Solutions to Stokes' differential equation.
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Modified Hankel Functions Of Order One Third and Their Derivatives

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Solutions to Stokes' differential equation:

From Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives:

Its only singularity is an irregular singularity at infinity. The equation occurs in the description of simple cases of diffraction and of refraction of waves. The general solution of [Stokes' equation] can be written in terms of Bessel functions of order one-third. The tabulation of these Bessel functions for complex arguments would make possible the computation of solutions of [Stokes' equation] for complex arguments. The direct tabulation of solutions of [Stokes' equation] should, however, be preferred to that of Bessel functions of order one-third. Unlike Bessel's equation, [Stokes' equation] has no singularity in the finite complex plane and its solutions are single-valued, whereas the Bessel functions of order one-third are not.


using ModifiedHankelFunctionsOfOrderOneThird

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

The functions h₁ and h₂

An independent pair of solutions, valid for all values of , is


where .

The contours of integration and are


with . We take .


Two solution approaches are used. 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

Stokes' equation may be solved in a power series of , valid in the entire complex plane,


Asymptotic expansion

The asymptotic expansions can be used to estimate , , and their derivatives, although in general with less accuracy than the power series. Two expansions are required depending on the value of arg z. The existence of two expressions of different forms which represent asymptotically the same integral function is an example of Stokes' phenomenon.

The expansion for for is


See the source for the full sets of solutions.


The 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.


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