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ComplexBessel

Complex Bessel Function Numerical Evaluation Note

Optimized Code

DoubleDouble |ν| \leq 256

Im(z) < 0

Complex Bessel functions are analytically connected so that they are complex conjugate symmetrically about the real axis.

bessel conj

DLMF 10.11
DLMF 10.34

Re(z) < 0, Im(z) ≥ 0

Complex Bessel functions are defined to be discontinuous on the negative real axis.

besselj minus rez
bessely minus rez
besseli minus rez
besselk minus rez
hankel1 minus rez
hankel2 minus rez

DLMF 10.11
DLMF 10.34

Re(z) ≥ 0, Im(z) ≥ 0, Asymptotic Expansion

hankel coef
hankel omega

besselj asymp
bessely asymp
besseli asymp
besselk asymp
hankel1 asymp
hankel2 asymp

DLMF 10.17
DLMF 10.40

Numerical Evaluation Stability

BesselJ, BesselY BesselI BesselK
besseljy stat besseli stat besselk stat
besseljy convergence besseli convergence besselk convergence

Connection Formula

besselji
besselyk
besselyk
bessel itoj
bessel ktoy
bessel ktoy minus
bessel ytok

DLMF 10.27

Reference

DLMF
Wolfram Math World

Author

T.Yoshimura