Modified Hankel Functions of Order One Third

Calculate modified Hankel functions of order one-third and their derivatives.

These special functions are solutions to Stokes' differential equation:

\[\frac{\mathrm{d}^2u}{\mathrm{d}z^2} + zu = 0\]

The modified Hankel functions are linearly related to Airy functions but are not as well known. One application of the modified Hankel functions of order one-third is as a solution to the wave equation for electromagnetic fields between the boundaries of the earth-ionosphere waveguide. It may also occur in other instances of diffraction and refraction of waves.

The direct computation of solutions to Stokes' equation is preferred to using Airy or Hankel functions. Unlike Bessel's equation, Stokes' equation has no singularity in the finite complex plane and its solutions are single-valued ^[SCL1945].

Usage

using ModifiedHankelFunctionsOfOrderOneThird

h1, h2, h1prime, h2prime = modifiedhankel(z)

Solutions

Two solution approaches are used, as presented in ^[SCL1945]. If abs2(z) < 36, a power series solution is used. Otherwise, an asymptotic expansion is performed because of floating point limits in the power series.

Power Series Solution

$h₁$, $h₂$, $h₁'$, and $h₂'$ are computed from auxiliary functions

\[h_1(z) = g + \frac{i\sqrt{3}}{3}(g - 2f), \qquad h_2(z) = g - \frac{i\sqrt{3}}{3}(g - 2f)\]

where

\[f(z) = a_0 + a_1z^3 + a_2z^6 + \cdots + a_mz^{3m} + \cdots\]

\[g(z) = z(b_0 + b_1z^3 + b_2z^6 + \cdots + b_mz^{3m} + \cdots)\]

\[f'(z) = -z^2(c_0 + c_1z^3 + c_2z^6 + \cdots + c_mz^{3m} + \cdots)\]

\[g'(z) = d_0 + d_1z^3 + d_2z^6 + \cdots + d_mz^{3m} + \cdots\]

where

\[a_0 = \frac{2^{1/3}}{\Gamma\left(\frac{2}{3}\right)} \qquad a_m = -\frac{a_{m-1}}{(3m-1)3m}\]

\[b_0 = \frac{2^{1/3}}{3^{2/3}\Gamma\left(\frac{4}{3}\right)} \qquad b_m = -\frac{b_{m-1}}{3m(3m+1)}\]

\[c_0 = \frac{a_0}{2} \qquad c_m = - \frac{c_{m-1}}{3m(3m+2)}\]

\[d_0 = b_0 \qquad d_m = - \frac{d_{m-1}}{(3m-2)3m}\]

Asymptotic Solution

For $h₁$ on $-2π/3 < \arg z < 4π/3$:

\[h₁(z) ≈ α z^{-1/4} \exp(2/3 i z^{3/2} - 5πi/12) \left( 1 + \sum_{m=1} (-i)^m C_m z^{-3m/2} \right)\]

\[h₁'(z) ≈ α i z^{1/4} \exp(2/3 i z^{3/2} - 5πi/12) \left( 1 + \sum_{m=1} (-i)^m C_m z^{-3m/2} \right) \\ - α/4 z^{-5/4} \exp(2/3 i z^{3/2} - 5πi/12) \left( 1 + \sum_{m=1} (-i)^m C_m z^{-3m/2} \right) \\ - 3/2 α z^{-1/4} \exp(2/3 i z^{3/2} - 5πi/12) \left( \sum_{m=1} (-i)^m m C_m z^{-3m/2 - 1} \right)\]

where

\[C_m = \frac{(9-4)(81-4)\cdots (9[2m-1]^2-4)}{2^{4m}3^m m!}\]

and

\[h₁(z) ≈ h₁(z) + α z^{-1/4} \exp(-2/3 i z^{3/2} - 11πi/12) \left( 1 + \sum_{m_1} (i)^m C_m z^{-3m/2} \right)\]

\[h₁'(z) ≈ h₁'(z) - α z^{1/4} \exp(-2/3 i z^{3/2} - 11πi/12) \left( 1 + \sum_{m_1} (i)^m C_m z^{-3m/2} \right) \\ - α/4 z^{-5/4} \exp(-2/3 i z^{3/2} - 11πi/12) \left( 1 + \sum_{m_1} (i)^m C_m z^{-3m/2} \right) \\ - 3/2 α z^{-1/4} \exp(-2/3 i z^{3/2} - 11πi/12) \left( \sum_{m=1} i^m m C_m z^{-3m/2 - 1} \right)\]

for $-4π/3 < \arg z < 0$.

And for $h₂$ on $-4π/3 < \arg z < 2π/3$:

\[h₂(z) ≈ α z^{-1/4} \exp(-2/3 i z^{3/2} + 5πi/12) \left( 1 + \sum_{m=1} (i)^m C_m z^{-3m/2} \right)\]

\[h₂'(z) ≈ -α z^{1/4} \exp(-2/3 i z^{3/2} + 5πi/12) \left( 1 + \sum_{m=1} (i)^m C_m z^{-3m/2} \right) \\ - α/4 z^{-5/4} \exp(-2/3 i z^{3/2} + 5πi/12) \left( 1 + \sum_{m=1} (i)^m C_m z^{-3m/2} \right) \\ - 3/2 α z^{-1/4} \exp(-2/3 i z^{3/2} + 5πi/12) \left( i^m m C_m z^{-3m/2 - 1} \right)\]

and

\[h₂(z) ≈ h₂(z) + α z^{-1/4} \exp(2/3 i z^{3/2} + 11πi/12) \left( 1 + \sum_{m=1} (-i)^m C_m z^{-3m/2} \right)\]

\[h₂'(z) ≈ h₂'(z) + α z^{1/4} \exp(2/3 i z^{3/2} + 11πi/12) \left( 1 + \sum_{m=1} (-i)^m C_m z^{-3m/2} \right) \\ - α/4 z^{-5/4} \exp(2/3 i z^{3/2} + 11πi/12) \left( 1 + \sum_{m=1} (-i)^m C_m z^{-3m/2} \right) \\ - 3/2 α z^{-1/4} \exp(2/3 i z^{3/2} + 11πi/12) \left( \sum_{m=1} (-i)^m m C_m z^{-3m/2 - 1} \right)\]

for $0 < \arg z < 4π/3$.

The coefficient $\alpha = 2^{1/3} 3^{1/6} \pi^{-1/2}$.

References