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
37 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
vote
1answer
50 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
21 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
47 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
599 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
33 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: ...
1
vote
1answer
34 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 that uses higher hypergeometric functions. I can't believe that a derivative can ...
6
votes
2answers
125 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
vote
0answers
12 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
31 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
80 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 ...
0
votes
1answer
34 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
37 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
votes
1answer
17 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
786 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
15 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 ...
1
vote
1answer
67 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
20 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: ...
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
89 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
39 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
vote
1answer
56 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 ...
5
votes
1answer
51 views

Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: ...
0
votes
1answer
46 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
58 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
0answers
16 views

Whittaker function integral representation

Let $W_{k,m}(z)$ denote the Whittaker $W$ function, which by definition satisfies $$ W_{k,m}^{''}(z)+\left(-\frac{1}{4}+\frac{k}{z}+\frac{1/4-m^2}{z^2}\right)W_{k,m}(z) =0, $$ where $W_{k,m}^{''}(z)$ ...
0
votes
2answers
64 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
413 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
193 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 ...
4
votes
1answer
41 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
386 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
45 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! ...
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_{- ...
3
votes
0answers
73 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
52 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
44 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
88 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} ...
0
votes
3answers
43 views

Is there a way to combine functions so that you combine their derivatives?

Suppose $y,z$ are functions. What manipulation: "$?$" to the functions would yield the following? (if any) $$y?z=y\cdot z\\~\\ \frac {d(y?z)}{dx}=\frac{dy}{dx}\cdot\frac{dz}{dx}\\~\\ \frac ...
2
votes
2answers
67 views

Proof to $\int_{0}^{\infty}\sin(t)t^{z-1}\,\mathrm{d}t= \sin\left ( \frac{\pi z}{2} \right )\Gamma(z)$

I tried to check the source of the proof to the equation $$\int_{0}^{\infty}\sin(t)t^{z-1}\,\mathrm{d}t= \sin\left ( \frac{\pi z}{2} \right )\Gamma(z),\qquad -1<\Re z < 1$$ but it only has a ...
0
votes
1answer
12 views

Completeness Relation for Tricomi Confluent Hypergeometric Function

Consider the Kummer differential equation $$ \frac{d}{dz}\left[z^be^{-z}\frac{dw}{dz}\right]=az^{b-1}e^{-z}w,\quad z\in\mathbb{R}. $$ It is an eigenvalue problem of Sturm-Liouville type with weight ...
1
vote
1answer
87 views

What are these theta functions appearing in Sloane's database

Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then $$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 ...
0
votes
1answer
19 views

Expressing trigonometric function in terms of integral of Bessel function

I am trying to show that, \begin{align*} \frac{1-\cos x}{x} = \int_{0}^{\pi/2}J_1(x\cos\theta)\,\mathrm{d}\theta \end{align*} I did the following but cannot figure out how to continue. ...
1
vote
0answers
24 views

prove the following problem based on beta function [closed]

$\int_{0}^{1} \frac {x^{m-1}\cdot (1-x)^{n-1}}{(a+x)^{m+n}}dx = \frac{B(m,n)}{a^{n}\cdot (1+a)^{m}}$
3
votes
1answer
110 views

Changes of variables to get an Elliptic Integral of the First Kind

I'm working with a non-linear second order ODE which has an analytical solution in terms of the Jacobi elliptical function $sn(u|k^2)$. The equation is $y''=y(\gamma - \frac{y^2}{2})$ where $\gamma$ ...
5
votes
1answer
112 views

Fundamental period of the Weierstrass $\wp$ elliptic function?

Consider the Weierstrass $\wp$ elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia ...
0
votes
2answers
99 views

Elliptic Integrals of the First Kind

Suppose I have $$F(\phi(x), k) = x$$ where the elliptic integral of the first kind is defined to be $$F(\phi, k) = \int_{0}^{\phi} \frac{1}{\sqrt{1-k^2\sin(\theta)}} \, d\theta $$ How could I invert ...