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

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0
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0answers
14 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 ...
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
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
votes
1answer
48 views

Let $z=\ln \tan\frac xy.$ What is $z_x$ and what is $z_y$?

Let $$z=\ln \tan\frac xy.$$ What is $z_x$ and what is $z_y$? Thanks ahead:) What I have tried: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
2
votes
2answers
334 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
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 ...
30
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1answer
446 views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
2
votes
1answer
47 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 ...
1
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0answers
30 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
28 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 $$\frac{1-e^X}{X}=K?$$
2
votes
1answer
209 views

Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
5
votes
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
votes
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
7
votes
1answer
541 views

median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
4
votes
2answers
198 views

Evaluating $\int_{0}^{\frac{\pi}{2}} \arctan( a \sin x) \ dx$ using the Taylor expansion of $\arctan (x)$

I was wondering if it's possible to show that for $a >0$, \begin{align}\int_{0}^{\pi/ 2} \arctan (a \sin x) dx &= 2 \sum_{k=0}^{\infty} \frac{\left(\frac{\,\sqrt{\vphantom{\Large A}\,1 + ...
4
votes
1answer
219 views

Integral involving bessel function/gaussian/rational function

I'd like to solve: $$\int_0^{\infty}\quad J_1(ak)\,\frac{b+k^2}{(k-\alpha_1)(k-\alpha_2)(k-\alpha_3)(k-\alpha_4)}\,\exp(-ck^2)\,\,dk$$ Is there any specific rule for it? Thanks!
0
votes
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 ...
8
votes
0answers
334 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
0
votes
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: $L(x,N)\lt+\infty$ ...
0
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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) ...
8
votes
2answers
2k views

How to evaluate Riemann Zeta function

How do I evaluate this function for given $s$? $$\zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
0
votes
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 ...
8
votes
1answer
220 views

Can it be shown that $Y_0(\lambda_n a)J_0(\lambda_n a) - J_0(\lambda_n a)Y_0(\lambda_n a) \ne 0$?

Background I am currently looking into the task of describing a transient temperature field $\theta(r,t)$ across the thickness $a \leq r \leq b$ of an infinitely long and hollow cylinder exposed to a ...
13
votes
1answer
263 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
3
votes
1answer
99 views

How to prove this identity for ${}_3F_2$ (Generalized Hypergeometric Function)?

This may look like homework, but it is not. I've found this identity (using Mathematica): $$ {}_3F_2 \left( \matrix{1,1,1 \\ 2, e} ; 1 \right) = (e-1) \psi^{\prime}(e-1), $$ valid for $e$ with ...
2
votes
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 ...
0
votes
2answers
79 views

positively homogeneous function

A function $f:X→\mathbb{R}$ is said to be positive homogeneous of degree $k\in\mathbb{R}$ if $f(tx)=t^kf(x)$ for every $x\in X$ and every $t\in\mathbb{R}_{++}$. For $X=\mathbb{R}_+$, the sample ...
0
votes
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
votes
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
votes
2answers
319 views

Does sinc function have any special inverse function defined?

We know that $y=xe^x$ cannot be solved for $x$ using elementary functions. The Lagrange inversion theorem can be used for finding a "new" function that would be the inverse function of the above ...
2
votes
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 ...
2
votes
5answers
170 views

Reference Book on Special Functions

Now I'm studying the topic that uses the special functions frequently, so I find myself in need for some good reference book on the properties and equalities of the special functions. The optimal one ...
1
vote
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
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 ...
0
votes
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 ...
5
votes
1answer
261 views

solution of Lagrange differential equation are square integrable

I was recently posing myself this question. Given the Lagrange DE $$[(1-x^2)u']'+\lambda u=0,$$ where $\lambda$ is a real parameter and $x\in[-1,1]$, it is well known that, if $\lambda=n(n+1)$ for ...
0
votes
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
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 $1/k^2$ ...
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 ...
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 ...
1
vote
2answers
213 views

solution of another definite integral

Does the following integral converge or not? \begin{align} && \sum_{k=0}^{\infty} (-\varphi)^k \binom{\frac1\varphi+k}{k}\int_{-\infty}^\infty\beta x^n e^{-\beta x(k+1)}dx&& ...
0
votes
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 ...
12
votes
1answer
182 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
30
votes
3answers
504 views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ but have problems ...
15
votes
2answers
272 views

Integral $\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz$ involving Dawson's integrals

I need you help with evaluating this integral: $$I=\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz,\tag1$$ where $F(x)$ represents Dawson's integral: $$F(x)=e^{-x^2}\int_0^x ...
37
votes
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 ...
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 ...