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
6 views

proving the output of cantor pairing function is un correlated if one bit of input is changed

Pairing function $\pi :N \times N\rightarrow N$ is defined as: $\pi(a_{1},a_{2})=\frac{1}{2}(a_{1}+a_{2})(a_{1}+a_{2}+1)+a_{2}$. Exp 1. If $a_{1},a_{2}$ are input then output is $\pi_{1}$ Exp 2. If ...
2
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4answers
51 views

How to express $2x^3-x^2-3x+2$ as a linear combination of Legendre polynomials

I have used the formula \begin{align}p_0(x)&=1\\ p_1(x)&=x\\ p_2(x)&=\frac12(3x^2−1)\\ p_3(x)&=\frac12(5x^3−3x) \end{align} $$2x^3-x^2-3x+2=Ap_3(x)+Bp_2(x)+Cp_1(x)+Dp_0(x)$$ EDIT- \...
6
votes
3answers
333 views

Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
0
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1answer
30 views

The function $\zeta(\frac{1}{2}+it) \left[ \sqrt{2}\left( \cos(t\log 2)+i\sin(t\log 2) \right) -2 \right]$ has a numerical root

Using the complex exponentiation (this is the MathWolrd's Page) one can deduce for $t>0$ $$2^{\frac{1}{2}+it}=\sqrt{2}e^0(\cos(t\log 2)+i\sin(t\log 2)),$$ since $a=2,b=0,c=\frac{1}{2}, d=t$ and $\...
17
votes
1answer
391 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 ...
1
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1answer
57 views

Integral of incomplete gamma function and limit of hypergeometric function

Let $a > 0$ and consider the integral $$\int_x^\infty \frac{\Gamma(a,t)}{t}\,dt$$ where $\Gamma(a,t)$ is the upper incomplete gamma function $$\Gamma(a,t) = \int_t^\infty x^{a-1} e^{-x} \, dx.$$ ...
0
votes
1answer
31 views

The Bessel function and finding expression

The Bessel Function $J_v$ of the first kind of order $v$ can be defined by the series expresion $$J_v(x)=\sum_{n=0} ^{\infty} \frac{(-1)^n}{n!\Gamma{(1+v+n)}}\left(\frac{x}{2}\right)^{2n+v}$$ (i) if ...
0
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1answer
18 views

What is the gamma term in the modified Bessel function of second kind of Order Zero?

In my version of Schaums mathematical handbook (4th edition page 156), it gives the following for $K_0$ $$ K_0= -\big(ln(x/2)+\gamma\Big) I_0(x)+\frac{x^2}{2}+\frac{x^4}{2^2\cdot4^2}\left(1+\frac{1}{...
2
votes
1answer
53 views

What is the motivation behind the Bessel function of second kind

I am studying Bessel function and found the good reference by G.N. Watson At some point in page 58 he introduces the following expression due to Hankel: \begin{eqnarray} \lim_{\nu \to n} \frac{J_{\...
0
votes
1answer
43 views

Is the derivative of a Bessel function really that complicated?

In this blog entry, they give this ridiculous complicated expression for the first derivative of the Bessel function $J_n(x)$ that uses higher hypergeometric functions. I can't believe that a ...
3
votes
0answers
67 views

Solving a functional equation $2 f(2x)=f(x)(1+\cos(x))+f(x+\pi)(1-\cos(x))$

I am trying to solve the following functional equation, which appears in some of my physics calculations : $f(x)=\frac{1}{2}\left(f(\frac{x}{2})(1+\cos(\frac{x}{2}))+f(\frac{x}{2}+\pi)(1-\cos(\frac{x}...
3
votes
1answer
139 views

What are some common ways to express in mathematical notation the indefinite integral of the $\Gamma$ function? [closed]

I checked on WolframAlpha (I am not good at calculating integrals) and it said there are no elementary functions to express it and gave me a horribly complex (I think... I didn't examine it very ...
0
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1answer
86 views

How is the following integral related to confluent hypergeometric functions?

I am solving an integral that appears in a physics paper. $$ -\int_0^{\infty}dt\,\frac{e^{-t}}{t}\bigg[\bigg(1+\frac{3}{N}t\bigg)^N-1\bigg] $$ The paper does not give the full solution, it only gives ...
1
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1answer
56 views

hyperbolic sum and elliptic integral 2

I try to show that $$\sum _{k=1}^{\infty } k^{36} \text{sech}(\pi k)=\frac{41222060339517702122347079671259045}{137438953472}+\frac{i \left(\psi _{e^{\pi }}^{(36)}\left(1-\frac{i}{2}\right)-\psi _{...
0
votes
0answers
29 views

Incomplete Gamma Asymptotics

my question is simple : if $a_n$ and $z_n$ are both real positive sequences tending to $+\infty$, what is the asymptotic ($n \to +\infty$) behaviour of $\Gamma(a_n,z_n)$ when 1) $a_n \neq z_n$ and $\...
2
votes
1answer
55 views

How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...
2
votes
2answers
604 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 ...
0
votes
2answers
43 views

Approximation of $\ln(x+1)$ with $\Psi$ function

I found the following approximation for the function $$f=\ln(x+1)$$ $$f\simeq\Psi\left(x+\dfrac{3}{2}\right)-2+\gamma+\ln(2)$$ where $\Psi(x)$ is the 'Digamma' function: $\Psi(x)=\dfrac{\dfrac{d}{dx}\...
7
votes
2answers
165 views

Integral ${\large\int}_0^\infty\big(2J_0(2x)^2+2Y_0(2x)^2-J_0(x)^2-Y_0(x)^2\big)\,dx$

I'm interested in the following definite integral: $$\int_0^\infty\big(2J_0(2x)^2-J_0(x)^2+2Y_0(2x)^2-Y_0(x)^2\big)\,dx,\tag1$$ where $J_\nu$ and $Y_\nu$ are the Bessel functions of the first and the ...
1
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0answers
28 views

On series of the kind $\sum_{n=1}^\infty\frac{1}{n^2}\cdot\frac{1}{(1+nx)^s}$ and Frullani's theorem

I would like to know if my computations, in this post are not required justifications for the computations unless if there is a mistake in my claims, were rights and solve a question as if are known a ...
2
votes
1answer
38 views

Upper bound for ratio of modified Bessel functions of second kind

I was wondering if someone has an idea if for $0 < x < y$, and $0< \nu \leq \frac{1}{2}$, one can obtain an upper bound for the ratio $$ \frac{K_{\nu}(x)}{K_{\nu}(y)} $$ Thanks.
4
votes
2answers
89 views

Solving $y^y = x$ for large $x$

I was playing around with recurrence relations and noticed that $\sqrt x$ has the fun property that $$\frac{x}{f(x)} = f(x)$$ ($\sqrt{x}$ and its negation are the only functions $f(x)$ that satisfy ...
1
vote
1answer
39 views

Question on the definition of dirac delta function

I was reading Wikipedia on the Dirac delta function https://en.wikipedia.org/wiki/Dirac_delta_function and there is a part that says: $\delta$ function is expressed by $$ \delta(x) = 1/2 \pi \int_{- \...
3
votes
0answers
40 views

Are these Infinite Series Representations of Special Functions?

I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series (...
0
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1answer
21 views

Show that $\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$ is convergent for $\Re s > -1$

I am struggling to understand the example in Special Functions p. 621, that states, that $$\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$$ is ...
30
votes
1answer
789 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 ...
0
votes
1answer
18 views

Plot a Floquet solution to Mathieu equation

In wikipedia https://en.wikipedia.org/wiki/Mathieu_function#Floquet_solution I want to know how the Floquet solution is plotted. One way I am thinking is to write Floquet solution in terms of the ...
2
votes
1answer
72 views

If $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ holds for $0<z<1$, then also for $0<\operatorname{Re}(z)<1$

In Special Functions p. 10, it has proven that $$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},$$ for $0<z<1$. Then it says that this equality implies for $0<\operatorname{Re}(z)<1$. I do ...
0
votes
1answer
20 views

Limit of trig functions

We have to evaluate $$\lim_{x\to 2} \frac{\cos^x a +\sin^x a -1}{x-2}.$$ I am working on it for hours I tried using series , replacing $\cos a$ by $t$ and $\sin a$ by $\sqrt{1-t^2}$ but not got any ...
-1
votes
1answer
18 views

Limit through a figure

If a circular arc of radius 1 subtends an angle of x radians . The centre of the circle is o and the point c is the intersection of two tangents lines at a and b . Now let T(x) be the area of the ...
0
votes
1answer
21 views

Limit of a function containing $\Psi(x)$

The taylor series expansion of the function $$f(x)=\ln(1+x)$$ around zero is: $$f(x)=\sum_{k=1}^\infty\dfrac{(-1)^{(k+1)}}{k}x^k$$ Putting $x=1$ we have the alternating series: $f(1)=1-\dfrac{1}{2}+\...
0
votes
1answer
16 views

Find a recurrence relationship for the following :

Find a recurrence relationhip for $a_{n}$: $a_{n}=\dfrac {2n+1}{2}\int^{1}_{-1}f\left( x\right) P_{n}\left( x\right) dx$ Where $f\left( x\right)= e^{-x}$ I have done it many times and keep ...
2
votes
1answer
108 views

Bessel's Differential Equation - textbook queries:

In order to ask this question I must first give some background information as written in my text book: Given Bessel's Differential equation: $$x^2y^{\prime\prime}+xy^{\prime}+(x^2-p^2)y=0$$ ...
0
votes
1answer
42 views

Particular values of the Riemann zeta function.

On the wikipedia, near the bottom of the "Specific Values" section, there is a statement that bothers me. $$\zeta(-13)=\zeta(-1)$$ Firstly, it is well noted that the summations must be evaluated ...
1
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1answer
58 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where $\...
0
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1answer
47 views

Having trouble evaluting error function integrals

I am trying to evaluate $$I = \int_1^{\infty } \left(\frac{\operatorname{erf}\left(a -b\log (x)\right)}{2 x^2}-\frac{\operatorname{erf}\left(a + b\log (x)\right)}{2 x}\right) \, dx$$ Let $\log (x) = ...
5
votes
1answer
62 views

Anti-treta function in terms of standard special functions

Define treta$^*$ function as $$ \tau(\alpha_1,\alpha_2,\alpha_3) = \iint_{0< x_1< x_2<1} x_1^{\alpha_1-1}(x_2-x_1)^{\alpha_2-1}(1-x_2)^{\alpha_3-1}\, d(x_1,x_2).\tag{1} $$ Similarly to the ...
0
votes
2answers
65 views

How to Find the Global Minimum and Maximum of this Multivariable Function?

We have the set $$M=\{(x,y,z)\in\mathbb R^3: x^2 + y^2 = z \wedge x+y+z=12\}$$ and the function $$F(x,y,z) = xy+ z^2.$$ How can we find the global maximum and global minimum of F on M and prove ...
3
votes
2answers
414 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
204 views

Deriving the Normalization formula for Associated Legendre functions: Stage $4$ of $4$

The question that follows is the final stage of the previous $3$ stages found here: Stage 1, Stage 2 and Stage 3 which are needed as part of a derivation of the Associated Legendre Functions ...
3
votes
1answer
67 views

About prefactor in book's Gamma function identity

In "Mathematical Methods for Physicists" (Arfken & Weber, 7th ed.), exercise 13.1.16 says the following, Prove that $$|\Gamma (\alpha+i\beta)|=|\Gamma(\alpha )|\prod_{n=0}^{\infty}\left[1+\...
4
votes
1answer
42 views

An always increasing function

Suppose I wanted a function $f(x)$ such that the following properties are had. $f(x)$ maps $\mathbb{R}\to\mathbb{R}$. $f(a)>f(b)$ if $a>b$. The function may or may not be continuous, but it ...
1
vote
1answer
399 views

Identity using q-Pochhammer symbols

Prove - $$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$ where $(a;q)$ are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ...
0
votes
0answers
38 views

How to evaluate I(y) = $\int_0^t e^{ax^b} e^{-cx^d} x^f dx$ in terms of special functions?

To put the above in the proper context, I am trying to solve a Bernoulli equation of the second order: $\frac{dy}{dt} = -\frac{A}{p-q}(e^{-pt}-e^{-qt})y-Be^{-rt}y^2$ where constants A, B, p, q, r ...
2
votes
0answers
46 views

Summation Involving Hermite Polynomials

From the generating formula for Hermite polynomials we know that $$ e^{2xz - z^2} = \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n!} \, . $$ The sum $$ \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n! \sqrt{n+\...
2
votes
1answer
110 views

Finding $f(x)$ such that $\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$

Does there exist any method to find the function $f(x)$ which satisfies $$\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$$ For example $$\int_{- \infty}^{\infty}\frac{\sin(x)}{x}dx=\sum_{k=-\infty}^{\infty}\...
3
votes
0answers
74 views

How to compute the following integral?

Someone has an idea to calculate the following integral $$I_{a,b,\alpha} = \int_{1}^{+\infty} e^{-at} \,(1-t^{-1})^b \log^{\alpha}(1-t^{-1}) \, dt; \quad a,b>0, -1<\alpha<0.$$ Thank you in ...
0
votes
1answer
54 views

How to compute the integral $I_{\alpha} $?

Someone has an idea to calculate the following integral $$I_{\alpha} = \int_{0}^{+\infty} t^{-\alpha} (1-a)^{t} dt; \quad 0<a,\alpha<1.$$ Thank you in advance
1
vote
1answer
46 views

hypergeometric transformation

I came across the following ${}_3F_2$ hypergeometric polynomial: $$ {}_3F_2\left(\left.\begin{array}{c} 1,1,-n\\ 2, -1-2n \end{array}\right| -x\right) $$ for some large $x > 0$. I am wondering ...
0
votes
0answers
93 views

Analytic Bound on The Riemann Zeta Function

Given the canonical infinite product representation (Weierstrass form) of the gamma function, $$\Gamma(z)= \left [ze^{\gamma z}\prod_{m=1}^{\infty} \left ( 1+ \frac{z}{m} \right)e^{-z/m} \right ]^{-1} ...