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
24 views

Reducing integral

let $$I=\int \frac{dx}{\sqrt{mx^3-x^2+n}}$$ How do we reduce $I$ to an elliptic Integral of the first Kind ? where $m,n>0$ are constants.
1
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0answers
26 views

How to compute $\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$

For a project I want to get a closed form solution of $$\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$$ Here $p \in \mathbb{N},\; d\ge3, \; d\in\mathbb{N}$ and $P_n^d$ is the associated ...
0
votes
0answers
11 views

Associated Legendre polynomial expansion of $\exp(\xi)$

For a project I need to compute the coefficients of the Associated Legendre polynomial expansion of the $\exp$ function. That is I need to find $b_n$ such that $$\exp(x^Ty) = \exp(\xi) = ...
4
votes
1answer
41 views

How I can simplify this inequality or how I can solve it?

How I can simplify this inequality or how I can solve it: $$\left\lceil\dfrac{\ln(t+2)}{\ln 2}\right\rceil-\left\lfloor\dfrac{\ln(t+1)}{\ln2}\right\rfloor>1$$ where $t$ is a positive integer. ...
1
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0answers
12 views

Integral involving the Legendre Polynomials

I'm trying to compute the following integral: $\int_a^{+\infty} \frac{dt}{(P_\lambda(\tanh{t}))^2}$ and to be honest I have no idea on how I should attack this problem. Is there any reference that I ...
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1answer
24 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} ...
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0answers
24 views

Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
0
votes
0answers
49 views

Definite integral involving Error function

Let us write $$\mathrm{erf}(x)=\frac{2}{\sqrt {\pi}}\int_0^x e^{-t^2}dt $$ for the usual Gauss error function. Given natural numbers $m,n,k$ I am interested in computing the integral ...
1
vote
0answers
46 views

How to write $H_{n+2}$ in terms of $H_n$ [closed]

How can I write: $H_{n+2}$ in terms of $H_n$ ?? I have no clue, which is why I havent tried anything.
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1answer
22 views

On the continuity of $xf(x)$ and $x^2f(x)$, where $f$ is the Dirichlet function

Let $$f(x) = \begin{cases}1\qquad x\in\mathbb{Q}\\ 0\qquad x\notin\mathbb{Q} \end{cases}$$ Then how do I show that $xf(x)$ is continuous in $0$ and that $x^2f(x)$ is differentiable there as well? ...
0
votes
0answers
8 views

Normalization for argument of maximum function

is it possible to normalize the maximum function of a certain argument ? Means: Is that $\theta_{ML} = arg \max\limits_{\theta} \{ \sum \limits_{n=1}^{N} |w_n w^*_{n+N}| - \Big( \frac{SINR + ...
4
votes
1answer
171 views

Evaluating $\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$

The title says it all, really - I am looking for $$\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$$ where $0<r<1$ and $x$ is in a domain where the integrand is real. It came up ...
5
votes
2answers
64 views

deriving the sum of $x^n/(n+2)^2$

I am writing a research paper and I have stumbled upon an issue. I have to evaluate $$\sum_{n=1}^{\infty} \frac{x^n}{(n+2)^2}$$ Here is what I did: $$ \sum_{n=1}^{\infty} x^{n-1} = ...
2
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0answers
24 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)= 4\text{Li}_2(\alpha^3)+3\text{Li}_2(\alpha^2)-6\text{Li}_2(\alpha)+\tfrac{7}{5}\zeta(2)\tag1$$ ...
16
votes
0answers
91 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
4
votes
0answers
29 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
8
votes
1answer
92 views
+100

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} ...
1
vote
2answers
99 views

Extremely difficult log integral, real methods only

$$\int_{0}^{1}\frac{x^2 + x\log(1-x)- \log(1-x) - x}{(1-x)x^2} dx$$ I tried this: $$M_1 = \int_{0}^{1} \frac{1}{1-x} \cdot \left(\frac{x^2 + x\log(1-x) - \log(1-x) - x)}{x^2}\right) dx$$ $$M_1 = ...
2
votes
0answers
38 views

Identities for hypergeometric functions ${}_2F_1$ with z=1/2

Is there a closed form (or approximation) for a hypergeometric function of form: $_2F_1(1,b+c;c;\frac{1}{2}) \quad \text{where} \; b,c \in \mathbb{N}$ ? I researched all identities in ...
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votes
2answers
44 views

What is the limitation of $\Gamma(\frac{n+1}{2})/\Gamma(\frac{n}{2})$?

What is the limitation of $$\frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}$$ as $n \rightarrow \infty$? If it diverges, could I get its divergence rate? $\Gamma(x)$ is the ...
1
vote
2answers
43 views

Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$

How to prove $$\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$$ I havent started, any hints?
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0answers
29 views

Absolutely continuous function and differentiability

let $m(E)=0$ in $[a,b]$ Can we define a absolutely continuous function on $[a,b]$ which is not differentiable only at $E$.
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0answers
31 views

How to compute this limit involving Gamma functions

I am investigating the following limit $$\lim_{u\to \infty}\frac{\Gamma \left(\frac{1}{2 H},\frac{H 2^{\left(\frac{1}{2H}\right)}}{u^{\frac{1}{H}}\left(1-H\right)^{\left(2 - \frac{1}{H}\right)}} ...
1
vote
1answer
12 views

Identity relating hypergeometric function and Legendre polynomial

In my notes I have written down the following relation: $_2F_1(a,a+\frac{1}{2};c;z)=2^{c-1}z^{(1-c)/2}(1-z)^{-a+(c-1)/2}L_{2a-c}^{1-c}\big(\frac{1}{\sqrt{1-z}}\big)\ ,$ where $_2F_1(a,b;c;z)$ is the ...
2
votes
0answers
40 views

improper integrals in q-calculus

In quantum calculus is this equality possible for improper integrals? $\lim_{x\to\infty}\int_0^xf(t)d_qt=\int_0^\infty f(x)d_qx$
0
votes
0answers
16 views

Upper bound of $\int_{z_1}^{z_2}x^k e^{-x}dx$

How can I find a good upper bound to this quantity ? $$\int_{z_1}^{z_2}x^k e^{-x}dx,\ \ \ \ z_{1,2}\in \mathbb {Im}, k\in \mathbb N$$ The integrand makes me thinking about an incomplete gamma ...
0
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0answers
25 views

Can we find a function $f$ verifying these equalities?

Let $(u_{n})$ be an increasing positive sequence. My question is: Can we find an increasing function $f$ verifying these equalities: $$f(2^{u_{n}}\sin(θ/(2^{u_{n}})))=u_{n}$$ ...
3
votes
1answer
38 views

Closed form of the sequence $(3n)\times (3n-1) \times \cdots \times 6 \times 5 \times 3 \times 2$?

I need the closed form of the sequence $(3n)\times (3n-1) \times \cdots \times 6 \times 5 \times 3 \times 2$? The title is fairly self-explanatory. I know that a closed form of this exists using the ...
1
vote
0answers
22 views

Sources on Jacobi Elliptic Functions

I'm interested in learning more about the Jacobi Elliptic Functions and the associated theta functions. For instance, what was the initial motivation for defining them? What are some applications? Is ...
4
votes
3answers
104 views

General derivative of $\text{sinc}(x)$

I'm trying to find a general formula for the $n$-th derivative of the following function $$ \frac{\sin(x)}{x} $$ I've computed the first five derivatives so far without finding any 'structure' in it. ...
1
vote
1answer
29 views

Iterative function with $z_{n+2}$

I'm currently playing arround with my custom fractal renderer and on Math SE in this answer Américo suggested the following function: $z_{n+2}=z_{n+1}^{3}+c^{z_{n}}$ But to get the first value I'd ...
0
votes
1answer
14 views

Do square integrable Legendre functions always have inteter degrees and order?

I am considering solutions to the generalized Legendre differential equation, e.g. the first kind, $P_{\nu}^{\mu}(x)$, on the interval $(-1, 1)$. Then I am wondering, for this function to be square ...
0
votes
0answers
34 views

Integration by parts with Bessel function $j_0$

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
6
votes
2answers
72 views

How to solve $\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$

Could you help me to prove $$\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$$ where ${\rm B}(a,b)$ is Beta function.
3
votes
0answers
25 views

Can derivative of Hurwitz Zeta be expressed in Hurwitz Zeta?

Can the derivative of Hurwitz Zeta function by the first argument be expressed in terms of Hurwitz Zeta and elementary fuctions? There is a formula which expresses Hurwitz Zeta through its ...
1
vote
1answer
29 views

Gamma of 3z using triplication formula:

I have to demostrate the gamma function for 3z as you see below: Using the multiplication formula demostrate gamma(3z) Gamma functions of argument $3z$ can be expressed using a triplication ...
9
votes
1answer
248 views

Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$

How can we prove that: $$\int_0^1\frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx=\frac{\pi^2}{8}\zeta(3)-\frac{27}{16}\zeta(5) $$
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0answers
42 views
5
votes
1answer
74 views

Pretty lower bound on the gamma function

According to http://functions.wolfram.com/06.05.29.0006.01, for every $x\geq 2$ it is $$ \left( \frac{x}{e}\right)^{x-1} \leq \Gamma(x) \leq \left( \frac{x}{2}\right)^{x-1}, $$ where $\Gamma$ is the ...
1
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0answers
23 views

Series in closed form

Let $\alpha \in (0,1)$ and $z\geq 0$. Define the function: $$ \theta_{\alpha}(z)=\sum_{j=0}^{\infty}\left(\frac{z}{\alpha j + 1}\right)^j. $$ Can $\theta_{\alpha}(z)$ be written in closed form? Are ...
4
votes
0answers
22 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
2
votes
0answers
24 views

Simplified expression of $ _2F_1((K-1)a,K,Ka,x) $

Is there any simplified expression of this Hypergeometric function $ _2F_1((K-1)a,K,Ka,x) $ Thanks!
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0answers
12 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
1
vote
1answer
50 views

Recognize as a special function?

Is the following function a special function of some kind $$ f(x) = \int_0^x (1+e^{-t})^{b}\,dt, $$ where $b>1$?
1
vote
0answers
19 views

Large-z limit of the *other* second derivative of the Laguerre polynomial

I'm trying to find the asymptotic behavior of the second derivative of the Laguerre polynomial (more precisely, the associated analytic function), $\frac{\partial}{\partial \nu^2}L_{\nu}(z)$, as $z\to ...
2
votes
0answers
42 views

A list of numbers and

I have a real life problem that math may be able to solve. I am no mathematician so if you have any insight please use the simplified version. This problem is way beyond me. My gut tells me there is ...
5
votes
1answer
68 views

Sum involving zeros of Bessel function

I came across the following sum in my work involving the infinite sum of function of zeros of Bessel functions. $$ \displaystyle ...
9
votes
0answers
203 views

Is this similarity just a coincidence?

Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. But if we do the same only in one direction, we get ...
2
votes
0answers
24 views

Identifying a function that involves combinations of terms

I need to know if a function exists that partitions terms in such a way as seen below $$ \frac{d^n}{dx^n}[\frac{(x)_c}{n!}] $$ Note that $(x)_c$ is the falling factorial of x and $c \geq n$, This in ...
0
votes
1answer
19 views

Monotonicity of Modified Bessel Functions of the Second type

Given $n\geq1$ an integer, Is it known that $$ x\to x^nK_n(x) $$ is a decreasing function on $(0,\infty)$? I am looking for a reference or a proof.