All Questions
Tagged with ca.classical-analysis-and-odes special-functions
192 questions
0
votes
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views
Is there an asymptotic expansion for the reciprocal of the classical Euler beta function?
The classical Euler beta function can be defined by
$$
B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t
$$
for $\Re(p),\Re(q)>0$.
The beta function and the classical Euler gamma function $\...
- 1,091
2
votes
1
answer
160
views
Is the function $(z-1)(2^{z}-1)\zeta(z)$ logarithmically concave and convex in $z\in(0,\infty)$?
For proving that the sequence
\begin{equation}\label{first-proof-decreas-seq}
\frac{1}{(2k-1)(k+1)} \frac{2^{2k+2}-1}{2^{2k}-1} \biggl|\frac{B_{2k+2}}{B_{2k}}\biggr|
\end{equation}
is decreasing in $k\...
- 1,091
0
votes
0
answers
43
views
The reciprocal of the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function is a convex function
The classical Bernoulli numbers $B_j$ are generated by
\begin{equation}\label{Bernoulli-No-Generating}
\frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty ...
- 1,091
3
votes
1
answer
83
views
Curious asymptotics of real part of ratio between Lambert W branches
This question was inspired by the inactive thread How to find this value of $A$? but the focus there was on the divergence of the imaginary part.
It seems that for a given nonzero real $x$, $$\...
- 1,459
2
votes
1
answer
188
views
Incomplete integral of confluent hypergeometric function
I would like to prove the following: if $a,b,c > 0$, $0<x<1$, $y>0$, then,
$$
\frac{1}{\Gamma(b)}\int_0^{\infty} s^{b-1} e^{-s} \,\mbox{$ {}_1\mathrm{F}_1 (a\,; c\,; x s + y\hspace{1pt})$}\...
- 391
4
votes
1
answer
168
views
Uniform decay of $J'_{\nu}(x)$ for $x\gg1$
I need a uniform decay estimate for the derivative $J'_{\nu}(x)$
of the Bessel functions.
By `uniform' I mean an estimate independent of $\nu$, at least for a range of orders like $\nu\ge0$.
For $J_{\...
- 9,007
1
vote
0
answers
102
views
Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$
I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
- 224
5
votes
3
answers
845
views
Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$
I uploaded this question here and here from my old account.
Let $\psi^{(1)}$ be the trigamma function defined by
\begin{equation}
\tag{1}
\psi_1(z) = -\int\limits_0^1 \frac{x^{z-1} \ln x}{1-x} \, dx.
\...
- 224
1
vote
1
answer
197
views
Derivation of indefinite integral involving hypergeometric function
I am doing a project on projectile motion and I ended up with this integral:
$$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$
where $g, c,$ and $m$...
- 21
0
votes
1
answer
250
views
Asymptotic behavior of the polylogarithm function and generalisation
So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper:
$$
\sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...
- 11
4
votes
1
answer
150
views
Eigenvalues of the modified Mathieu equation with normalizable solution
The Mathieu equation (https://en.wikipedia.org/wiki/Mathieu_function) is $y''+(a-2q\cos(2z))y=0$. The modified Mathieu equation is obtained by replacing $z$ with $\pm iz$:
$$y''-(a-2q\cosh(2z))y=0.$$
...
- 385
1
vote
1
answer
188
views
T functions arising from derivatives of incomplete Gamma function
Here the derivatives of the incomplete gamma functions are described via:
$$
T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right|...
- 13
0
votes
0
answers
164
views
How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?
The classical Euler gamma function can be defined by the integral
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0.
\end{equation*}
Its ...
- 1,091
7
votes
2
answers
529
views
Weak convergence related to Hermite polynomial?
I am reading Griffiths's quantum mechanics book; in the section about harmonic oscillators, he wrote out the amplitude of wave function, and compared with the classical harmonic oscillators. He ...
- 637
2
votes
1
answer
215
views
An integral transform computation
In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2.
they note that
\begin{align}
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds
= 2^{-\nu/2} \pi^{-...
- 21
10
votes
7
answers
875
views
$\int_L^\infty \exp(- t - y/t) \, dt = \text{?}$
Let $y>0$, $L>0$. Has the (special?) function given by
$$f(y,L) = \int_{L}^\infty e^{- t - y/t} \, dt$$
been studied? Are there precise, simple bounds?
Let me try to attempt to reinvent the ...
- 20.2k
3
votes
0
answers
89
views
Transforming a Fuchsian equation with four finite singularities to the Heun equation
I keep seeing it claimed that the general second-order Fuchsian equation with four singularities can be transformed to the Heun equation, but I have never seen anyone explicitly write out the steps, ...
1
vote
0
answers
197
views
Infinite series involving generalised hypergeometric functions
I've recently stumbled into hypergeometric functions while trying to evaluate the integral:
$$
\int \exp \big( x^2 + bx + c \big) {\rm erf} ( x ) \operatorname{d\!}x
$$
Essentially, working from an ...
- 41
0
votes
0
answers
111
views
The upper bound of hyperbolic cosine function in complex plane
I want to find the upper bound of the following function :
\begin{equation}
M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right|
\end{equation}
where
\begin{equation}
\begin{aligned}
& 4 \...
5
votes
2
answers
468
views
Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$?
By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$):
$$J_0(u)J_0(v)+2\sum_{n=1}^\infty J_n(u)J_n(v) \cos(n\alpha) = J_0 \left( \sqrt{u^2+v^2-2uv \...
- 1,035
1
vote
1
answer
346
views
Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?
It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential ...
- 1,091
5
votes
1
answer
198
views
Compositional inverse of Bessel function
Was ever studied a function $f$ which solves $J_0(f(x))=x$? Integral representations, natural domains of existence and whatever.
- 109k
1
vote
1
answer
259
views
Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?
Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
- 399
0
votes
0
answers
130
views
Solution to the integral of Bessel $\int^1_0 x \sin(a x) J_1 (b x) dx$
I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral:
\begin{equation}
\int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...
3
votes
1
answer
213
views
Is there a theory of "elementary closed form solution" at the operator level for differential equations?
We begin by considering the usual general first order linear equation of the form
$$ a_0 y' + a_1 y + a_2 = 0 $$
Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
- 2,689
1
vote
1
answer
155
views
Solution to non-autonomous delay differential equation?
If you define a special function called the Lambert W function, you can explicitly solve the classic delay differential equation $x'(t) = x(t - a)$ by supposing the solution is some $\exp(\lambda t)$ ...
- 217
1
vote
2
answers
99
views
Asymptotics of Bessel functions in the discrete parameter
Is there any information on the asymptotics of $J_n(z)$ as $n\to \pm\infty$ for fixed $z$ (real or imaginary)? I originally wanted to ask about the modified Bessel functions $I_n(z)$, but found out ...
- 1,143
2
votes
2
answers
509
views
Definite integral of the square root of a polynomial ratio
I found myself with the following integral
$$ \int_{b_1}^{b_2} \sqrt{\frac{(b-b_1)(b_2-b)(b_3-b)}{(b_4-b)}} \ db $$
with $ b_1 < b_2 < b_3 < b_4 $. I know that
$$ \int_{b_1}^{b_2} \frac{db}{\...
0
votes
1
answer
128
views
Generalization of identity for terminating hypergeometric function
Let ${}_2F_1(a,b;c;z)$ be the ordinary hypergeometric function for $z \in \mathbb{C}$
\begin{equation}
{}_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{z^k}{k!} \frac{(a)_{k} (b)_k}{(c)_k}\,,
\end{...
- 679
1
vote
0
answers
127
views
About writing solutions of linear ODE's: Is this statement correct?
A motivating example: Consider the Hypergeometric equation
$$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$
it has a solution given by the Gauss's Hypergeometric function
$$_2F_1(a,b;c;z)...
- 480
2
votes
0
answers
130
views
Motivation behind the Bohr-Mollerup Theorem relating the Gamma function
In Wikipedia, it states about the Bohr-Mollerup Theorem:
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
If anyone knows, ...
- 143
1
vote
1
answer
276
views
Exponential decay bound on integral
I have an integral of the form
$$ \int_R^{\infty} e^{-x} x^n \vert L_m^{\alpha}(x) \vert^2 \ dx,$$
where $L_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$
I would to get a nice ...
- 73
3
votes
0
answers
186
views
Is factorial the restriction of some elementary function?
Hölder's theorem says that Gamma function is very non-elementary, but it does not exclude the possibility that factorial is the restriction of some elementary function to natural numbers. The answer ...
- 623
2
votes
1
answer
196
views
Gamma function and the somewhat extended version of Bohr-Mollerup theorem
The Gamma function $\Gamma$ is defined by
\begin{equation*}
\Gamma(x)=\int_{0}^\infty t^{x-1}e^{-t} \,\mathrm{d}t,
\end{equation*}
for $x>0$. It satisfies the well-known functional equation
$$\...
- 143
5
votes
2
answers
434
views
How to integrate the multinomial over a ball in $\mathbb{R}^{n}$?
I got an interesting question. Consider this integral:
$$ \int_{B(0,1)}\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m \mbox{d}x, \quad m,n\in \mathbb{N}, \ a_{i}>0, \ i=1,2,\ldots,n.$$
It is clear that ...
- 51
2
votes
1
answer
341
views
What functions do we need to solve linear second order differential equations with polynomial coeficients? [closed]
.
Final edit: The problem I had in mind is properly asked in THIS MO QUESTION, so I'll vote to close the present post e recommend anyone interested in the topic to visit that link.
.
.
.
.
.
Below is ...
- 480
1
vote
0
answers
68
views
Airy-type integrals (with different power $\neq 3$)
I am looking for integrals closely related to the Airy function
\begin{eqnarray} &&
A_1= \int _0^\infty x \sin \Phi dx
\nonumber \\ &&
A_2= \int _0^\infty \cos \Phi dx
\nonumber \\&...
0
votes
2
answers
215
views
Asymptotic for eigenvalues for the following ode?
Consider the following Sturm-Liouville problem,
$$(\sqrt{\sin \theta} Y')' + \lambda \sqrt{\sin \theta} Y =0$$
where $Y(\theta):[0,\pi] \to \mathbb{R}$ with boundary conditions $Y'(0)=Y'(\pi)=0.$
I ...
- 537
2
votes
1
answer
150
views
How to compute this limit involving the associated Legendre function?
I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the ...
- 537
8
votes
3
answers
1k
views
Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?
This seems such a simple question that I fear I must have missed some elementary maths.
I am looking for a way to solve $x+x^a = y$ by reference to an already defined function, $a,x,y > 0$ are real....
- 83
11
votes
1
answer
911
views
How to determine if you've discovered a new identity for a special function
Often times, we consult resources, like Abramowitz and Stegun's Handbook of Mathematical Functions https://www.math.ubc.ca/~cbm/aands/, NIST's database on special functions https://www.nist.gov/...
- 243
3
votes
0
answers
175
views
On analogues of Weber's formula
Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that
$$
\int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta})....
- 7,193
2
votes
1
answer
314
views
Two questions about an integral involving double product of Bessel functions
Let us define the following integral :
$$W_n(r)=r\int_0^{+\infty} J_1(rt)[J_0(t)]^n dt,$$
with $r>0$ a real number and $n\in\mathbb{N}$ and where $J_0(x)$ and $J_1(x)$ are Bessel functions of the ...
- 21
2
votes
0
answers
81
views
Uniform bound on a certain family of hypergeometric functions
We have the following problem, which we can't solve.
Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
- 5,562
1
vote
1
answer
300
views
Asymptotic behavior of an ODE
Consider the following ODE eigenproblem of $y(x)$
\begin{equation}
y'' + [\varepsilon + b^2 x - (a + \frac{b^2}{2}x^2)^2 ] y=0
\end{equation}
with eigenvalue $\varepsilon$, real constants $a,b$. ...
- 155
2
votes
2
answers
155
views
Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type
Let $J_\nu$ be a Bessel function of the first kind and let $\{\lambda_{n, \nu}\}_{n\ge 1}$ be a sequence of its zeroes. I claim that
$$
\inf_{n\ge 1}\bigg|\sqrt{\lambda_{n,\nu}} J_{\nu+1}(\lambda_{n,\...
- 421
3
votes
1
answer
183
views
On integral representation of Whittaker $W$ functions
According to NIST, the integral representation of Whittaker $W$ functions
$$
W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(%
\frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
- 373
2
votes
1
answer
194
views
Generalized Selberg integral
I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions
$$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\...
- 83
3
votes
2
answers
71
views
Optimal scaling of Lipschitz estimates in generalized geometric series
If we did not know it before, then wikipedia teaches us the generalized geometric series
$$\sum_{n \ge 0} \binom{n+k}{n} (1-\mu)^{n} \mu^k = \frac{1}{\mu}.$$
We can then study for $0 <\varepsilon &...
- 173
3
votes
0
answers
269
views
definite integral with incomplete gamma function and exponential
While working with electron density computations in quantum chemistry, I encountered the following improper integral:
$$
I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
- 31