# 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).

4,215 questions
Filter by
Sorted by
Tagged with
16 views

### From incomplete beta function sum $\frac1{(\text B(a,b)c)^2}\sum_{k=0}^\infty\frac{\text B_y(2a+r+k,b)(1-b)_k}{(a+k+r)k!}$ to hypergeometric function.

The goal is to integrate Inverse Beta Regularized $\text I^{-1}_{z}(a,b)$ to a constant power with respect to $z$ twice for a future identity. Notice the Incomplete Beta function $\text B_z(a,b)$ and ...
• 5,229
48 views

### Asymptotic behavior of the integral $\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$

Which method can I use to study the asymptotic behavior as $\rho \to \infty$ of the integral for $q \geq 0$? $$\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$$ I wish to study this behavior to ...
40 views

### Integrating $\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$

Let $\alpha>1$, $n \in \mathbb{N}$ and $q\geq0$. Which methods is possible to use to solve this integral? $$\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$$ I tried using the computer for especific ...
42 views

• 6,540
1 vote
46 views

• 731
35 views

### For what reason Bessel functions of the first kind can be differenciated in relation to the variable $q$?

For that reason the modified bessel function $I_q(\rho)$ defined as $$I_{q}(\rho)=\sum_{m=0}^{\infty}\frac{(\rho/2)^{2m+q}}{m!\Gamma(m+q+1)},$$ where in the above $\rho$ is fixed, can be ...
21 views

• 5,229
64 views

• 1,809
77 views

### Upper bound for Meijer-G function

What I need is a monotonically decreasing function that forms an upper bound for the following function: - G_{0,6}^{4, 0}\biggl({-\atop -\frac{1}{2},\frac{1}{2},\frac{5}{6},\frac{7}{6}...
44 views

### Mittag-Leffler function recurrence relation

The general Mittag-Leffler function $$E_{a,b}(z)=\sum_{h=0}^{\infty}\frac{z^h}{\Gamma(ha+b)}$$ satifies the recurrence $$E_{a,b}(z)=zE_{a,b+a}(z)+\frac1{\Gamma(b)}.$$ I am having a hard time in ...
• 6,557
1 vote
32 views

• 5,229
1 vote
46 views

### special function that is between arcsinh(x) and arctan(x)? [closed]

I am looking for a function between arcsinh(x) and arctan(x) for x>0. The function should be calculated using +, -, *, /, elementary functions, or well know special functions. But no numerical ...
• 199
1 vote
19 views

### Extension of reflection formula for polygamma function

So I've seen the reflection formula for polygamma functions: $\Psi(z) - \Psi(1-z)=-\pi \cot{\pi z}$. Is there an extension for arguments that sum not to $1$ but to some other (not necessarily integer) ...
1 vote
81 views

### Can you suggest a software to solve this equation?

Can you please suggest a free software or website that would allow me to approximate, numerically, the first $n$ roots of the equation $2J_{0}(2\alpha)+2\alpha J'_{0}(2\alpha)=0$ ? I'm trying to find ...
• 284
1 vote
22 views

### Intersection of modified Bessel functions with different scaling

Let $0< s_1 < s_2$ and $0<Z_1<Z_2$, consider functions $F_j:[0, \infty)\to [0, \infty)$ defined by $$F_j(t) = \frac{1}{Z_j} I_0(2\sqrt{ts_j}),$$ where $j = 1,2$ and $I_0$ is the modified ...
1 vote
79 views

### Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(n!)}$

WolframAlpha gives the evaluation as $\text{Ei}(1)-\gamma$. It does not offer a step-by-step solution. Where does this come from?
102 views

### Does $(1-\cos(x))/x^2$ have or deserve a name, like $sinc$ for $\sin(x)/x$?

i think it's the “dual” of $\mathrm{sinc}(x)$, that is, $$f(x) = \begin{cases} \frac 1 2, & \text{if x=0} \\ \frac{1-\cos{x}}{x^2}, & \text{otherwise} \end{cases}$$ they share some ...
36 views

### Evaluating an integral with derivatives of Associated Legendre polynomials

I came across the following integral $$\int_{-1}^{+1} (1-x^{2}) \frac{\partial P_{lm}(x)}{\partial x} \frac{\partial P_{km}(x)}{\partial x} dx$$ where $P_{lm}(x)$ is an associated Legendre polynomial, ...
• 361
20 views

### How to find $xy=F(t)$ from $- \operatorname{Ei}{\left(x y e^{i \pi} \right)} = A t^{2} + B t + C$?

I have the integral equation $$- \operatorname{Ei}{\left(x y e^{i \pi} \right)} = A t^{2} + B t + C$$ where $\operatorname{Ei}$ is the exponential integral and $A$, $B$, and $C$ are (finite) arbitrary ...
• 1,316
17 views

### Double sum in terms of generalized Mittag-Leffler functions

I have the series $$S = \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^n y^m}{\Gamma(1+\alpha n + \beta m)}\frac{(n+m)!}{n!m!}$$ which originates from a fractional calculus problem. One can see that $S$ ...
• 927
14 views

### Inverse Laplace transform of $\frac{1}{s}\frac{1}{1 + (k/s)^a + (l/s)^b}$

I want to express the Inverse Laplace transform (arising from a fractional calculus problem) $$F(t) = \mathcal{L}^{-1}\Big\{ \frac{1}{s}\frac{1}{1 + (k/s)^a + (l/s)^b}\Big\}(t)$$ in terms of ...
• 927
1 vote
72 views

### Integral of $e^{-(Ax^2 + By^2)}$ around the unit sphere

I'm trying to calculate $\mathcal{I} = \int_{S^2} e^{-(Ax^2 + By^2)} dS$ for real constants $A$ and $B$, where $S^2$ is the two-dimensional sphere.. I have external constraints on these constants ...
• 262
28 views

### An elegant computation of series

I try to compute the following series, $$S(\omega,k)=\sum_{p=0}^{\infty}\frac{(-k)^p(p+k)^2}{(p+k)^2+\omega^2}\frac{1}{p!},$$ where $\omega>0$, $k>0$. To be honest, I have no idea how to compute ...
16 views

### Calculate time until accrued compound interest reaches a fixed amount.

Suppose I start with an amount $a_i$ of each of several assets indexed $i\in \mathbb{I}$. The fixed interest rate associated with each asset is $r_i$. After some time $\Delta t$ the amount held of ...
21 views

### Derivative of Bessel function of order 1

what is the derivative of the given Bessel function? d/dx(xJ1(x)) , where x=A.z A is constant and z is variable. If one Bessel function is J1(z) and the other is J1(Az), Is it possible to write the ...
• 11
19 views

### Does the Meier-G function have a lower and an upper bound?

The Meijer G-function is given by It serves as a solution to certain differential equations, and has been shown to generalize a variety of elementary and transcendental functions. Is there a lower ...
• 1,316
45 views

### Simplifying a ratio of incomplete beta functions

Can the following be simplified? $$\frac{\int_0^a t^{x+1} (1-t)^{y} dt}{\int_0^a t^{x} (1-t)^{y} dt} \qquad \big((x,y)\in(0,\infty)^2; a\in(0,1]\big)$$ Note: If it helps to assume that $0<x<y$ ...
24 views

### Simplifying a ratio of "two-sided incomplete Beta functions"

Let $0\leq L < R\leq 1$ and $0\leq s \leq T$. Can the following ratio be $\text{simplified?}$ $$\frac{\int_L^R z^{s+1}(1-z)^{T-s}dz}{\int_L^R z^{s}(1-z)^{T-s}dz}$$ If it helps to assume that $s$ ...
89 views

### Strange summation? $\sum _{k=-n}^{n+1} \frac{(-1)^k}{x-k}$

I'm mainly concerned with the bounds of summation here. I've never personally seen such a summation before, but I came across this summation in "Special Functions" by Andrews Askey Roy on ...
1 vote
Any recommendations to help me to solve this integral $$\int_{-\sqrt{x}+y}^{1-y} \left(1-(t+y)^2\right)^{r} \left(1-x(t-y)^2 \right)^{r} dt$$ where $\{r,x,y\} \in \mathbb{R} , \, |y|<1+x, ... • 55 1 vote 0 answers 70 views ### Double sum over Gauss hypergeometric function. I've been dealing with sums and integrals over hypergeometric functions quite a bit lately, and the latest problem is the following double sum: F(x,y;\alpha,t)=\sum_{n,m=0}^\infty\... • 171 0 votes 1 answer 29 views ### It is$x(t) = e^{c_1-t}\cdot\theta(c_1-t)$a solution to$\dot{x} = -|x|$with$\theta(t)$the unitary step function? It is$x(t) = e^{c_1-t}\cdot\theta(c_1-t)$a solution to$\dot{x} = -|x|$with$\theta(t)$the unitary step function? I am trying to understand solutions of finite duration to differential equations. ... • 789 1 vote 1 answer 70 views ### How to "formally" prove that$x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$solves$\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$? How to "formally" prove that$x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$solves$\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$? (with$\theta(t)$the standard unitary step function). I have found the ... • 789 2 votes 0 answers 38 views ### Finding a simple differential equation to define an inverse of$\,_2\text F_1(a,b;c;z)$with respect to$z$with the Gauss Hypergeometric function. An “Inverse Gauss Hypergeometric function” with respect to$z$in terms of a differential equation would define many special case inverse functions. Define: $$\,_2\text F_1(a,b;c;z)=\sum_{n=0}^\infty \... • 5,229 3 votes 1 answer 59 views ### Laplace transforms of products of modified Bessel Functions I am dealing with integrals of the form$$\int_0^\infty e^{-t}I_0(xt/a)^a\ \mathrm{dt}$$where$I_0(x)$is the modified Bessel function of the first kind. Clearly this is just a Laplace Transform$\...
I need to solve the following equation for x: $a\log(\frac{x}{a})-b\log(\frac{a-x}{2b})=x-d$. I know that the solution of $\log(a x+b)=c x+d$ can be determined using the Lambert function https://en....