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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2answers
16 views

How do you shift a sigmoidal curve to the right?

How do you shift the function $1$ $/ ( 1 + e ^ {-x} )$ to the right without altering the shape of the curve?
1
vote
2answers
407 views

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ ...
1
vote
1answer
21 views

Sums Involving the Mobius Function

Are there any good approximations for the following sums in terms of $n$? $$\sum_{k=1}^{n}\mu(k)$$ $$\sum_{k=1}^{n}\mu(k)\log^m(k)$$ $$\sum_{k=1}^{n}\frac{\mu(k)}{k}.$$ I realize that the third sum ...
0
votes
1answer
27 views

Mean and variance of truncated generalized Beta distribution

The generalized Beta probability density function is given by: $$f(x) = \frac{(x-A)^{\alpha - 1} (B-x)^{\beta - 1}}{(B-A)^{\alpha + \beta - 1} \mathrm{B}(\alpha ,\beta)}$$ for $A<x<B$, and ...
3
votes
1answer
15 views

bessel function maximizer

I try to find global maximum for $ \frac{J_2(x)}{x^2} $ I suspect it happens at x=0 ( plotting the graph) where the value of the function is $ \frac{1}{8} $ I know local maximizers are at zeros of ...
0
votes
1answer
55 views

Integrate square root of 4th grad polynomials

During some calculations for a program I came upon this Integral which I am not able to solve. I already tried Matlab but it didn't help me. Here is the Integral: $$\int\left(\sqrt{\sum_{0}^{5} 9 ...
1
vote
0answers
41 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 ...
76
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 ...
2
votes
2answers
555 views

How to prove this generating function of Legendre polynomials?

How to prove this generating function of Legendre polynomials? $$\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$ I found 2 proofs and they are different from each other and I don't ...
16
votes
1answer
465 views

Fourier Transform $J^3_0(x)$.

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $$ analytically. We can also write $$ \mathcal{I}(\omega)=\mathcal{FT}\big(J^3_0(x)\big) ...
7
votes
1answer
210 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 ...
1
vote
0answers
16 views

Zoo of sigmoid integrals (computational convenience)

In many areas in computational science (e.g. neural networks, fuzzy logic ... ) there is special interest in function like sigmoid ( erf, arctan, tanh ... ) which are kind of blured version of ...
0
votes
1answer
22 views

Discontinuity of the indicator function

Consider the function $q(x,\theta)=1\{ x \in \{x \text{ s.t. } \theta+x_i>0 \text{ }\forall i \}\}$ where 1 is the indicator function taking value 1 if the condition inside $\{ \}$ is satisfied and ...
2
votes
1answer
24 views

Proof of the Unsöld's Theorem (the sum of spherical harmonics)

There is an identity concerning spherical harmonics that plays a pretty important role in atomic physics. Thanks to wikipedia (http://en.wikipedia.org/wiki/Spherical_harmonic) I know that its name is ...
1
vote
1answer
274 views

Equation containing modified bessel functions and exponential function

I'm trying to find a approximation solution for the following equation: ${e^{ - x}}\left[ {{I_o}\left( x \right) + {I_1}\left( x \right)} \right] = C$ where $I_0$ and $I_1$ is the modified Bessel ...
0
votes
0answers
12 views

Solution to Legendre eq in trig form

Okay I'm having a little trouble in answering this question... so the general solution is $y(x) = AP_n(x) + BQ_n(x)$ umm then what do I do?
24
votes
0answers
402 views

Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
0
votes
1answer
23 views

Continuous superposition of bump functions

I am trying to "model" Fig 2 with a superposition of a bump function. I understand that bump functions are bounded and can be often differentiated. The bump function I have used is shown in Fig 1. My ...
0
votes
1answer
21 views

Bessel function with shifted argument

Is there any standard practice which may represents $J_m(a\pm kx)$ in terms of $J_m(kx)$
0
votes
0answers
34 views

Question about an exponential funtion

Now we have a function: $f(x)=e^x, x\in \mathbb R$ Question: 1) Assume that $x>0$, discuss the number of the intersects between $f(x)$ and $y=mx^2,m>0$ under different situation. 2)Assume ...
1
vote
0answers
19 views

What is the product of bessel functions of first and second kind when their arguments are same and tends to zero?

As we know, $\lim_{x \to 0} J_m(x)=0$ where $m\geq 1$ and $\lim_{x \to 0} Y_m(x)=\infty$ then what would be $\lim_{x \to 0}J_m(x)Y_m(x)$. Matlab shows the product is finite and $< 1$. What should I ...
0
votes
0answers
65 views

Integrate $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

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 IMO training ...
4
votes
1answer
252 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 ...
5
votes
1answer
79 views

Closed form of $\operatorname{Li}_2(\varphi)$ and $\operatorname{Li}_2(\varphi-1)$

I am trying to calculate the dilogarithm of the golden ratio and its conjugate $\Phi = \varphi-1$. Eg the solutions of the equation $u^2 - u = 1$. From Wikipdia one has the following ...
2
votes
1answer
71 views

Gamma Function Contour Integration

So, I've been trying to prove the following integral related to the gamma function, and I'm really banging my head against the wall over this: ...
1
vote
1answer
261 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 ...
5
votes
2answers
147 views

Property of sum $\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}$

Is it true that for all $n\in\mathbb{N}$, \begin{align}f(n)=\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}\end{align} is always rational. I have calculated via Mathematica, which says ...
0
votes
0answers
13 views

Intuition about the where the beta distribution has its maximum

I've stumped myself trying to develop an intuition about why the beta distribution should have its maximum where it does. I can differentiate easily enough, and I can manage a simple argument based on ...
0
votes
0answers
15 views

Move Hill equation curve horizontally without changing its shape?

I have a normal Hill function of: $y = \dfrac{x^\lambda}{h^\lambda + x^\lambda}$; where $\lambda$ is Hill coefficient, and $h$ indicates the infection point. I am concerning if we could add another ...
2
votes
1answer
62 views

Shifted integral for a Bessel Function

I have an integral of the kind $\int_{-\infty}^\infty e^{- d \cosh(x+i a)} dx $ where $d, a \in \mathbb{R}$. Now, I know that $\int_{-\infty}^\infty e^{- d \cosh{x}} dx = 2 K_0(d)$ and I would ...
2
votes
1answer
576 views

Integral representation of modified bessel function

How can I obtain $$I_n(x)= \frac{1}{2\pi}\int_{0}^{2\pi}e^{in\theta}e^{x \cos\theta}\,d\theta ,$$ from the integral $$J_n (x) =\frac{1}{2\pi} \int_{0}^{2\pi}e^{-in\theta}e^{ix \sin\theta}\,d\theta .$$ ...
0
votes
1answer
18 views

Interpolate the number of arrangements in a set

I am working the integral $$\int_0^\infty e^{x(k-\alpha) - e^x} dx$$ where $k$ is a positive integer and $\alpha$ a positive real. WolframAlpha shows that for $\alpha=0$ and $k=1,\ldots,7$ the ...
3
votes
0answers
68 views

integral involving incomplete gamma function

Need to evaluate the integral $$ \int_a^b e^{1/x}\,\Gamma(m,1/x)\,dx $$ or equivalently $$ \int_{1/a}^{1/b} y^{-2}\,e^{y}\,\Gamma(m,y)\,dy, $$ where $m$ is an integer, and $0<a<b<\infty$. The ...
2
votes
1answer
112 views

help with exponential integral and square root

can somebody help me integrating this: where m, p and q are positive constants. I tired change of variables and searched for the solution but could not find it. Thanks Note: this is the result ...
0
votes
3answers
97 views

Gaussian-Like integral

What is the integral of this? $$\int_0^\infty xe^{-(ax^2+bx)}\,\mathrm{d}x$$ $a$ and $b$ are positive integers.
0
votes
0answers
30 views

Help with a double integral involving the modified Bessel function of the first kind

Consider the function $f(t)$ given by the double integral: $$f(t)=\frac{2}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}\frac{I_{0}\left(2\sqrt{y x} \right )-1}{e^{y}-1}\cos(xt)dydx$$ Where $I_{0}(\cdot)$ is ...
1
vote
0answers
8 views

mathieu function of non integer order asymptotics

I have an asymptotic expression for the integer mathieu functions: $se_\nu(q,z)$ and $ce_\nu(q,z)$, where $\nu$ is an integer. I would like to use these expression for the case $\nu$ real. My question ...
7
votes
1answer
160 views
2
votes
2answers
664 views

How are the “real” spherical harmonics derived?

How were the real spherical harmonics derived? The complex spherical harmonics: $$ Y_l^m( \theta, \phi ) = K_l^m P_l^m( \cos{ \theta } ) e^{im\phi} $$ But the "real" spherical harmonics are given ...
11
votes
1answer
335 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
0
votes
0answers
13 views

Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
2
votes
2answers
83 views

Rewrite an Integeral according to elliptic function of the second kind

When I try to review the solution process of solving the 2-D Ising model, I get interested in a integral, which is considered to be impossible to get a closed-form. The integeral is ...
0
votes
0answers
32 views

Product of two Whittaker functions

According to 6.669.3 of Gradshteyn and Ryzhik the following identity $$ W_{a,b}(z_1)\,W_{a,b}(z_2) = \frac{2\sqrt{z_1z_2}}{\Gamma(1/2+b-a)\,\Gamma(1/2-b-a)}\int_0^\infty ...
2
votes
2answers
88 views

Real roots plot of the modified bessel function

Could anyone point me a program so i can calculate the roots of $$ K_{ia}(2 \pi)=0 $$ here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D My conjecture ...
0
votes
0answers
37 views

Calculating the limit with integrals of Bessel function

I have $\alpha,\alpha_{0}$ complex numbers. Now I would like to calculate the following: $$\lim\limits_{\epsilon\rightarrow ...
0
votes
1answer
20 views

Interesting special functions identity involving the inner product of real spherical harmonics with a cosecant weight function

In spherical coordinates $\Omega=(\theta,\phi)\in[0,\pi]\otimes[0,2\pi]$, define the inner product $$C_{L_1m_1}^{L_2m_2}:=\left\langle Y_{L_1m_1},\rho,Y_{L_2m_2}\right\rangle=\int ...
0
votes
1answer
37 views

Definite integral involving bessel functions of first and second kind

Is there any standard solution of the integral: $\lim_{\epsilon \to 0} \int_{\epsilon}^{a} J_m(k_1\rho)Y_m(k_2\rho)\rho \, d\rho$. where the integer $m\geq0$ and $a<\infty$
4
votes
0answers
178 views

How do I express $\int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz$ in terms of named functions?

Recently I derived an expression for a particular probability density function. The expression contains the integral $$ f(t,v,a) = \int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz = 2a ...
1
vote
3answers
49 views

search for closed form solution of definite integral

Integrate/hint for this definite integral $$\int_0^\infty(\log\theta)^n\frac{1}{\theta^{k+2}}\text{d}\theta,$$ where $n$ and $k$ are positive integers. It is a simplified form of my earlier question ...
0
votes
1answer
48 views

Definite integral of product of two bessel functions of different order and different argument

What is the solution of the integral: $\int_0^a J_m(k_2\rho)J_{m+1}(k_1\rho)d\rho$ where the integer $m\geq0$