Hypergeometric functions

The confluent and Gauss hypergeometric functions, with the kernels scipy.special uses (Boost.Math for real hyp1f1, as in SciPy).

Usage

sc_hyp0f1(v, z)

sc_hyp1f1(a, b, x)

sc_hyp2f1(a, b, c, z)

sc_hyperu(a, b, x)

Arguments

v, a, b, c: Numeric vectors of parameters.
z, x: Numeric vector; complex too, except for sc_hyperu().

Value

A double vector, or a complex vector for complex input.

Details

sc_hyp0f1(): \({}_0F_1(;v;z)\), complex.
sc_hyp1f1(): Kummer's \({}_1F_1(a;b;x)\), complex x.
sc_hyp2f1(): the Gauss \({}_2F_1(a,b;c;z)\), complex z.
sc_hyperu(): Tricomi's \(U(a, b, x)\) for real x >= 0.

Only the last argument may be complex; the parameters are real.

Arguments are recycled: each must have length one or the common length. NA gives NA; NaN is evaluated as SciPy does. The result takes its dimensions, or else its names, from the arguments as base R arithmetic does, and so does each element of a list result. Conditions the kernels raise, such as a pole, are handled as sc_seterr() says.

Branch cuts

sc_hyp2f1() has a branch cut on the real axis from 1 to infinity, as in SciPy. sc_hyp0f1() is entire in z, and sc_hyp1f1() in x.

Other packages

In gsl, for real arguments: hyperg_0F1(), hyperg_1F1(), hyperg_2F1() and hyperg_U(). In hypergeo: hypergeo(A, B, C, z), the Gauss function for complex z.

Examples

sc_hyp0f1(0.5, 1)
#> [1] 3.762196
sc_hyp1f1(1, 2, c(0.5, 1 + 1i))
#> [1] 1.297443+0.0000000i 1.378025+0.9093307i
sc_hyp2f1(1, 1, 2, 0.5)
#> [1] 1.386294
sc_hyperu(1, 2, 3)
#> [1] 0.3333333