The file BesselFunctions.mth defines functions for approximating Bessel and Airy functions.  These functions are described in Chapters 9 and 10 of Abramowitz and Stegun [1965].  See the introduction to ExponentialsIntegrals.mth7RN4D for a discussion of accuracy.  The function definitions in the file are automatically loaded when any of its functions are first used.


BESSEL_J(n, z) approximates to the Bessel function of the first kind Jn(z) for integer or fractional n.


BESSEL_J_LIST(n, z) simplifies to a vector of Bessel functions of the first kind J0(z) through Jn(z) for integer n.  It uses a recurrence relationship, so the values become less accurate for Ji(z) as i approaches n.


BESSEL_J_SERIES(n, z, m) approximates to m+1 terms of a series approximation for the Bessel function of the first kind Jn(z).


BESSEL_J_ASYMP(n, z) simplifies to a one-term asymptotic approximation for the Bessel function of the first kind Jn(z) for large magnitude z.


BESSEL_Y(n, z) approximates to the Bessel function of the second kind Yn(z) for noninteger n.


BESSEL_Y_SERIES(n, z, m) approximates to m+1 terms of a series approximation for the Bessel function of the second kind Yn(z) for integer n.


BESSEL_Y_ASYMP(n, z) simplifies to a one-term asymptotic approximation for the Bessel function of the second kind Yn(z) for large magnitude z.


BESSEL_I(n, z) approximates to the modified Bessel function of the first kind In(z) for integer n.


BESSEL_I_SERIES(n, z, m) approximates to m+1 terms of a series approximation for the modified Bessel function of the first kind In(z) for complex n.


BESSEL_I_ASYMP(n, z) simplifies to a two-term asymptotic approximation for the modified Bessel function of the first kind In(z) for complex n and large magnitude z.


BESSEL_K(n, z) approximates to the modified Bessel function Kn(z) for complex n and |phase z| < π/2.


BESSEL_K_ASYMP(n, z) simplifies to a two-term asymptotic approximation for the modified Bessel function Kn(z) for complex n and large magnitude z.


SPHERICAL_BESSEL_J(n, z) simplifies to a closed form for the spherical Bessel function of the first kind jn(z) for integer n.


SPHERICAL_BESSEL_J_LIST(n, z) simplifies to a vector of spherical Bessel functions of the first kind j0(z) through jn(z) for integer n.  It uses a recurrence relationship, so the values become less accurate for ji(z) as i approaches n.


SPHERICAL_BESSEL_Y(n, z) simplifies to a closed form for the spherical Bessel function of the second kind, yn(z) for integer n.


AI_SERIES(z, m) approximates to m+1 terms of a series approximation for the Airy function Ai(z).


BI_SERIES(z, m) approximates to m+1 terms of a series approximation for the Airy function Bi(z).


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