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

learn more… | top users | synonyms

0
votes
1answer
10 views

Help with function $f_r(x^q)=q^rx^{q-1}$

Let $r,q$ be a positive integers. I am looking for a function $f_r(x^q)$ such that it is satisfied $$ f_r (x^q)=q^r x^{q-1}$$ (without explicit dependence on $q$ of course, and for $r>1$). I ...
79
votes
9answers
4k views

Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is ...
1
vote
1answer
74 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} ...
0
votes
1answer
33 views

Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$ where $F(x)$ is ...
17
votes
2answers
321 views

$\int_0^\infty\text{Ci}(x)^3\mathrm dx$

Is there a closed form for this integral: $$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$ where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
2
votes
1answer
41 views

Proof or source for this Hurwitz Zeta function identity?

I need a proof or source for this identity: $ \zeta '\left(z,\frac{q}{2}\right)-2^z \zeta '(z,q)+\zeta '\left(z,\frac{q+1}{2}\right)=\zeta(z,q)2^{z}\ln 2$ Here the derivative means the derivative by ...
6
votes
1answer
109 views

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite ...
1
vote
1answer
37 views

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
25
votes
1answer
1k views

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Show that : $$ \int_{0}^{\pi/2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over ...
2
votes
2answers
28 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
1
vote
0answers
11 views

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
3
votes
1answer
46 views

Does $\int { y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$ have a closed form

I am trying to solve the following indefinite integral $$F_Y(y) = \int {y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$$ Where $J_0$ is the Bessel function of the first kind. I tried ...
1
vote
1answer
95 views

Trying to solve $\int{-2\exp{\left(z\cos^2 \theta \frac{\left(a^2 - 1\right)}{2a^2}\right)}}d\theta$

I am trying to solve this integral which has come up as part of some other work, but it is proving to be much harder than I had originally thought. For $0 < |a| \le 1$ being some constant, I am ...
18
votes
3answers
376 views

How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?

I need some help with solving this integral involving Bessel function: $\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
12
votes
1answer
286 views

How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
0
votes
0answers
25 views

Intepolate from linear to step function

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me ...
1
vote
1answer
283 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
0
votes
0answers
26 views

Definite integral with doable improper case

Is there a way to evaluate one or both of the following integrals: $$ \int_{a_1}^{a_2} e^{ib_1(x+b_2 \sqrt{1+x^2})}dx \quad \text{and}\quad \int_{a_1}^{a_2}\frac{x}{\sqrt{1+x^2}} e^{ib_1(x+b_2 ...
1
vote
2answers
79 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 ...
0
votes
1answer
58 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
3
votes
1answer
67 views

Difficult infinite integral involving a Gaussian, Bessel function and complex singularities

I've come across the following integral in my work on flux noise in SQUIDs. $$\intop_{0}^{\infty}dk\, e^{-ak^{2}}J_{0}\left(bk\right)\frac{k^{3}}{c^{2}+k^{4}} $$ Where $a$,$b$,$c$ are all positive. ...
0
votes
0answers
45 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
0
votes
1answer
39 views

closed form expression for an infinite sum

Is there any closed form expression for the infinite sum $\sum_{n≥0}q^{n^2}u^n$ where both q and n are variables and $n \in N∪0$ ?
2
votes
1answer
77 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 ...
13
votes
2answers
427 views

Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$

I am asking this question out of curiosity. $$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha ...
4
votes
2answers
60 views

2D Integral of Bessel Function and Gaussians

I've run into the following integral, and I'm not sure how to evaluate it. $$F(k)=\int ...
1
vote
0answers
55 views

Analytical evaluation of integral

I would like to evaluate the following integral analytically, but Mathematica does not give me an answer: $$ \int_0^1 dr \ e^{(1-2r)x^2} \left[p(r,x) Y_0\left(2x^2\sqrt{r-r^2}\right)+q(r,x) ...
0
votes
1answer
35 views

Sigmoid function in neural network

I am studying a doctoral thesis on control-theory and have trouble understanding the notions and the notation introduced there. I am doing this out of interest on the subject, so I haven't had a ...
2
votes
1answer
16 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
4
votes
3answers
87 views

Proving $\int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1$.

By testing in maple I found that $$ \int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1 $$ Does there exists a proof for this? I tried rewriting it as an series but no luck ...
0
votes
2answers
172 views

Complementary error function in matlab

Please I really want to know how to verify the following relation in MatLAB $\text{erfc}(x)\overset{x\rightarrow\infty}{\longrightarrow}\dfrac{e^{-x^2}}{x\sqrt{\pi}}$
7
votes
4answers
158 views

Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$ I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0. $$ I am ...
0
votes
0answers
24 views

Keep a Function Positive via Mod

I have a function $F$ and I want it to remain positive--i.e., $$- F=F,\quad F=F $$ Would sticking a $\mod 2$ in front of $F$ do this? That is, because $$-1\mod 2=1\mod 2=1 $$ Then let ...
16
votes
1answer
362 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
6
votes
3answers
95 views

A closed form expression for $\int_0^{\infty} (t^2+t^4)^n e^{-t^2-t^4}\,dt$

I was doing some computations for research purposes, which led me to this integral: $$I(n) = \int_0^{\infty} (t^2+t^4)^n e^{-t^2-t^4}\,dt.$$ This is very suggestively written so as to employ a ...
1
vote
1answer
59 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 ...
0
votes
0answers
39 views

answer to polygamma function [duplicate]

I have an expression related to polygamma function. I just need to know whether it is greater than zero or less than zero. In the experssion a>b (both integers) and c is positive real any help ...
0
votes
0answers
15 views

Evaluation of polygamma function?

I have an expression related to polygamma function. I just need to know whether it is greater than zero or less than zero. In the experssion a>b (both integers) and c is positive real any link ...
14
votes
1answer
563 views

Integral$=-\frac{4}{3}\log^3 2-\frac{\pi^2}{3}\log 2+\frac{5}{2}\zeta(3)$

Hi I have been trying to prove this $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
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
126 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
347 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 ...
30
votes
1answer
454 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
49 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
vote
0answers
33 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
31 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
212 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
261 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
31 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