Functions exported from ModifiedHankelFunctionsOfOrderOneThird:
ModifiedHankelFunctionsOfOrderOneThird.modifiedhankel — Methodmodifiedhankel(z)Return $h₁$, $h₂$, $h₁'$, and $h₂'$, the first and second modified Hankel functions of order 1/3 and their derivatives.
These functions solve Stokes' equation $d²u/dz² + zu = 0$ as a power series in $z$ for abs2(z) < 36 and an approximate asymptotic expansion otherwise. The asymptotic solution is necessary because the $z³ⁱ$ in the power series blows up as $i → ∞$.
For very large arguments z, the exponential function called within the asymptotic expansion may overflow and return Inf, Nan, or 0 values. No warnings will be thrown from this module or Base for floating point issues.
For more information, see [SCL1945].
See also: powerseries, asymptotic
Examples
julia> h1, h2, h1prime, h2prime = modifiedhankel(complex(2.687, -0.648));References
Private functions:
ModifiedHankelFunctionsOfOrderOneThird.asymptotic — Methodasymptotic(z)Return $h₁$, $h₂$, $h₁'$, and $h₂'$, the first and second modified Hankel functions of order 1/3 and their derivatives using an asymptotic expansion, useful for abs2(z) > 36.
For more information, see [SCL1945].
See also: modifiedhankel, powerseries
References
ModifiedHankelFunctionsOfOrderOneThird.powerseries — Methodpowerseries(z)Return $h₁$, $h₂$, $h₁'$, and $h₂'$, the first and second modified Hankel functions of order 1/3 and their derivatives using a power series with 20 terms.
For more information, see [SCL1945].
See also: modifiedhankel, asymptotic
References
- SCL1945The Staff of the Computation Library (1945), Tables of the modified Hankel function of order one-third and of their derivatives. Cambridge, MA: Harvard University Press.
- SCL1945The Staff of the Computation Library (1945), Tables of the modified Hankel function of order one-third and of their derivatives. Cambridge, MA: Harvard University Press.
- SCL1945The Staff of the Computation Library (1945), Tables of the modified Hankel function of order one-third and of their derivatives. Cambridge, MA: Harvard University Press.