H1v - [ v,z ] Hankel function of real order v and complex argument z (1st kind). Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
H2v - [ v,z ] Hankel function of real order v and complex argument z (2nd kind). Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
Jv - [ v,z ] Bessel function of real order v and complex argument z. Argument z is considered in the cut plane -pi < arg(z) <= pi.
Yv - [ v,z ] Neumann function of real order v and complex argument z. Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
Iv - [ v,z ] Modified Bessel function of real order v and complex argument z. Argument z is considered in the cut plane -pi < arg(z) <= pi.
Kv - [ v,z ] Modified Bessel function of the third kind of real order v and complex argument z. Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
jv - [ v,x ] Spherical Bessel function of real order v and complex argument z. Argument z is considered in the cut plane -pi < arg(z) <= pi.
yv - [ v,x ] Spherical Neumann function of real order v and complex argument z. Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
h1v - [ v,x ] Spherical Hankel function of real order v and complex argument z (1st kind). Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
h2v - [ v,x ] Spherical Hankel function of real order v and complex argument z (2nd kind). Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
Ai - [ x ] First Airy function, solution of the differential equation y"=xy. The argument can be complex
Bi - [ x ] Second Airy function, solution of the differential equation y"=xy. The argument can be complex
Aip - [ x ] Derivative of the first Airy function, solution of the differential equation y"=xy. The argument can be complex
Bip - [ x ] Derivative of the second Airy function, solution of the differential equation y"=xy. The argument can be complex
Struve - [ v,x ] Computes the Struve function of real order v and real argument x. Negative x is rejected unless v is an integer.
H1e - [ v,z ] Exponentially scaled Hankel function of real order v and complex argument z (1st kind). Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
H2e - [ v,z ] Exponentially scaled Hankel function of real order v and complex argument z (2nd kind). Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
Je - [ v,z ] Exponentially scaled Bessel function of real order v and complex argument z. Argument z is considered in the cut plane -pi < arg(z) <= pi.
Ye - [ v,z ] Exponentially scaled Neumann function of real order v and complex argument z. Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
Ie - [ v,z ] Exponentially scaled modified Bessel function of real order v and complex argument z. Argument z is considered in the cut plane -pi < arg(z) <= pi.
Ke - [ v,z ] Exponentially scaled modified Bessel function of the third kind of real order v and complex argument z. Argument z must be nonzero and is considered in the cut plane -pi < arg(z) <= pi.
Ae - [ x ] Exponentially scaled first Airy function of complex argument.
Be - [ x ] Exponentially scaled second Airy function of complex argument.
Aep - [ x ] Exponentially scaled derivative of first Airy function of complex argument.
Bep - [ x ] Exponentially scaled first derivative of second Airy function of complex argument.
lgam - [ z ] Natural logarithm of gamma function. Argument z is considered in the cut plane -pi < arg(z) <= pi.
rgam - [ x ] Returns one divided by the gamma function of the argument.
binomial - [ a,k ] Binomial coefficient, a is real k must be a non negative integer.
beta - [ x,y ] Beta function or Euler's integral of the first kind. The arguments can be complex
lbeta - [ x,y ] Natural logarithm of beta function. Arguments are considered in the cut plane -pi < arg(z) <= pi.
Psi - [ z ] Logarithmic derivative of the gamma function. The argument can be complex
igam - [ a,x ] Incomplete gamma integral; both arguments must be real and positive.
igamc - [ a,x ] Complemented incomplete gamma integral; both arguments must be real and positive.
igami - [ a,x ] Inverse of complemented imcomplete gamma integral of real arguments.
ibeta - [ a,b,x ] Incomplete beta integral; the domain of definition is 0<=x<=1, a>0 and b>0.
ibetai - [ a,b,x ] Inverse of incomplete beta integral; the domain of definition is a>0 and b>0.
hyp1f1 - [ a,b,x ] Confluent hypergeometric function 1F1 with real arguments.
hyp2f0 - [ a,b,x ] Hypergeometric function 2F0 with real arguments.
hyp2f1 - [ a,b,c,x ] Gauss hypergeometric function 2F1 with real arguments.
hyp1f2 - [ a,b,c,x ] Hypergeometric function 1F2 with real arguments.
hyp3f0 - [ a,b,c,x ] Hypergeometric function 3F0 with real arguments.
LegendreE - [ x,k ] Legendre's canonical incomplete elliptic integral of the second kind with real arguments.
LegendreF - [ x,k ] Legendre's canonical incomplete elliptic integral of the first kind with real arguments.
LegendreEc - [ k ] Legendre's complete elliptic integral of the second kind with real argument.
LegendreEc1 - [ k ] Associated Legendre's complete elliptic integral of the second kind with real argument.
LegendreKc - [ k ] Legendre's complete elliptic integral of the first kind with real argument.
LegendreKc1 - [ k ] Associated Legendre's complete elliptic integral of the first kind with real argument.
LegendreP - [ x,n,k ] Legendre's canonical incomplete elliptic integral of the third kind with real arguments.
LegendrePc - [ n,k ] Legendre's complete elliptic integral of the third kind with real arguments.
LegendrePc1 - [ n,k ] Associated Legendre's complete elliptic integral of the third kind with real arguments.
Rf - [ x,y,z ] Carlson's incomplete elliptic integral of the first kind with real arguments.
Rd - [ x,y,z ] Carlson's incomplete elliptic integral of the second kind with real argument.
Rj - [ x,y,z,p ] Carlson's incomplete elliptic integral of the third kind with real argument.
sn - [ u,k ] Jacobian elliptic functions sn(u,k) of real arguments.
cn - [ u,k ] Jacobian elliptic functions cn(u,k) of real arguments.
dn - [ u,k ] Jacobian elliptic functions dn(u,k) of real arguments.
phi - [ u,k ] Amplitude of jacobian elliptic functions phi(u,k) of real arguments.
Dawson - [ x ] Dawson's Integral of real argument.
FresnelC - [ x ] Fresnel integral C(x) of real argument.
FresnelS - [ x ] Fresnel integral S(x) of real argument.
dilog - [ x ] Dilogarithm function of real argument.
Zeta - [ x ] Riemann zeta function of real argument. x must be positive
Zeta2 - [ x,q ] Riemann zeta function of two arguments. It is the sum, for k integer ranging from 0 to infinity, of (k+q)^-x where q is a positive integer and x > 1.
Ei - [ n,x ] Exponential integral Ei. n in an integer, x is real.
Si - [ x ] Sine integral of real argument.
Ci - [ x ] Cosine integral of real argument.
Shi - [ x ] Hyperbolic sine integral of real argument.
Chi - [ x ] Hyperbolic cosine integral of real argument.
Plm - [ l,m,x ] First kind Legendre polynomials and associated functions of degree l and integer order m. Plm(l,m,x) = (-1)^m (1-x^2)^(m/2) d^m( Pn(n,x) )/dx^m where l and x must be real and Pn(n,x) is the Legendre polynomial.
Qlm - [ l,m,x ] Second kind Legendre functions of degree l, integer order m and argument 0<=x<1.
plm - [ l,m,x ] Normalized first kind Legendre polynomials and associated functions of integer degree l and integer order m.
Ylm - [ l,m,theta,phi ] Spherical harmonic of integer degree l, integer order m, latitude theta in [-PI,PI] and longitude phi.
Yl - [ l,theta,phi ] Sequence of spherical harmonic of integer degree l, integer order m=0..l, latitude theta in [-PI,PI] and longitude phi.
round - [ x ] Round real x to nearest or even integer number.
signum - [ x ] Sign of x.
csgn - [ x ] Complex sign of x.
sfact - [ n ] Semifactorial of integer n.
mask - [ x ] Masks and unmasks the partial loss of precision error. If called with x = 0 that error message is disabled, if called with x != 0 that error message is enabled. Returns the previous state; default is unmasked.