}}}} e^{- \frac{\omega^2}{4\Lambda^2}}...
- 619
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Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx$ have a known closed form? [duplicate]
I am studying the definite integral
$$
I = \int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx .
$$
The integral does converge:
as $x \to 0$, $\ln(1+x) \sim x$ and $\ln(1-x) \sim -x$, so the ratio tends to $-...
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150
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analytic continuation of the series $\sum_{n=0}^{\infty}\frac{n^2}{\sqrt{a^2+n^2}}$
I would like to understand how to make sense of the following divergent series, or at least to identify the appropriate analytic continuation of its general term:
\begin{align}
\sum_{n=0}^{\infty}\...
- 315
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Does the polylogarithm $\mathrm{Li}_k(x)$ solve a first or second order ODE/ADE?
I have been working with the Polylogarithm on several problems and I think it might help if I knew an algebraic/ordinary differential equation (ADE/ODE) which it satisfies (and just for the sake of it!...
- 2,556
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Definite integral of the two Bessel functions / (x-a)
In my work, an integral of the following type arose:
$$\int_0^\infty dx J_l(a x) J_l(b x) \frac{1}{x - c + i 0}.$$
Here $J$ is the Bessel function of the first kind. Assume that $a$, $b$, and $c$ are ...
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373
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Prove that $\int_0^1\operatorname{Li}_2\left(\frac{1-x^2}{4}\right)\frac{2}{3+x^2}\,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}$
I would like to prove that
$$\int_0^1 \operatorname{Li}_2 \left(\frac{1-x^2}{4}\right) \frac{2}{3+x^2} \,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}.$$
It’s known that $\Im \operatorname{Li}_3(e^{2πi/3})= \...
- 307
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Analytic sum of an alternating series$\sum\limits_{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}$
I recently came across the following series with a positive real number $a$:
\begin{align}
S(a) = \sum _{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}
\end{align}
Does anyone know if ...
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