Computation of Asymptotic Expansions of Turning Point Problems via Cauchy’s Integral Formula: Bessel Functions

Abstract

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 a 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.

Appendix: Exponential-Form Expansions for Airy Functions

First, the Airy functions $\mathrm {Ai}_{j}\left( u^{2/3}\zeta \right)$ satisfy

$$\begin{aligned} d^{2}w/d\zeta ^{2}=u^{2}\zeta w. \end{aligned}$$

(A.1)

Letting $\xi =\int \zeta ^{1/2}d\zeta =\frac{2}{3}\zeta ^{3/2}$ (as in (2.1)), we then have that the functions $V=\zeta ^{1/4}\mathrm {Ai}_{j}\left( u^{2/3}\zeta \right)$ satisfy

$$\begin{aligned} \frac{d^{2}V}{d\xi ^{2}}=\left\{ u^{2}-\frac{5}{36\xi ^{2}}\right\} V. \end{aligned}$$

An asymptotic solution of the form (2.4) now applies. In particular, on identifying solutions of (A.1) that are recessive at $\zeta =+\infty$, we have that there exists a constant c(u) such that

$$\begin{aligned} \mathrm {Ai}\left( u^{2/3}\zeta \right) \sim c(u) \zeta ^{-1/4} \exp \left\{ -u\xi +\displaystyle \sum _{s=1}^{\infty } (-1)^{s} \frac{e_{s}(\xi )}{u^{s}}\right\} \end{aligned}$$

as $u^{2/3}\zeta \rightarrow \infty$ in the sector $\left| \arg \left( \zeta \right) \right| \le \pi -\delta$ ($\delta >0)$.

The coefficients in this expansion are given by (2.5) - (2.7), with E and F replaced by e and f, respectively, and $\phi (\xi )=-\frac{5}{{36}}\xi ^{-2}$. In particular, $e_{s}(\xi )=\int _{\infty }^{\xi }f_{s}(t)dt$, where

$$\begin{aligned} f_{1}(\xi )=-\frac{5}{72\xi ^{2}},\quad f_{2}(\xi )=-\frac{5}{72\xi ^{3}} \end{aligned}$$

and

$$\begin{aligned} f_{s+1}(\xi )=-\frac{1}{2}f_{s}^{\prime }(\xi )-\frac{1}{2} \displaystyle \sum _{j=1}^{s-1}f_{j}(\xi )f_{s-j}(\xi )\quad (s\ge 2). \end{aligned}$$

Thus

$$\begin{aligned} f_{s}(\xi ) =-\frac{a_{s}}{\xi ^{s+1}},\quad e_{s}(\xi ) =\frac{a_{s}}{s\xi ^{s}} \quad (s=1,2,3,\ldots ) , \end{aligned}$$

where $a_{1}=a_{2}=\frac{5}{72}$, and

$$\begin{aligned} a_{s+1}=\frac{1}{2}(s+1) a_{s} +\frac{1}{2} \displaystyle \sum _{j=1}^{s-1}a_{j}a_{s-j}\quad (s\ge 2) . \end{aligned}$$

(A.2)

From the well-known leading term

$$\begin{aligned} \mathrm {Ai}\left( u^{2/3}\zeta \right) \sim \frac{e^{-u\xi }}{2\pi ^{1/2}u^{1/6}\zeta ^{1/4}} \quad \left( u^{2/3}\zeta \rightarrow \infty \right) , \end{aligned}$$

we obtain $c(u) =1/\left( 2\pi ^{1/2}u^{1/6}\right)$, and hence we deduce that

$$\begin{aligned} \mathrm {Ai}\left( u^{2/3}\zeta \right) \sim \frac{1}{2\pi ^{1/2}u^{1/6}\zeta ^{1/4}} \exp \left\{ -u\xi +\displaystyle \sum _{s=1}^{\infty }(-1)^s \frac{a_{s}}{su^{s}\xi ^{s}}\right\} , \end{aligned}$$

(A.3)

which is uniformly valid for $\left| \arg \left( \zeta \right) \right| \le \pi -\delta$ ($\delta >0$). Expansions for $\mathrm {Ai}_{\pm 1}\left( u^{2/3}\zeta \right)$ can be obtained directly from this.

Next, from differentiating (A.1), we find that $y=\zeta ^{-1/2}\mathrm {Ai}_{j}^{\prime } \left( u^{2/3}\zeta \right)$ satisfy

$$\begin{aligned} \frac{d^{2}y}{d\zeta ^{2}}=\left\{ u^{2}\zeta +\frac{3}{4\zeta ^{2}}\right\} y. \end{aligned}$$

Thus, again with $\xi =\frac{2}{3}\zeta ^{3/2}$, we have that ${\tilde{V}}=\zeta ^{-1/4}\mathrm {Ai}_{j}^{\prime }\left( u^{2/3}\zeta \right)$ satisfy

$$\begin{aligned} \frac{d^{2}{\tilde{V}}}{d\xi ^{2}}=\left\{ u^{2}+\frac{7}{36\xi ^{2}}\right\} {\tilde{V}}. \end{aligned}$$

Similarly to (A.3), on using [10, (9.7.6)], we deduce that

$$\begin{aligned} \mathrm {Ai}^{\prime }\left( u^{2/3}\zeta \right) \sim -\frac{u^{1/6}\zeta ^{1/4}}{2\pi ^{1/2}} \exp \left\{ -u\xi +\displaystyle \sum _{s=1}^{\infty }(-1)^{s} \frac{{\tilde{a}}_{s}}{su^{s}\xi ^{s}}\right\} \end{aligned}$$

(A.4)

for $\left| \arg (\zeta )\right| \le \pi -\delta$, where ${\tilde{a}}_{1}={\tilde{a}}_{2}=-\frac{7}{72}$, and subsequent terms also satisfy (A.2). Expansions for $\mathrm {Ai}_{\pm 1}^{\prime }\left( u^{2/3}\zeta \right)$ can be obtained directly from this.