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

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29 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
1
vote
1answer
12 views

Kronecker delta function notation

Can someone please help me, what does $\delta_{i-j-1}$ stand for? I have a matrix with elements $z_{ij}=\delta_{i-j-1}$ where $\delta_k$ is the Kronecker delta function (that's how it's written in the ...
2
votes
1answer
33 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
0
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0answers
15 views

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
2
votes
1answer
48 views

How many answers can be created using the elementary arithmetic operators?

If I gave you an amount of $n$ numbers, how many anwswer will you be able to create using the elementary arithmetic operators ($+, -, \times, /$)? These are the rules: All numbers ...
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0answers
31 views

Is there a name for this property of a real function?

Let $M=\sup_{x \in [0,1]^n} f(x)$ where $f:[0,1]^n \rightarrow \mathbb{R}$ is differentiable twice, and write $x=(x_1, \dots, x_n)$. Let $M_{x_i=0}=\sup_{x \in [0,1]^n:x_i=0} f(x)$ and ...
2
votes
2answers
29 views

Lambert Function as a solution

The solution to the equation $$Xe^X=K$$ is given by $$X=W(K)$$ where $W$ is the Lambert function. Is it possible to adapt this such that we can find a solution for $$\frac{1-e^X}{X}=K?$$
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0answers
29 views

Prove an inequality involving $Si(x)$ and $Si(2x)$

How Is it possible to prove the following inequality? $$xSi(2x)-2Si(x)*\sin(x)\lt x^2$$ for $x\in\mathbb{R}$ Thanks
5
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2answers
246 views

Curious gamma identity

I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc). The identity is as follows: $$\Gamma(t) = x^t ...
0
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0answers
23 views

Conjecture about the harmonic number

I would like to know if is it possible to prove or disprove the following conjecture: Given the following limit: $$L(x,N)=\lim_{N\to\infty}\left(H^{(-x)}_N-NH_N\right)$$ we have: $L(x,N)\lt+\infty$ ...
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2answers
16 views

Can b=0 in the confluent hypergeometric function U(a,b,z)?

I am confused about the possible values of b in the confluent hypergeometric function of the second kind U(a,b,z). Specifically can b=0? I know that the U function can be expressed as $$U(a,b,z)=\pi ...
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0answers
30 views

Rogers-Ramanujan Continued Fraction

How to calculate Rogers-Ramanujan Continued Fraction $R(e^{-2\pi{\sqrt{5}}})$ ?
3
votes
1answer
27 views

Calculate integral with $\Gamma$ and $B$

The integration is like: $$\int_{a}^{b}\left(\frac{b-x}{x-a}\right)^{p}dx$$ with $0<p<1$ Answer is $(b-a)p \frac{\pi}{\sin p\pi}$ Apparently, we can reversely construct $$\Gamma(1-p) ...
0
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0answers
17 views

Constant Coefficient Legendre Equation via Change of Variables?

In the introduction to this old book by Craig on ode's it is said that The theory of linear differential equations may almost be said to find its origin in Fuchs's two memoirs published in 1866 ...
2
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2answers
121 views

Compute the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$ or $I=\int_{d}^1 y^{-a}(1−y)^{b-1} dy$

I need to calculate the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$, where $a$, $b$ are REAL NUMBERS and $b>0$. (my goal is to determine the definite integral $I=\int_{d}^1 y^{-a}(1−y)^{b-1} ...
2
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8answers
699 views

What are the most important functions every mathematician should know? [closed]

I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think ...
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1answer
22 views

Partial fraction that contain special function

In the following equation; $$ \frac{e^{\frac{(2c+5x)}{3x}} \mathop{E_{n}}\nolimits\!\left(x\right)}{(a+x)(b+x)} $$ 1- Can I apply the partial fraction to the above equation as the following: $$ ...
5
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1answer
100 views

Prove that $\exists\,! \,\lambda \in (1/5,1/4)$ such that $\frac{1}{2\pi}\int_0^{2\pi}e^{\sin x}\,\mathrm{d}x=e^{\lambda}$

The following question came up in chat Prove that $\exists\,! \,\lambda \in (1/5,1/4)$ such that $\displaystyle\frac{1}{2\pi}\int_0^{2\pi}e^{\sin x}\,\mathrm{d}x=e^{\lambda}$ Now the integral ...
1
vote
1answer
47 views

$\int xtanx$ and the Clausen Function

I have been attempting to evaluate $\int x \tan x \;\mathrm{d} x$. My first instinct was integration by parts, which produces $-x \ln|\cos x|+\int \ln|\cos x| \;\mathrm{d} x$. I have read online ...
2
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1answer
16 views

Prove convexity of log modified bessel function

I need to prove that the modified bessel function of the second kind is log convex in the square of the argument. Specifically I'm interested in showing, $\log \mathcal{K}_0(\sqrt{x})$ (zero order) is ...
0
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0answers
38 views

Solving integral that contain upper incomplete gamma function, exponential, and powers

I have this integration formula; $ f=\int\limits_{0}^{\infty}\frac{e^{-b~z}}{\sqrt{z}} \Big(\frac{\beta}{\beta+z}\Big)\Big(\frac{\beta+z}{z}\Big)^L ...
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0answers
31 views

Ramanujan Class Invariant $G_{125}$ and $ G_{5}$

How to calculate the Ramanujan Class Invariant $G_{125}$ and $G_{5}$?
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0answers
12 views

closed form of integral of special function? $\int_{0}^{1} e^{a\, q^{1/k}\, {}_1 F_{2}(1,1/k,1 + 1/k;q)} d q$

Take the following integral, defined by hypergeometric functions: $$\int_{0}^{1} e^{a\, q^{1/k}\, {}_1 F_{2}(1,1/k,1 + 1/k;q)}d q$$ (there is a similar formulation Lerch). I think the series ...
1
vote
2answers
59 views

$f(x)=\log x\Leftrightarrow f^{-1}(x)=e^x$. Why $e=2.73\cdots$?

$$f(x)=\log x\Leftrightarrow f^{-1}(x)=e^x.$$ Ok, $\log x$ is defined as the function $f(\cdot)$ such that: $f'(x)=\dfrac{1}{x}$. How to get, from this, the inverse of it $f^{-1}(x)$? And why ...
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0answers
33 views

Exponential Integral Function representation

It is well known that the exponential integral function can be represented by: $$ e^{x}\mathop{E_{n}}\nolimits\!\left(x\right)\leq\frac{1}{x+n-1}, $$ For n=1,2,3,… and $x>0$ , If we have ...
1
vote
1answer
34 views

How to update the probabilities so that it still sum up to $1$?

At time $t$, I have a probability vector $\mathbf{\pi}^{t}=\left({\pi}_{1}^{t}, \cdots, {\pi}_{n}^{t} \right)$. I would like to construct a function $f(\cdot)$ and update the vector ...
1
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0answers
82 views

What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?

In a consideration of summation of a series $$ s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients $a_k$ are somehow related to $1/k^2$ ...
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0answers
9 views

What is the sigmoid *squashing* function?

I've just read the following The basic unit ("neuron" i) performs the following computation to update its state $y_i$: it computes a weighted sum $v_i$ of its inputs $x:j$ which is passed ...
5
votes
2answers
80 views

Hyberbolic and Circular (Trig) Functions: Why no parabolic? [duplicate]

There are circular (trig) functions which determine all the points on a unit circle: and which relate to the area swept out by an angle subtended on the circle. -- These functions can of course be ...
4
votes
3answers
65 views

How to show that $\int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt = \pi^{1/2} \frac{ \Gamma(s-1/2)}{\Gamma(s)}$

I want to compute the integral $$ \int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt $$ for $s \in \mathbb{C}$ such that the integral converges ($\mathrm{Re}(s) > 1/2$ I think) in terms of the Gamma ...
37
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4answers
2k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
4
votes
1answer
96 views

Alternating second power Euler sum $\sum_{k\geq 1} \frac{(H'_k)^2}{k^2}$

Question: Evaluate $$\sum_{k\geq 1} \frac{(H'_k)^2}{k^2}$$ Where we define the alternating harmonic number $$H'_k=\sum_{n=1}^k\frac{(-1)^n}{n}$$ I remember seeing a closed form involving a ...
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0answers
13 views

Sturm Liouville eigenvalues eigenfunctions

The equation/Sturm Liouville problem is: $$u'' + \lambda u = 0, \quad 0≤x≤\frac{\pi}{2}, \quad u'(0) = 0, \quad u(\frac{\pi}{2}) = 0 $$ I want to find the eigenvalues and eigenfunctions and the ...
0
votes
1answer
30 views

Orthogonality of associated Legendre polynomials

Let $P_n(x)$ be the $n$-th degree Legendre polynomial. Let $k$ be a nonnegative integer less than or equal to both $n,m$. How to prove that $$ \int_{-1}^1 (1-x^2)^k D^kP_n(x) D^kP_m(x)\,dx = ...
12
votes
3answers
149 views

Why does this function start swinging up and down so weirdly

Please have a look at the function: $$f(x) = \left(x + \frac{1}{x^x}\right)^x - x^x$$ You may see the plot on Wolfram Alpha. Why does it have such a weird behaviour from $x = 13$? It starts swinging ...
1
vote
0answers
39 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
4
votes
1answer
303 views

Generalized Legendre differential equation

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ which is ...
0
votes
1answer
38 views

Satisfaction of Bessel equation by any other function.

Is it possible that any function $y(x)$ other than Bessel group of functions, satisfy Bessel's equation? $$x^2 \dfrac{d^2 y}{d x^2} + x \dfrac{d y}{d x} + (1-n^2/x^2) y = 0.$$
2
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0answers
28 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
3
votes
0answers
23 views

Hypergeometric function with negative $b$ and $a>c>0$

Recall the definition of the hypergeometric function $$_2F_1(a,b,c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{n!(c )_n}x^n$$ where $(a)_n$ is defined to be $a(a+1)\cdots(a+n-1)$. We suppose that none ...
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2answers
71 views

Basic question on complex integration

I have a very basic question on complex integration. How is the definite integral $$ \int_{z_1}^{z_2}{f(z)dz} $$ $z \in \Bbb{C}$ to be interpreted in the absence of a specific path over which ...
2
votes
1answer
31 views

composite function problem

If I have the following expression: $$g(f(x))-g(x)=1,$$ it is possible to express $f(x)$ in terms of the $g(x)$: $$f(x)=g^{-1}(1+g(x)).$$ Is it possible to express $g(x)$ in terms of $f(x)$?
1
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0answers
22 views

Identity involving the hypergeometric function

Let $n$ be an integers greater than one and $p,q$ be real numbers.How do I prove the following identity: \begin{equation} F_{2,1}\left[ \begin{array}{cc} \frac{3}{2} - n && 2-n \\ & ...
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1answer
52 views

Copulas and their properties

I am working with the following copula, and have a few questions about it: $C(x,y) = xy + \theta (1-x)(1-y)xy$ Here $\theta \in [-1,1]$ and $x,y \in [0,1]$ First, I am trying to show this copula is ...
0
votes
0answers
20 views

Integrating products of Hankel and Riccati Bessel functions

I want to do the integral: $$ \int_0^\infty dr h_l^+(kr)\hat j_l(kr) $$ where $h_l^+$ is the type 1 Hankel function, $\hat j_l$ is the type 1 Riccati-Bessel function. I would like a algebraic ...
0
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0answers
13 views

Simplify $\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$

I would like to rewrite this integral $$\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$$ (where $a>\mathbf{R^+}$ and $J_{\frac{3}{2}}$ is the bessel function ...
2
votes
1answer
73 views

An integral that might be related to the modified Bessel function of second kind

It is known that the modified Bessel Function $K_z(a)$ ($a>0$)can be expressed as a Fourier transform $$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$ Can ...
3
votes
0answers
44 views

Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
0
votes
0answers
40 views

How to get the asymptotic formula of generalized Bessel function?

How to get the asymptotic formula of generalized Bessel function? $$J_{\nu}^{(\mu)}(z)=\frac{2}{\sqrt{\pi}\Gamma(\nu+1-1/\mu)}\Big(\frac{z}{2}\Big)^{\mu \nu/2} \int_{0}^{1} ...
2
votes
0answers
38 views

In the space $L^2 [0,1]$ to solve for all values ​​of the complex parameters $\lambda$ and $b:$ [closed]

In the space $L^2 [0,1]$ to find a solution of the integral equation for all values ​​of the complex parameters $\lambda$ and $b$: $x (t)-λ\int_0^1 t^2s^2x(s) \, ds = 4t + bt^2$