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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1answer
39 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
152 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 ...
2
votes
0answers
39 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
32 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 ...
0
votes
2answers
72 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
35 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 $f(x)$?
4
votes
0answers
183 views

The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school training ...
1
vote
0answers
24 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 \\ & ...
1
vote
1answer
157 views

How does one calculate the amount of time required for computation?

For example, to compute the zeroes of the Riemann zeta function using the Euler-Maclaurin summation method one has to do O(T) work. The Euler-Maclaurin summation method for zeta is given by $ ...
1
vote
1answer
192 views

Hankel curve formula for gamma function proof

I recently read a proof for the following formula which I don't understand completely. For $Re(z)>0$: $\Gamma(z)=\frac{1}{e^{2\pi iz}-1}\int_{C_\delta}e^{-\zeta}\zeta^{z-1}d\zeta$ , where ...
11
votes
1answer
510 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
1
vote
1answer
69 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 ...
2
votes
1answer
247 views

Integration and $\gamma(n,x)$

It isn't hard to prove that: $$\int_0^x e^{-t} {t^n} dt = n! \cdot e^{-x}\left( e^x-\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$ Or put in a different way: $$\int_0^x e^{-t} \frac{t^n}{n!} dt = ...
3
votes
1answer
130 views

How to reproduce the Mathematica solution for $\int(\cos x)^{\frac23}dx$?

I entered this integration problem to Mathematica Online Integrator an got a solution I would never have been able to find manually. $$\int\root 3 \of{\cos(x)^2}\,dx=\frac{(-3\cos(x)\root 3 ...
6
votes
5answers
4k views

Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
6
votes
2answers
177 views

How prove this $p(x)>0$ if $p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$

let the polynomials $$p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$$ and such $$a_{0}+\sum_{a_{i}<0}(1-\dfrac{i}{n})\binom{n}{i}a_{i}>0$$ and ...
0
votes
0answers
44 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
1answer
128 views

Integral involving the Spherical Bessel Function of the First Kind

How can I prove the equation below using Spherical Bessel Function Recurrence Relation? (where $ j_{n}(x) $ means Spherical Bessel function of first kind) Definition using BesselJ function: $$ ...
2
votes
1answer
40 views

Upper bound on the modified Bessel function of the first kind

I am looking for an upper bound tighter than $e^x$ for the modified Bessel function of the first kind. Does there exist $\epsilon\in (0,1)$ such that $I_0(x)\le e^{x^{1-\epsilon}}$ for all $x\ge 0$ ? ...
0
votes
0answers
22 views

Solving non-linear, but separable and autonomous, matrix ODE $H'(z) = A H(z)^k + S$

Start with the non-linear scalar ODE: $H'(z) = A H(z)^k + S$. You can separate and integrate this to find something like: $$z + C_1 = \int_0^{H}\frac{1}{S - A q^k}dq$$ From this, you can use the ...
3
votes
1answer
141 views

Integral Representation of Bessel Function (K)

There is an integral representation for the modified Bessel function of the second (or third depending on who you talk to) kind (denoted $K_\nu$) that says: $$K_\nu(z) = ...
3
votes
2answers
200 views

Evaluation of another definite integral

I have a definite integral that I am trying to solve. Any hint or reference is urgently sought. , where $r$ is any positive integer while $\psi$ and $\nu$ are positive real numbers.
4
votes
0answers
52 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( ...
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 $b$: $x (t)-λ\int_0^1 t^2s^2x(s) \, ds = 4t + bt^2$
1
vote
1answer
89 views

Define Beta function in terms of recursive relation?

The Beta function has recursive relation: $$ Beta(x,y) = Beta(x, y+1) + Beta(x+1, y) \! $$ Is there some least redundant condition, if added, will make the recursive relation an equivalent ...
1
vote
2answers
392 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
10
votes
1answer
161 views

Integral $\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3$

Hi I am trying to prove this below. $$ I:=\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3 $$ where $$ \zeta_3=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I am ...
12
votes
1answer
194 views

On Continuous Replicative Functions

Knuth, in The Art of Computer Programming Vol. 1, defines a replicative function as a function $f$ such that $$f(x)+f\left(x+\frac{1}{n}\right)+\cdots +f\left(x+\frac{n-1}{n}\right)=f(nx)$$ whenever ...
4
votes
1answer
80 views

Elliptic integral $\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk$

Question: Prove that $$\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk=\frac{1}{16\pi}\Gamma^4\left( \frac{1}{4}\right)$$ My attempt Start by the transformation $$k \to \frac{2\sqrt{k}}{1+k}$$ ...
3
votes
1answer
94 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
1
vote
1answer
60 views

Continuous non differentiable functions :)

I was searching for functions like Weierstrass (continuous but differentiable nowhere), but I haven't found any. If you could tell me some that would be great. Also, I would like to find some ...
5
votes
3answers
123 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
8
votes
1answer
148 views

Algebraic structures whose Hilbert-Poincaré series are special functions

Are there good examples of algebraic structures whose Hilbert-Poincaré series are that of special functions? I'm particularly interested in cases where complex analytic reasoning about those series ...
1
vote
1answer
51 views

Equailty involving Elliptic integrals and hypergeometric function

How to prove the following $$\,_2F_1\left(-1/2,-1/2,1,k^2 \right)=\frac{2}{\pi}\left(2E+(k^2-1)K \right)$$ where we define The complete integral of first kind $$K=K(k) = \int^1_0 ...
3
votes
1answer
82 views

Expressing upper incomplete gamma function of half-integer order in terms of gamma function?

N. M. Temme, "Special Functions" (Wiley 1996) gives the following expression that expresses the upper incomplete gamma function in terms of the ordinary gamma function, for integer orders: $$ ...
26
votes
9answers
7k views

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ ?

It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ ...
4
votes
1answer
108 views

Infinite sum of Bessel Functions

I came across the following sum in my work involving the infinite sum of a product of Bessel functions. Does anyone have any idea of how to express this in a simpler form? 'a' and 'b' are positive ...
3
votes
0answers
76 views

Log Log Integrals III

The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
2
votes
0answers
49 views

Log Log Integrals II

The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
3
votes
0answers
34 views

integral involving hypergeometric function $\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{y})}{y}\,dy$

I arrived at the following result $$\tag{1}\int^\infty_0 z^{p-1} E^2(z)\,dz=\frac{\Gamma(p)}{p}\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{z})}{z}\,dz$$ where the exponential integral $E(z)$ is defined ...
0
votes
1answer
37 views

Reduction of DEs to Bessel equation

A question in my textbook asks me to write down the general solution to: $\frac{d}{dx}(x^2\frac{dR(x)}{dx}) + [k^2x^2 - n(n+1)]R(x) = 0$ in terms of Bessel functions. Now two similar questions ...
3
votes
2answers
67 views

Prove: $\sin (3\pi/2 - x) = -\cos(x)$

I know the sine of $3\pi/2$ is $-1$. So i plug it in the function making it. $\sin(-1-x) = -\cos x$. However, I don't know where to go from here.
1
vote
1answer
41 views

BesselJ function on negative real numbers

I have evaluated Bessel$J_v(x)$ with some real $v$ and negative real $x$ in MATHEMATICA. I cannot understand how the result is complex (non-real). I look at the series definition of BesselJ and I ...
7
votes
2answers
143 views

Infinite Series :$ \sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2}$

Prove that: $$\sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2} = \frac{-\pi^{\frac{3}{2}}}{12}\left( \pi^2+6\gamma(1-2\log ...
2
votes
1answer
134 views

How can I express such function as known functions or power series?

$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$ $$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$ ...
9
votes
1answer
193 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see ...
4
votes
0answers
169 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
0
votes
0answers
50 views

Is $2\delta(x) \neq \delta(x)$?

$2\delta(x) \neq \delta(x)$ since, by definition, Can this been seen graphically though? If so, how? If not, why is it that they are mathematically different but graphically the same? Btw, I ...
0
votes
1answer
25 views

Solutions in terms of Bessel functions

I came across this question that asked to express solutions to: $$x^2y'' + xy' + (4x^2 - v^2)y = 0,\quad 0\le x<\infty$$ in terms of Bessel functions subject to the boundary conditions y(x) is ...
0
votes
1answer
19 views

Integral Bessel recurrence relation

I want to show that $\int x^vJ_{v-1}(x)dx = x^vJ_v(x) + C$. Now I know the recurrence relations of the Bessel equation/function and the one I need to use is $x^vJ_v(x) = x^vJ_{v-1}(x)$ I'm just ...