Compute the principal branches of the (modified) Bessel functions of the first and second kind. The Bessel functions of the first and second kind solve Bessel's equation $$z^{2} \dfrac{\text{d}^{2} w}{\text{d} z^{2}} + z \dfrac{\text{d} w}{\text{d} z} + (z^{2} - \nu^{2}) w = 0$$ and are given by $$\begin{aligned} J_{\nu}(z) &= (\tfrac{1}{2} z)^{\nu} \sum_{k = 0}^{\infty} (-1)^{k} \frac{(\frac{1}{4} z^{2})^{k}}{k! \Gamma(\nu + k + 1)} \\ Y_{\nu}(z) &= \frac{Y_{\nu}(z) \cos(\nu \pi) - J_{-\nu}(z)}{\sin(\nu \pi)} \end{aligned}$$ The modified Bessel functions of the first and second kind solve the modified Bessel's equation $$z^{2} \dfrac{\text{d}^{2} w}{\text{d} z^{2}} + z \dfrac{\text{d} w}{\text{d} z} + (z^{2} - \nu^{2}) w = 0$$ and are given by $$\begin{aligned} I_{\nu}(z) &= (\tfrac{1}{2} z)^{\nu} \sum_{k = 0}^{\infty} \frac{(\frac{1}{4} z^{2})^{k}}{k! \Gamma(\nu + k + 1)} \\ K_{\nu}(z) &= \frac{\pi}{2} \frac{I_{-\nu}(z) - I_{\nu}(z)}{\sin(\nu \pi)} \end{aligned}$$
arb_hypgeom_bessel_j(nu, x, prec = flintPrec())
acb_hypgeom_bessel_j(nu, z, prec = flintPrec())arb_hypgeom_bessel_y(nu, x, prec = flintPrec())
acb_hypgeom_bessel_y(nu, z, prec = flintPrec())
arb_hypgeom_bessel_i(nu, x, prec = flintPrec())
acb_hypgeom_bessel_i(nu, z, prec = flintPrec())
arb_hypgeom_bessel_k(nu, x, prec = flintPrec())
acb_hypgeom_bessel_k(nu, z, prec = flintPrec())
An arb or acb vector
storing function values with error bounds. Its length is the maximum
of the lengths of the arguments or zero (zero if any argument has
length zero). The arguments are recycled as necessary.
numeric, complex, arb, or
acb vectors.
a numeric or slong vector indicating the
desired precision as a number of bits.
The FLINT documentation of the underlying C functions: https://flintlib.org/doc/arb_hypgeom.html, https://flintlib.org/doc/acb_hypgeom.html
NIST Digital Library of Mathematical Functions: https://dlmf.nist.gov/10
Classes arb and acb;
arb_hypgeom_gamma_lower and
arb_hypgeom_beta_lower for the “incomplete” gamma
and beta functions.