User:Akanksh

Special Funtions

Complete Fermi Dirac Integrals

$F_{j}(x):=(1/\Gamma (j+1))\int _{0}^{\infty }dt(t^{j}/(\exp(t-x)+1))$

Incomplete Fermi Dirac Integrals

$F_{j}(x,b):=(1/\Gamma (j+1))\int _{b}^{\infty }dt(t^{j}/(\exp(t-x)+1))$

Airy Functions and Derivatives

$Ai(x)=(1/\pi )\int _{0}^{\infty }\cos((1/3)t^{3}+xt)dt$
$Bi(x)=(1/\pi )\int _{0}^{\infty }(e^{(}-(1/3)t^{3})+\sin((1/3)t^{3}+xt))dt$

Clausen Functions

$Cl_{2}(x)=-\int _{0}^{x}dt\log(2\sin(t/2))$

Normalized Hydrogenic Bound States

$R_{n}:=2(Z^{3/2}/n^{2}){\sqrt {(n-l-1)!/(n+l)!}}\exp(-Zr/n)(2Zr/n)^{l}L_{n-l-1}^{2l+1}(2Zr/n)$
Where:$L_{n}^{k}(x)=(-1)^{k}(d^{k}/dx^{k})L_{(}n+k)(x)$

Legendre Forms

$F(\phi ,k)=\int _{0}^{\phi }dt1/{\sqrt {(}}(1-k^{2}\sin ^{2}(t)))$

$E(\phi ,k)=\int _{0}^{\phi }dt{\sqrt {(}}(1-k^{2}\sin ^{2}(t)))$

$\Pi (\phi ,k,n)=\int _{0}^{\phi }dt1/((1+n\sin ^{2}(t)){\sqrt {(}}1-k^{2}\sin ^{2}(t))$

Carlson Forms

$RC(x,y)=1/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1)$

$RD(x,y,z)=3/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1/2)(t+z)^{(}-3/2)$

$RF(x,y,z)=1/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1/2)(t+z)^{(}-1/2)$

$RJ(x,y,z,p)=3/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1/2)(t+z)^{(}-1/2)(t+p)^{(}-1)$

Gamma Functions

$\Gamma (x)=\int _{0}^{\infty }dtt^{x-1}\exp(-t)$

Psi (Digamma) Function

$\psi ^{(n)}(x)=(d/dx)^{n}\psi (x)=(d/dx)^{n+1}\log(\Gamma (x))$

Riemann Zeta Function

$\zeta (s)=\sum _{k=1}^{\infty }k^{-s}$

Hurwitz Zeta Function

$\zeta (s,q)=\sum _{0}^{\infty }(k+q)^{-s}$

Eta Function

$\eta (s)=(1-2^{1-s})\zeta (s)$

Transport Functions

$J(n,x):=\int _{0}^{x}dtt^{n}e^{t}/(e^{t}-1)^{2}$