Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
Questions tagged [special-functions]
831 questions
16
votes
2 answers
Sum of trigonometric functions
Do somebody know the closed form of the following sum (m is an integer)
$$f(\beta)=\sum _{k=1}^{2m+1} \sin^{2 m+1}\left[\frac{-\beta+k \pi
}{1+2 m}\right]$$
If instead of $n=1+2m$ we put $n=2m$, then the above sum do not depends on $\beta$ and is…
david
- 161
- 1
- 4
16
votes
2 answers
Is there a known solution to $f(x) = (1-x)f(x^2)$?
The functional equation $f(x) = (1-x)f(x^2)$ (with $f(0)=1$) has a simple solution that can be expressed as a rapidly converging infinite product
$$f(x) = \prod_{n=0}^\infty (1 - x^{2^n}) = (1-x)(1-x^2)(1-x^4)(1-x^8)\cdots$$
and as a more slowly…
Alan Stern
- 161
9
votes
1 answer
$\mathrm{Li}(x)$ vs $x/\log x$
I need an explicit lower bound for $\mathrm{Li}(x)$ in terms of $x$ and $\log x$. Say, Wikipedia gives
$$
\mathrm{li}(x) >\frac x{\log x}+\frac x{(\log x)^2}
$$
for $x>e^{11}$, see the logarithmic integral entry, and so
$$
\mathrm{Li}(x)=…
Yuri Bilu
- 1,130
7
votes
2 answers
On a polynomial related to the Legendre function of the second kind
The Legendre function of the second kind, $Q_n(z)$, along with the usual Legendre polynomial $P_n(z)$, are the two linearly independent solutions of the Legendre differential equation.
$Q_n(z)$ can be expressed in the following…
7
votes
2 answers
Proving a hypergeometric function identity
While playing around with the fractional calculus, I got stuck trying to show that two different ways of differintegrating the cosine give the same result. DLMF and the Wolfram Functions site don't seem to have this "identity" or something that can…
7
votes
2 answers
When is ArcTan a rational multiple of pi?
Is there a characterisation for which $x\in\mathbb{R}$ the value $\arctan(x)$ is a rational multiple of $\pi$?
Or reformulated: What is the "structure" of the subset $A\subseteq\mathbb{R}$ which fulfils
$$ \arctan(x) \in \pi\mathbb{Q}…
Daniel Krenn
- 567
6
votes
1 answer
Modern comprehensive account of the Barnes G-Function
I previously put this forward on the math.stackexchange community with little luck:
I am looking for a comprehensive account of the properties and applications of the Barnes G-Function. Everything from recurrence relations, proof of the infinite…
Ismail Bello
- 315
5
votes
2 answers
generalization of (Rogers) dilogarithm
Let $C$ and $S$ be abbreviations for $\cosh$ and $\sinh$, and consider the following
function:
$$f(x,y) = \int_{-y\le r+l \le y} \frac{ (C(x)S(l)C(r) - C(l)S(r))(C(x)C(l)S(r)-S(l)C(r)) }
{(C(x)C(l)C(r) - S(l)S(r))^2-1} dl dr$$
If $y=\infty$, this…
Danny Calegari
- 2,542
4
votes
2 answers
Are there noteworthy functional properties of the exponential integral?
I noticed that the following function
$$\mbox{Ei}(x) := - \int_{-x}^{\infty}\frac{e^{-t}}{t} \,\mathrm d t$$
occurs increasingly in different areas of physics and in mathematics.
I am wondering if anyone knows of any interesting functional or…
John Joe
- 141
4
votes
1 answer
Jacobi elliptic functions with modulus on the unit circle
I am gathering some available informations on Jacobi elliptic functions $sn(z,k)$, $cn(z,k)$, $dn(z,k)$ with $k\in\mathbb{C}$, $|k|=1$. I can not find much on them in standard references (Abramowitz&Stegun, DLMF, wiki,...). It seems that something…
Twi
- 2,188
4
votes
2 answers
Recognize this sum
I have strong feeling that the function,
$$
f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)},
$$
is a known special function (here $\Gamma(x)$ is the usual extension of the factorial). Is this the case?
Daniel Parry
- 1,286
4
votes
1 answer
Is there any heartbeat like function?
I'm looking for a function, where the result is something like this:
I tried to figure it out myself, but I have no idea how to manage it.
f(x) = ...
Thanks in advance,
never_shown
- 51
3
votes
2 answers
A calculation involving Lerch Transcendents
The Lerch Transcendent is defined here as
$$\Phi(z,s,a):=\sum_{k=0}^\infty \frac{z^k}{(k+a)^s}.$$
I am interested in the case $z=\frac 12,$ $s=1.$ The following limit showed up in estimating uniform distributions on an interval of length 1 under the…
Hedonist
- 1,269
3
votes
2 answers
What is known about this power series?
In the course of some calculations, I came across the following powers series.
For fixed $C>1$ let
$$
f_C(u)=\sum_{k=0}^\infty\frac{u^k}{C^{k^2}}.
$$
This series converges for all $u\in\mathbb C$, hence $f_C$ is an entire function.
Can it be…
user1688
3
votes
1 answer
hyperbolic functions and Gauss hypergeometric series
If the Gauss hypergeometric function
$F(1, 3/2, 5/2; z^2) = 3 [\tanh^{-1} (z) –z]/z^3$
what is the corresponding result for $F(1,15/8,23/8;z^8) $?
M V S
- 33