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
24 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
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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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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
66 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 ...
2
votes
1answer
72 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: ...
0
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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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
21 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 $a<\infty$
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 $m\geq0$
1
vote
0answers
14 views

Weighting values using curves

I am in the field of computer programming but I think my question is maths based. I have tried several google searches on this matter but nothing has come up with what I am looking for, probably ...
0
votes
0answers
21 views

Incomplete gamma function and hypergeometric function to Meijer-G

can somebody help me to convert the incomplete gamma function and the hypergeometric function (in the forms shown below and as a function of z) into a form of Meijer-G function?
6
votes
1answer
66 views

Integral $I=\int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx$

Hello can you please help me solve this integral $$ \int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx=-\frac{2\pi}{3}(\pi^2+24\ln 2). $$ I am trying to work through all logarithmic integrals. Note, ...
0
votes
0answers
42 views

Fourier expansion of the complexified Gram series

Consider the Riemann's R function, also known as the Gram series: $$\text{R}(x)=1+\sum_{k=1}^{\infty}\frac{\left(\log x\right)^{k}}{kk!\zeta(k+1)}$$ Now consider the form: $$\text{R}\left(e^{2\pi ix} ...
5
votes
1answer
88 views

Integral computation of $\int_0^\pi \mathrm d t \sin(a\cos t/2) \mathrm{sinh}(b\sin t/2)$

I'm having trouble computing an integral. $$ I=\int_0^1 \frac{\mathrm{d}x}{2x(1-x)}\left(x-\cosh\left(\frac{t\sqrt{1-x}}{\tau}\right)+\sqrt{1-x}\text{ ...
0
votes
2answers
45 views

An example of a homeomorphism on $[0,1]^2$ with constant Jacobian determinant $\pm1$

Let $T(x,y):=(t_1(x,y),t_2(x,y))$ be a continuous bijection, namely a homeomorphism on $[0,1]^2$. I am trying to find a $T$ such that $\det(J_T)=1$. (*) The trivial cases are $T(x,y)=(x,y)$, ...
3
votes
2answers
47 views

Prove that $\lim\limits_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0$.

I conjecture that for any $\epsilon>0$, we have $$ \lim_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0 $$ where $\Gamma(x,a) = \int_a^\infty t^{x-1}e^{-t} \mathrm{d}t$ denotes the ...
1
vote
0answers
74 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&& ...
1
vote
0answers
38 views

finding value of definite integral

Does the following definite integral exist here $\alpha > 0$ and $n$ is positive integer.
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 ...
7
votes
2answers
94 views

Does this really converge to 1/e? (Massaging a sum)

Short version: can we prove that $$\sum_{k=0}^n (-1)^k \binom{n}{k}^2 \frac{k!}{n^{2k}} \to \frac1e$$ as $n \to \infty$? Long version: First, consider $$a_n = \sum_{k=0}^n \frac{(-1)^k}{k!}$$ It is ...
0
votes
1answer
19 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
40 views

Function returning number of subsets of size $k$ of a set of size $n$.

I am looking for a function that returns the number of subsets of size $k$ of a set of size $n$. Ideally, the function is commonly used. I took a look at the binomial coefficient. However, there ...
2
votes
1answer
38 views

Beta function proof

Show that : $$\beta \left( x,n\right) =\dfrac {\left( n-1\right) !}{x\left( x+1\right) \left( x+2\right) ....\left( x+n-1\right) }$$ My attempt : $$\beta \left( x,n\right) =\dfrac {\Gamma \left( ...
3
votes
1answer
42 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: $$ ...
2
votes
2answers
44 views

Integration by using special functions

$$\int ^{\pi }_{0}\dfrac {dt}{\sqrt {3-\cos t}}$$ How can you solve the following equation by using alpha/gamma functions and putting $$\cos t=1-2\sqrt {u}$$
0
votes
0answers
19 views

Superadditivity ; How to prove Superadditive function?

How can I prove this superadditive function? It's quite confused
2
votes
1answer
49 views

How to prove this integral representation of Bessel function?

I'm trying to prove this integral representation: $$J_\nu = \frac{({x/2})^\nu}{\Gamma(\nu+1/2)\sqrt{\pi}}\int_{-1}^{1}(1-t^2)^{\nu-1/2}e^{itx}dt$$ I think countour integration would be useful, but ...
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 $J_m(kx)$
1
vote
1answer
40 views

Bessel function with shifted argument

Is there any standard practice which may represents $J_m(a\pm kx)$ in terms of $J_m(kx)$ where $a$ is any constant and $m$ is integer $>-1$
0
votes
0answers
38 views

Is there a way to simplify Legendre-squared sum $\sum_{n} \frac{[P'_{n}(x)]^2}{n(n+1)}$

Is there a closed-form expression for $$F(x) = \sum_{n=2,{\rm even}}^{\infty} \frac{[P'_{n}(x)]^2}{n(n+1)}$$ where $P'_{n}(x)$ is the derivative of the $n^{\rm th}$ Legendre polynomial? Simple ...
0
votes
0answers
17 views

modern analysis: step functions with upper and lower sums [duplicate]

A function $f$ defined on $[a,b]$ is a step function if there is a partition $P$ such that $f$ is constant on each subinterval of $P$ a. Show that upper and lower sums are integrals of step ...
6
votes
0answers
90 views

analytic solution to definte integral

I am looking for Analytic solution to a definite integral. Or an approriate transformation to apply. the conditions on alpha, beta being positive real numbers while n is positive integer.the integral ...