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10 Bessel FunctionsNotation

§10.1 Special Notation

(For other notation see Notation for the Special Functions.)


integers. In §§10.4710.71 n is nonnegative.


nonnegative integer (except in §10.73).


real variables.


complex variable.


real or complex parameter (the order).


arbitrary small positive constant.




Γ(x)/Γ(x): logarithmic derivative of the gamma function (§5.2(i)).


derivatives with respect to argument, except where indicated otherwise.

The main functions treated in this chapter are the Bessel functions Jν(z), Yν(z); Hankel functions Hν(1)(z), Hν(2)(z); modified Bessel functions Iν(z), Kν(z); spherical Bessel functions jn(z), yn(z), hn(1)(z), hn(2)(z); modified spherical Bessel functions in(1)(z), in(2)(z), kn(z); Kelvin functions berν(x), beiν(x), kerν(x), keiν(x). For the spherical Bessel functions and modified spherical Bessel functions the order n is a nonnegative integer. For the other functions when the order ν is replaced by n, it can be any integer. For the Kelvin functions the order ν is always assumed to be real.

A common alternative notation for Yν(z) is Nν(z). Other notations that have been used are as follows.

Abramowitz and Stegun (1964): jn(z), yn(z), hn(1)(z), hn(2)(z), for jn(z), yn(z), hn(1)(z), hn(2)(z), respectively, when n0.

Jeffreys and Jeffreys (1956): Hsν(z) for Hν(1)(z), Hiν(z) for Hν(2)(z), Khν(z) for (2/π)Kν(z).

Whittaker and Watson (1927): Kν(z) for cos(νπ)Kν(z).

For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).