Computation of asymptotic expansions of turning point problems via Cauchy's integral formula: Bessel functions

Linear second order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy's integral formula is employed to compute the coefficient functions to high order of accuracy. The method employs a certain exponential form of Liouville-Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.


Introduction
In this talk we are dealing with asymptotic expansions for large u for the solutions of second order differential equations d 2 w dz 2 = (u 2 f (z) + g (z))w , in two cases: 1 In regions where f (z) does not have a zero (type I), 2 Around a simple zero of f (z) (type II), to which we add a third intermediate case.
The first type of expansions are also called Liouville-Green expansions and the second one uniform Airy-type expansions. We will see how to link these two separates cases through the intermediate case of LG Airy-type expansions.
We will show applications to the computation of Bessel functions and Laguerre polynomials in the complex plane.

Introduction
We say that an expansion of the form h(u) ∼ ∞ n=0 a n u −n , is an asymptotic expansion as u → ∞, if u N h(u) − N−1 n=0 a n u −n , N = 0, 1, 2, . . . , is a bounded function for large values of u, with limit a N as u → ∞, for any N. This can also be written as

Liouville-Green expansions
We start with the expansions of type I (Liouville-Green) for solutions of d 2 w dz 2 = (u 2 f (z) + g (z))w for large u, and z away from turning points (zeros of f (z)).

Change of variable:
dξ dz

This gives formal solutions
with coefficients that can be determined recursively by Symbolic differentiation is needed. Nested integration occurs. An alternative: expansion in exponential form.
For an approximation around the turning point, we set The natural approximants are Airy functions, solutions of y (x) = xy (x), and more specifically where cs and as are easily computable positive rational coefficients.
The coefficients satisfy (see Olver's book, chap. 11): Not easy! Instead of closed form expressions, normally one considers some approximations, like series around the turning point. Alternative: compute directly A(u, z) and B(u, z) by some other means.
The approach here is different. We start from Solve for the coefficients (using the Wronskian relation for Airy functions): We have A(u, z) and B(u, z). Problem: precisely we want to compute W j (u, z)!! But we know how to compute away from the turning point: use LG expansions.
Javier Segura (Universidad de Cantabria) Airy expansions via Cauchy's formula OPSFA14, 2017 10 / 23 Airy-type expansions around the turning points Using LG asymptotics, we can write: and B(u, z) ∼ 1 The coefficients Es and the constants as are those appearing before. The constantsãs correspond to the LG asymptotics for the derivative of the Airy function.

Theorem
For the differential equation assume u is positive and large, f (z) has a simple zero at z = z0, and f (z) and g (z) are analytic in a domain D containing z0. Further assume that f (z) does not vanish in the disk D (z0, ρ) := {z : 0 < |z − z0| < ρ} ⊂ D. Define variables ξ and ζ by and let Ai j (u 2/3 ζ) (j = 0, ±1) denote the Airy functions Ai(u 2/3 ζe −2πij/3 ). Then there exist three numerically satisfactory solutions of the differential equation, given by In these, the coefficient functions A(u, z) and B(u, z) are analytic at z = z0, and possess the previous asymptotic expansions in a domain that includes D (z0, ρ). The integration constants for the odd coefficients must be selected so that ζ 1/2 E 2j+1 (ξ) (j = 0, 1, 2, · · · ) is meromorphic as a function of ζ at ζ = 0.
We have LG expansions for the coefficients A(u, z), B(u, z), with which we can approximate But we can not compute close to the turning point with the LG expansions.
By now, we just have a LG Airy-type expansion (valid all around the TP, but not too close But because the coefficients A(u, z) and B(u, z) are analytic in a domain containing the turning point we can consider the Cauchy integral formula with A(u, t) and B(u, t) approximated by LG-asymptotics along the Cauchy contour.
For the numerical integration the trapezoidal rule gives an exponential convergence rate.

Bessel functions
With the transformations and coefficients shown previously for the Bessel equation, the computation of the coefficients is straightforward.
We show an example of application of the Cauchy integral.
Error in the computation of H (1) ν (νz) for z = 1 + 0.1i and various selections of the number of coefficients. The minimal possible error (close to 10 −26 ) is due to the discretization of the Cauchy integral with N = 500 points.
The trapezoidal rule has exponential convergence and the smallest reachable error roughly depends on the number of points over the Cauchy contour as 10 −N/20 . Laguerre polynomials w = z (α+1)/2 e −uz/2 L (α) n (uz), u = n + 1/2 satisfies The TPs coalesce when a = 0 (α = −(n + 1/2)). z1 coalesces with z = 0 when a = 1 (α = 0). We have considered two types of expansions: We compute expansions around z2 for As before, we should consider: 1 The computation of the LG expansions (away from z2). The transformation is the same, but with a different f (z) (and therefore a different new variable). This provides the coefficients for the next step as well as LG expansions for L (α) n (z) 2 The computation of the analytical (and slowly varying coefficients) A(u, z) and B(u, z) from the previous LG expansions. The expression for the coefficients is the same as before, but with different LG coefficients. This gives the LG Airy-type expansion away from the turning point.
3 The computation of the Airy-type expansion close to the turning point with Cauchy integrals. We don't give all the details of the expansions.
The LG expansion (with matching at +∞) is: The LG Airy-type expansion away from the turning point L (α) n (uz) = H(u, z) Ai(u 2/3 ζ)A(u, z) + Ai (u 2/3 ζ)B(u, z) with H(u, z) a known (lengthy but simple) function and A(u, z) and B(u, z) very similar to the Bessel case (but with different LG coefficients Es ).
And around the turning point Cauchy integrals are used.
Let us illustrate the use of these expansions with some numerical results.