SpecialFunctions.osax

bessel v : Bessel functions: J, Y, I, K and modified ones (j,y,i,k)

bessel real or array of real

kind string : a character in [J,Y,I,K,j,y,i,k]

index integer : order of the function: an integer (i1≥0) or a range {i1,i2} (0 ≤ i1 ≤ i2)

[scaled boolean] : default: false. J and Y are never scaled, i and k are always scaled

→ real : a real or an array of real

dawson v : the Dawson integral: \exp(-x^2) \int_0^x dt \exp(t^2)

dawson real or array of real

→ real : a real or an array of real

debye v : the Debye functions D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))

debye real or array of real

index integer : order of the function: an integer 1≤i≤6

→ real : a real or an array of real

EllipticE v : EllipticE integral \int_0^1 dt sqrt(1-x^2*t^2)/sqrt(1-t^2)

EllipticE real or array of real

→ real : a real or an array of real

EllipticK v : EllipticK integral \int_0^1 dt 1/(sqrt(1-t^2)*sqrt(1-x^2 t^2))

EllipticK real or array of real

→ real : a real or an array of real

expint v : Exponential integral functions \int_a^b dt f(t)/t^n (noted as [a,b,f(t),n]).

expint real or array of real

kind string : see table below

index integer : relevant for expint_En

→ real : a real or an array of real

kind	:		[	a	,	b	,	f(t)	,	n	]
expint_E1	:		[	1	,	\infty	,	\exp(-xt)	,	1	]
expint_E2	:		[	1	,	\infty	,	\exp(-xt)	,	2	]
expint_En	:		[	1	,	\infty	,	\exp(-xt)	,	index	]
expint_Ei	:	-PV{	[	-x	,	\infty	,	\exp(-t)	,	1	]	}
Shi	:		[	0	,	x	,	\sinh(t)	,	1	]
Chi	:	gamma+\ln(x) +	[	0	,	x	,	\cosh(t)-1	,	1	]
expint_3	:		[	0	,	x	,	\exp(-t^3)	,	0	]
Si	:		[	0	,	x	,	\sin(t)	,	1	]
Ci	:	-	[	x	,	\infty	,	\cos(t)	,	1	]
atanint	:		[	0	,	x	,	\arctan(t)	,	1	]

fractionalbessel v : Fractional Bessel functions: J, Y, I, K

fractionalbessel real or array of real

kind string : a character in [J,Y,I,K]

index real : order of the function : a positive number

[scaled boolean] : default: false. J and Y are never scaled

→ real : a real or an array of real

gegenbauer v : Gegenbauer polynomials also known as ultraspherical polynomials

gegenbauer real or array of real

index integer : order of the function: an integer (i1≥0) or a range {i1,i2} (0 ≤ i1 ≤ i2)

[parameter real] : parameter a>-1/2 (default 0)

→ real : a real or an array of real

hzeta v : the Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty (k+q)^{-s}

hzeta real : s≠1 is a real or an array of real

parameter real : parameter q>0

→ real : a real or an array of real

laguerre v : The Laguerre polynomials are defined in terms of confluent hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x).

laguerre real or array of real

index integer : order of the function: an integer (i1≥0) or a range {i1,i2} (0 ≤ i1 ≤ i2)

[parameter real] : parameter a>-1 (default 0)

→ real : a real or an array of real

LambertW0 v : Lambert's W functions, W(x), are defined to be solutions of the equation W(x) \exp(W(x)) = x. This function is defined for x>-1/e and has multiple branches for x < 0; however, it has only two real-valued branches. We define W_0(x) to be the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the other real branch, where W < -1 for x < 0.

LambertW0 real or array of real

→ real : a real or an array of real

LambertWm1 v : see LambertW0

LambertWm1 real or array of real

→ real : a real or an array of real

legendre v : Legendre polynomials

legendre real or array of real

index integer : order of the function: an integer (i1≥0) or a range {i1,i2} (0 ≤ i1 ≤ i2)

→ real : a real or an array of real

polygamma v : The polygamma functions of order m defined by (d/dx)^{m+1} \log(\Gamma(x)) (for m=0 : digamma function)

polygamma real or array of real

index integer : order of the function : an integer (≥0)

→ real : a real or an array of real

zeta v : the Riemann zeta function is defined by the infinite sum \zeta(s) = \sum_{k=1}^\infty k^{-s}.

zeta real : s≠1 is a real or an array of real

→ real : a real or an array of real



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