Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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13 views

$K$ which is of second category in itself.let $H = K \cap ( - K)$. Why $H$ is non empty interior

Let $X$ be topological vector space.Let $K$ be closed, convex, dense subset of $X$ and $K$ which is of second category in itself. Put $H = K \cap ( - K)$. Why does $H$ is nonempty interior?
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1answer
142 views

Infinite series of Hypergeometric function

Any ideas how to find a closed form for the sum given by: $$ \sum^\infty_{n=0} \frac{1}{n!} \frac{a^n b^{n+m}}{(m+n)^2 \Gamma(m+n)} {}_2F_2 \left(m+n,m+n;m+n+1,m+n+1;-b\right) $$ Given that both $a$ ...
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0answers
32 views

Barnes' double gamma function versus q-gamma function

According to wikipedia, the q-analog of the gamma function is closely related to a multiple gamma function defined by Barnes. Besides the fact that they are both generalizations of the Gamma function, ...
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1answer
54 views

Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
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1answer
78 views

How to prove this problem about supermodularity function?

The problem is as follows, and I have solved the subproblem (a), but haven't solved (b) yet. And for (b) the method I think about is proof by contradiction, but I get stuck before I could solve this. ...
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2answers
70 views

Eliminate the arbitrary funcion - PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
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1answer
22 views

Difference Between Lyapunov and Strong Lyapunov Function.

Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function. To show that $V$ is Lyapunov. I ...
2
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1answer
74 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
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2answers
65 views

Derivative of the Gamma function

How do you prove that $$ \Gamma'(1)=-\gamma, $$ where $\gamma$ is the Euler-Mascheroni constant?
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2answers
36 views

Integrals involving exponential functions and the gamma function [duplicate]

I'm having trouble evaluating this integral $$\int_0^\infty {e^{-ax^2}} \,dx $$ My guess is that it would evaluate into something like $$\int_0^\infty \frac 12e^{-s}s^{\frac 12} \ldots \,dx = \frac ...
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1answer
57 views

Best aproximation to an numerical solution using two aproximated functions

I want to find the best aproximation to a numerical solution. For that I want to use two aproximated functions (that I already know). If I plot them I see that one of them underestimates the original ...
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1answer
38 views

Formula for the Beta function for natural m, n

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function $B(x, y)$, it's symmetry $B(x,y) = B(y,x)$ aswell as the fact that $(x + y)B(x + 1, y) = xB(x, y) ...
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votes
1answer
21 views

Logarithmic algorithm performance

If I have an algorithm that on $T$ iterations gets me within $O(\log(T)/T)$ accuracy, what is a (preferably concise, closed form) lower bound on $T$ that gets me within $\epsilon$ accuracy? In other ...
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1answer
19 views

Logical comparison of two values with algebra

Suppose I have two real numbers A and B (A $\wedge$ B $\subset$ $\mathbb{R}$). I want to do some algebra over these number and get 1 if they are equal and get 0 if not. For example: In this ...
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1answer
61 views

Primitive of the function $(\sin x)/x$

I know that for some functions, for instance $f(x) = e^{-x^2}$, there does not exist a primitive. Does there is a primitive for the function $f(x) = \frac{\operatorname{sin}(x)}{x}$?
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1answer
40 views

An integral related to the derivative of Legendre polynomials

I want to calculate the integral $$ I=\int_{-1}^{1} \Big(\frac{\mathrm{d}P_{n+1}(t)}{\mathrm{d}t}\Big) \Big(\frac{\mathrm{d}P_{m+1}(t)}{\mathrm{d}t}\Big) \mathrm{d}t $$ where $P_n(t)$ is Legendre ...
1
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1answer
15 views

Integer functions $f_k(n)$ which return {0,1} depending on whether or not $k|n$

In computer programs and physics problems it is often nice to have mathematical functions that work when you want them to but sort of zero out when they don't apply. I'm thinking of how useful delta ...
2
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1answer
58 views

Asymptotic series of Confluent Hypergeometric function $U(a,1,z) $ as $z \to 0$

Consider the Confluent hypergeometric function $U(a,b,z)$, which is a solution of the Kummer's Equation : $$zw''+(b-z)w'-aw=0$$ it has the following integral representation when $- \pi/2 < \arg ...
0
votes
1answer
56 views

Evaluation of Spence's function.

Spence's function is defined as $${\rm Li}_2 (z)=- \int_0^z \frac{\ln(1-u)}{u} \, du $$ where $$z \in {\mathbb C} \setminus [1, \infty )$$ For $|z|<1 $ $${\rm Li}_2 (z)= \sum_1^ \infty \frac{ ...
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0answers
72 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
4
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3answers
106 views

Proving positivity of the exponential function

Question. Without using the semigroup property ($\mathrm{e}^{x}\mathrm{e}^{y}=\mathrm{e}^{x+y}$), how can we show that $\mathrm{e}^{x}>0$ for all $x\in\mathbb{R}$ only by using the series ...
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2answers
27 views

How can I show that $\left(a-n-1\right)!/\left(a-1\right)!=\left(-a\right)!\left(-1\right)^n/\left(-a+n\right)!$?

Is it possible to show that \begin{align}\frac{\left(a-n-1\right)!}{\left(a-1\right)!}\stackrel{?}{=}\frac{\left(-a\right)!\left(-1\right)^n}{\left(-a+n\right)!}\tag{1},\end{align} or, more ...
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1answer
14 views

Set of points at which a function coincides with its convexification is compact?

Let $f:[0,1]\rightarrow\ \mathbb{\bar{R}}$, and let $\tilde{f}$ be the convexification of $f.$ (i.e., $\tilde{f}$ is the pointwise supremum of all affine functions that lie everywhere below $f$.) Let ...
5
votes
1answer
77 views

Evaluation of the series $ \sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$.

The following series: $$\sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$$ came up as an intermediate step of calculating an integral. The answer according to Wolfram is $\displaystyle ...
0
votes
1answer
16 views

A low-discrepancy or quasirandom series which would guarantee all value sequences

I am trying to find a type of quasi-random sequence which would guarantee that it could produce all possible sequences of values within the possible value range, while still producing random-seeming ...
4
votes
2answers
113 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
2
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1answer
109 views

Limit involving incomplete gamma function

Let $\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t} dt$ be the incomplete gamma function. What is the limit $$ \lim_{p \to \infty} \frac{1}{\sqrt{p}} \Big[\Gamma(p+\tfrac12,x)\Big]^{\frac{1}{2p}} $$ as a ...
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0answers
25 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
1
vote
1answer
49 views

Product with $\Gamma$ function

Evaluate the product: $$\Gamma \left ( \frac{1}{n} \right )\Gamma \left ( \frac{2}{n} \right )\cdots \Gamma \left ( \frac{n-1}{n} \right )$$ A path to a solution: $$\begin{aligned}\Gamma\left ...
5
votes
2answers
76 views

How are Fox-H functions useful in math?

The Fox H-function, as far as I know, is the most general families of functions - encompassing an even larger family of functions than the already very general Meijer G-function. While I've known ...
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0answers
20 views

Upper bound for the ratio of Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex and $z$ is a positive real number. Do you know any results about it? Thank ...
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vote
1answer
31 views

How do I know if a fractional linear transformation exists?

I have a feeling I'm missing another obvious point about FLTs. How do I know if a specific fractional linear transformation exists? I think I can find specific transformations by using the ...
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0answers
41 views

Question about the function ξ(s)

We define When Π(s) is the gamma function and ζ(s) is the zeta function. I can prove that ξ(s) has order one. Let be ρ the zeros of ξ(s) (the non- trivial zeros of ζ(s)). Why ∑1/|ρ| diverges? And ...
1
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1answer
35 views

Quasiconvex functions

Let $\lambda$ be any real number and $f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ given by $$ f(x,y) = \begin{cases} \lambda &\mbox{if } \quad0<x<1, y=1, \\ 1+\lambda y & \mbox{if } ...
21
votes
3answers
450 views

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Let $P_s(x)$ denote the Legendre functions of the $1^{st}$ kind, i.e. the Legendre polynomial generalized to an arbitrary (not necessarily integer) order $s$. It can be expressed using the ...
0
votes
1answer
29 views

Polygamma reflection formula

How does one prove the polygamma reflection formula: $$\psi^{(n)}(1-z)+(-1)^{n+1}\psi^{(n)}(z)=(-1)^n \pi \frac{d^n}{d z^n} \cot \pi z $$ Do we have to invoke the power of contour integration and ...
1
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1answer
53 views

How to remodel sigmoid function so as to move stretch/enlarge it?

I have a question similar to this. I want the sigmoid to have asymptotes to $+1$ and $0$ in specific points $\frac{1}{A}$ and $-\frac{1}{A}$, as in the Figure (where $\frac{1}{A}=2$ and ...
2
votes
2answers
508 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 ...
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1answer
97 views

A limit related to an alternating series [closed]

Show that the limit $$ \lim_{N\rightarrow\infty}\bigg(\sum_{k=1}^{2N-1}(-1)^{k+1}\frac{\sin\big(\frac{\pi}{2N}\big)}{\sin\big(\frac{k\pi}{2N}\big)}\bigg) = 2\ln2 $$ holds. Hint: I think the identity ...
2
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1answer
179 views

Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

How would one deal with such an integral? $$I(k)\equiv \int_0^\infty r^n e^{-r(1+\mu)} e^{-{\mathrm i} kr}\:{}_1F_1({\mathrm i}/k+1;2;2{\mathrm i} kr) \, \mathrm{d} r$$ Here $n\in\{0,1\}$, $\mu\in ...
2
votes
2answers
101 views

Challenging integral

I am trying to find a close form representation for the following integral: $$ A(x;a,b,c)= \int_{0}^{x}\frac{\sin\left(a k+b k^{2}\right)+\sin\left(c k-b k^{2}\right)}{k}dk $$ for $0<x \ll ...
0
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0answers
22 views

What can we say about variational energies?

Suppose that $U \subset \mathbb{R}^d$ is open and let $V_{ij}^{kl}(r)$ $(1 \leq i,j,k,l \leq d)$ be functions on $V$ to $\mathbb{R}$ which are as smooth as the coming problem may require. For the ...
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2answers
81 views

Closed form of the integral $\int_0^1 \frac{x^n}{1+x}\, dx$

I am trying to evaluate the integral $$\int_0^1 \frac{x^n}{1+x}\, dx, \;\;\; n \in \mathbb{N}$$ in a closed form. I tried tackling it using Beta Form $\displaystyle \int_0^1 ...
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0answers
24 views

Taylor series of $(1-x)^b$ $_2F_1(a,b;c;x)$: when to stop?

Let $f(x)= (1-x)^b$ $_2F_1(a,b;c;x)$, where $0<x<1$ and $a=(K-1)d$, $b=K$, c=$Kd$ (with $a$, $b$ and $c$ are positive and $K>d$ ). I need to derive the Taylor series of the corresponding ...
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0answers
92 views

Compute $\int_{1}^{\infty} \frac{J^2_{n}(k)}{k^m} dk$

Question as the title showed,in which J means Bessel functions, n and m are positive integers. How to get the analytic result? Any comment is much appreciated. Many thanks in advance.How to simplify ...
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0answers
16 views

Can the regularized beta function be calculated for integer values of $a$ using identities?

I've been trying to create a cumulative Student's T Distribution calculator using javascript as fun side project. I've successfully created a gamma function approximator, and from there, a beta ...
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vote
2answers
55 views

Integrals with the special functions $Ci(x)$ and $erf(x)$

I'm looking for the solutions of the following two integrals: $$I_1=\int\limits_0^\infty dx\, e^{-x^2}Ci(ax)$$ and $$I_2=\int\limits_0^\infty dx\, e^{-ax}erf(x)$$ with ...
0
votes
0answers
31 views

Solution to a particular Wave Equation

Consider the partial differential equation \begin{align} \frac{1}{c^{2}} \, \frac{ \partial^{2} U}{\partial t^{2}} &= \frac{\partial^{2} U}{\partial x^{2}} + x \, \frac{\partial U}{\partial x} + ...
1
vote
1answer
48 views

Asymptotic behavior of the zeros of the digamma function

The gamma function has just one extremum on each interval $(k,k+1)$, where $k$ is a negative integer. These extrema occur at the zeros of the derivative of the gamma function. Let $z_n$ denote the ...
3
votes
1answer
55 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 ...