Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [special-functions]

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

0
votes
0answers
37 views

Finding $\lim_{x\to\infty}\int_0^x\frac{|\sin t|}tdt-\frac2\pi \ln x$

I want to evaluate $$\lim_{x\to\infty}\int_0^x\frac{|\sin t|}tdt-\frac2\pi \ln x.$$ Related question:Asymptotic approximation of $\int_1^x \frac{| \sin t|}{t}dt$ as $x \to \infty$ It shows the $\...
2
votes
2answers
46 views

Dilogarithm property

I'm working on dilogarithms (Don Zagier 'The dilogarithm function', Matilde Lalìn 'dilogarithm, a cool function' )and I have encountered this six term relation: $Li_2(x)+Li_2(y)+Li_2(z)=(1/2)[Li_2(-...
0
votes
0answers
90 views
+50

Modified Riemann Zeta function - where do the non trivial zeros live now?

I'm curious as to where the non trivial zeros of the Riemann Zeta function live after you modify it in the following way: Take the multivariate complex function: $\phi(s,t)=\zeta(t)^{log(s)}, $ ...
0
votes
0answers
15 views

The formula of inverse Meijier-G?

I have an expression like $G^{\,2,1}_{1,3} \left(\begin{array}{c} a_1\\ b_1,b_2,b_3 \end{array} \middle\vert\ z\right) $, $a_1,b_1,b_2$ and $b_3$ are fixed number. ...
1
vote
1answer
56 views

integral of definite Meijer's G-function [closed]

I am working on my research and i have one integral seems difficult to me given as: $$\int_0^\infty x^{-\omega}\exp(-\theta x)\large{G}_{1,2}^{1,1} \left( \beta x^{\alpha/2} \left| \begin{array}{cc} ...
0
votes
1answer
47 views

Asymptotic Expression involving generalized hypergeometric function

What is the asymptotic expression for $F\left( \gamma \right) = \frac{1}{{\Gamma \left( m \right)\Gamma \left( k \right)}}\left[ {\frac{1}{k}{{\left( {\frac{{km\gamma }}{{\overline \gamma }}} \right)...
6
votes
0answers
217 views

An $\operatorname{erfi}(x)e^{-x^2}$ integral

I want to find an elementary evaluation of $$I=\int_0^\infty \left(\frac{\sqrt\pi}2\operatorname{erfi}(x)e^{-x^2}-\frac1{1+2x}\right)dx$$ where $\operatorname{erfi}(x)=\frac{2}{\sqrt\pi}\int_0^...
3
votes
2answers
98 views

Evaluating $\int_0^\infty\operatorname{erfi}(x)e^{-x^2}\frac{dx}x$

I want to evaluate $$I=\int_0^\infty\operatorname{erfi}(x)e^{-x^2}\frac{dx}x$$ where $\operatorname{erfi}(x)=\frac{2}{\sqrt\pi}\int_0^x e^{t^2}dt$. I can only prove this integral converges. Not-...
0
votes
1answer
37 views

Compute a limit that involves a hypergeometric function.

Let $a\ge 0$,$P_1\ge 0$ and $x\ge 0$ be real numbers. When answering Looking for closed form solutions to linear ordinary differential equations with time dependent coefficients. we came across a ...
0
votes
0answers
15 views

Reduction of appearing dilogarithms

I have a series of dilogarithms with various arguments and I was just wondering if it is possible to tell if they are all independent (that is to say, cannot be transformed into each other through e.g ...
0
votes
1answer
29 views

Extend an asymptotic relation with integer parameter to real parameters?

A text I'm studying derives the following, for integer s>0 and $n \to \infty$: $$S(s,n):=\sum_{k=0}^{2n}(-1)^{n+k}\binom{2n}{k}^s \sim \frac{1}{\sqrt{s}}\big(2\cos\big(\frac{\pi}{2s}\big)\big)^{2ns+s-...
0
votes
1answer
58 views

Calculate $I_{m, n}=\int_{0}^{\pi} \cos^{m}(x)\, \sin^{n}(x) \, dx,$?

Does anyone have a reference which gives the explicit expression of the following integral $$I_{m, n}=\int_{0}^{\pi} \cos^{m}(x)\, \sin^{n}(x) \, dx,$$ for any positive integers $m,n.$ Thank you ...
0
votes
0answers
25 views

Solutions of hypergeometric equation when $c\in\mathbb{Z}$

Let $E(a,b,c;z)$ be the hypergeometric differential equation $$ z(1-z)w''+(c-(a+b+z)z)w'-abw $$ with $w$ the unknown. It is well known that if $z\notin\mathbb{Z}$, then $E(a,b,c;z)$ has linearly ...
3
votes
1answer
50 views

What is the integral of $e^{x\cos\theta + y\cos\theta}$

Knowing that $$\int_{0}^{2\pi} e^{x\cos \theta } d\theta = 2\pi I_0(x)$$ where $I_0$ is the modified Bessel function. Is there a way/trick to find an analytical expression for $$\int_{0}^{2\pi} e^{x\...
1
vote
1answer
28 views

Relation between dilogarithm and its complex conjugate

I am looking for a relation between the dilog and its complex conjugate, that is can I simplify the following summation of terms $$f(z) = \text{Li}_2(z) + (\text{Li}_2(z))^*?$$ I have looked through ...
2
votes
3answers
83 views

A function that is in $C^k(\mathbb R)$ but not in $C^{k+1}(\mathbb R)$

Is there an example of a family of functions, index by $k$, that is in $C_b^k(\mathbb R)$ but not in $C_b^{k+1}(\mathbb R)$ for arbitrary $k$? $C_b^k(\mathbb R)$ is the space of functions with ...
1
vote
0answers
26 views

Solution of the differential equation $\ddot{x}(t)+\sin(\omega t)x(t)=cos[\eta(t)]$

The differential equation: $$\ddot{x}(t)+\sin(\omega t)x(t)=cos(\eta t)$$ has an analytical solution involving Mathieu functions. This is valid if both $\omega$ and $\eta$ are constant. Suppose the ...
0
votes
0answers
9 views

Proving a proposition about special functions in finite dimensional Hilbert space

We call frame function any mapping $f : \{ x \in H, \| x \|=1 \} =:\mathbb{S}(H) \mapsto \mathbb{R} \cup \{ \pm \infty \}$ (here $H$ denotes a finite dimensional Hilbert space) satisfying the ...
0
votes
1answer
33 views

Can $\prod_{n=1}^\infty(1+e^{2\pi i n\tau})$ be expressed in terms of the Euler Phi Function?

I am wondering if $$(-e^{2\pi i \tau},e^{2\pi i \tau})_\infty=\prod_{n=1}^\infty(1+e^{2\pi i n\tau})$$ can be expressed in terms of Euler's Phi Function. Any help is appreciated.
0
votes
2answers
34 views

Assume $x_1$,…,$x_n$ are positive real numbers - Is there a measurable function such that $f(x_1\cdot … \cdot x_n)=max(f(x_1),…,f(x_n))$?

Assume $x_1$,...,$x_n$ are positive real numbers - Is there a measurable function $f:\mathbb{R_+^n}\to \mathbb{R}_+$ such that $f(x_1\cdot ... \cdot x_n)=max(f(x_1),...,f(x_n))$? Here with $x_1 \...
0
votes
1answer
59 views

Asymptotic form for a generalized hypergeometric function

${}_3{F_2}\left( {m+m_s,{m_s},{m_s};1 + {m_s},1 + {m_s}; - \frac{{{m_s}\overline \gamma }}{{m{\gamma _0}}}} \right)$, where $m$, $m_s$, $\gamma_0$, and $\overline \gamma$ are positive real numbers. ...
1
vote
0answers
21 views

Transfer foundamental functions into MeijerG function

How can we transfer $\frac{1}{{1 + \delta x}}G_{0,1}^{1,0}\left[ {\theta x\left| {_n^ - } \right.} \right]$ into the form like $G_{u,v}^{m,n}\left( {} \right)$, where $\sigma$, $x$, and $\theta$ are ...
0
votes
0answers
19 views

An integration about Hypergeometric function need to be solved in closed form

$\int_\rho ^{+\infty} {\frac{{{x^{{m_d} + {m_p} - 1}}}}{{{{\left( {{m_p}x + {m_{ps}}{\lambda _P}} \right)}^{{m_p} + {m_{ps}}}}}}{}_2{F_1}\left( {{m_d} + {m_{ds}},{m_d};{m_d} + 1; - ax} \right)} dx$, ...
0
votes
2answers
65 views

Looking for closed form solutions to linear ordinary differential equations with time dependent coefficients.

Let $a \in {\mathbb C}$ and $b\in {\mathbb C}$ and let $n\ge 1$ be an integer. Consider a following family of Ordinary Differential Equations (ODEs). We have: \begin{equation} \frac{d^2 y(x)}{d x^2}...
5
votes
1answer
38 views

Looking for a sigmoid-like function with convex segment around origin

I am looking for a function with some specific properties (this is for a probabilistic simulation). It should be a function that is runs though (and is symmetric) the origin and asymptotically ...
-1
votes
0answers
28 views

An integration need to be solved in closed-form

$\int_\rho ^\infty {\frac{{{x^{{m_d}L + {m_p} - 1}}}}{{{{\left( {x + \lambda } \right)}^{{m_p} + {m_{ps}}}}}}{}_2{F_1}\left( {{m_d} + {m_{ds}},{m_d}L;{m_d}L; - x} \right)} dx$, where $m_d$, $m_p$, $...
1
vote
2answers
39 views

How can I graph $f(x)=\lfloor{x^2}\rfloor$ when the domain is $ℝ^{-}$.

How can I graph $f(x)=\lfloor{x^2}\rfloor$ when the domain is $ℝ^{-}$. I know that by definition $(\lfloor{x}\rfloor=m) ≡ (m≤x<m+1)$, so it follows that $(\lfloor{x^2}\rfloor=m ≡ (m≤x^2<m+1) ≡ (...
3
votes
0answers
39 views

Proof of converting the product of Bessel function of the second kind and the sine function into Meijer $G$-function.

How can I derive the formula which converts the product of Bessel function of the second kind and the sine function into Meijer $G$-function, $$ \sin(\sqrt{z})Y_v(\sqrt{z})=\frac{1}{\sqrt{2}}G_{3,...
0
votes
0answers
47 views

Convert product of Bessel functions of the first and second kind into a Meijer G-function

There exists such an equality which converts the product of Bessel functions of the first and second kind into a Meijer G-funtion, $$x^\mu J_\nu(x)Y_\nu(x)=-\pi^{-1/2}G_{13}^{20}\left(x^2\left| \...
0
votes
1answer
34 views

integrating modified gamma functions

I have an integral that looks like an incomplete gamma function, with another factor in front \begin{equation} I = \int_x^y \frac{1}{a^{c+1}}[1-t]^bt^ce^{-\frac{t}{a^2}}dt \end{equation} Is there a ...
1
vote
1answer
66 views

Show $\sum_{k=0}^\infty \frac{1}{(k+a)\binom{n+k}{k}}=\frac{-\,n!}{(1-a)_n}\Big(\gamma+\psi(a)+\sum_{k=1}^{n-1}\frac{1}{k}\frac{(1-a)_k}{k!}\Big)$

The conjectured identity of the title, $$(C)\quad\sum_{k=0}^\infty \frac{1}{(k+a)\binom{n+k}{k}}=\frac{-\,n!}{(1-a)_n}\Big(\gamma+\psi(a)+\sum_{k=1}^{n-1}\frac{1}{k}\frac{(1-a)_k}{k!}\Big)$$ arose in ...
1
vote
0answers
65 views

Integral of the bessel function

i have a qustion on the integration of a modified bessel function. According to the reference: Werner Rosenheinrich,"TABLES OF SOME INDEFINITE INTEGRAL OF BESSEL FUNCTIONS OF INTEGER ORDER", 2017 $\...
2
votes
0answers
84 views

Conjecture on limit of a partial fraction expansion involving tan and cot

Let $q$ be an integer such that $q \equiv 1\mod{4}$ or $q \equiv 2\mod{4}$; i.e., q={... -7,-6,-3,-2,1,2,5,6,9...} I seek a proof of the following conjecture: $$(C)\quad \lim_{n \to \infty} \sum_{k=1}^...
4
votes
1answer
81 views

Calculating the first zero of Riemann Zeta Function by hand

There are lots of pages on MSE and other websites regarding few first zeros of The Riemann Zeta Function. On MSE [this] is by far the closest (or the only one I've found) explanation for finding the ...
2
votes
1answer
63 views

Elliptic integral of a quartic

I am trying to compute the following integral: \begin{equation} I(a) = \int_{-\infty}^{+\infty} \left[ 1-x^2 + \sqrt{(1-x^2)^2 + a} \right] dx \end{equation} where $a >0$. I am pretty confident ...
1
vote
0answers
84 views

Integration of $\int_0^l\int_0 ^{2\pi} \frac {\exp (i k\sqrt{ (r^2+r (vt) \sin \phi)})}{r^2 + r (vt) \sin \phi+d^2} \, r \,dr \, d\phi$.

I have faced some difficulties to do the following integral $$\int_0^l\int_0 ^{2\pi} \frac {\exp (i k \sqrt{(r^2 + r vt \sin \phi)})}{r^2 + r vt \sin \phi +d^2} \, r \,dr \, d\phi$$ Where, $vt$ ...
5
votes
0answers
33 views

Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...
2
votes
1answer
38 views

prove that $x∉ℤ⇒\lfloor{-x}\rfloor=-\lfloor{x}\rfloor-1$

prove that $x∉ℤ⇒\lfloor{-x}\rfloor=-\lfloor{x}\rfloor-1$ So, I have that $(\lfloor{-x}\rfloor=n)⇔(n≤-x<n+1)$ And I also have that $(-\lfloor{x}\rfloor-1=n)⇔(\lfloor{x}\rfloor=-n-1)⇔(-n-1≤x<-n)$...
0
votes
0answers
31 views

Dirac's delta function with vector argument and relation to Dirac's delta of the vector's magnitude

Is there some way to approximately express Dirac's delta function with argument from $\mathbb{R}^2$ as delta function of magnitude of the vector, in other words, is there some way (under certain ...
3
votes
1answer
36 views

The solutions of $(y')^2=P(y)$ with $\deg P \in \{3,4\}$.

Here is some background on my question. The Weierstraß function (as well as its translates) $\wp(x)$ solves the implicit first-order ODE $$(y')^2=4y^3-g_2y-g_3.$$ Differentiating gives the second-...
1
vote
0answers
38 views

Proving that the Gamma function infinite product definition extends its integral form definition

The integral form definition of the Gamma function is as follows. It is valid for all complex numbers with $\mathrm{Re}(z)>0$: $$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x} dx$$ See this Wikipedia page ...
2
votes
0answers
21 views

Definition of Poincaré rank

Dealing with the confluent Heun equation, something unimportant to me at the beginning got me curious lately: the Poincaré rank of an irregular singularity. In particular, that one in the confluent ...
1
vote
1answer
53 views

Integral Resulting in Hypergeometric Function

Any tips on how to prove this result? $$ \int \frac{x}{x^K + c} dx = \frac{x^2 {}_2F_1 \left(1,\frac{2}{K};\frac{K+2}{K};-\frac{x^K}{c} \right)}{2c}, $$ where $${}_2F_1 (a, b;c;z)$$ is the ...
0
votes
1answer
37 views

Kernel of Hankel Transform

I need to solve a cylindrical diffusion problem that is defined in $[1,\infty]$. I would like to use Hankel Transform that has is defined on $[0,\infty]$. So in order to apply Hankel transform in my ...
-2
votes
1answer
82 views

A specific integral

Consider the following integral $$\int_0^\pi e^{a\cos\theta} \sin^n\theta\ \mathrm{d}\theta, $$ where $a$ is a real number and $n$ is an integer. Is it possible to relate the integral given above to ...
2
votes
0answers
65 views

Sine integral and hyperbolic tangent.

Why does $\operatorname{Si}(x)$ resemble $\tanh(x)(1+\frac{\sin x}{x})$? How are both functions related? What is the simplest function that produces such a shape? What are the uses of such a function, ...
3
votes
2answers
68 views

Computing an infinite limit involving a double integral

$$\lim_{T\to \infty}\frac{\int_0^T\cos^2(s) \exp(-s)\int_0^s \cos(\cos a)\exp(a)\,\mathrm{d}a\,\mathrm{d}s}{T}$$ I tried computing this limit in maple but got the answer: 'undefined'. How to compute ...
0
votes
1answer
64 views

Sum of associated Legendre functions

I want to find the sums of two expressions involving the Schmidt-normalized associated Legendre functions. They are defined by \begin{align} S_l^0(x) &= P_l^0(x) \\ S_l^m(x) &= \sqrt{2 \frac{(...
4
votes
2answers
190 views

Show that $\int_0^1 4 \space\operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $

My mentor tommy1729 wrote $\int_0^1 4 \space \operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $ I wanted to prove it thus I looked at some methods for computing integrals and also ...
5
votes
1answer
62 views

What is the name of the function $D(a,x) = \frac{x^a e^{-x}}{\Gamma(a+1)}$?

Does the function $\dfrac{x^a e^{-x}}{\Gamma(a+1)}$ have its own specific name? Temme [1] introduced the function in (3.1) $$D(a,x) = \frac{x^a e^{-x}}{\Gamma(a+1)}.$$ It is the dominant part in many ...