An Extension of Laplace’s Method

Abstract

Asymptotic expansions are obtained for contour integrals of the form

$$\begin{aligned} \int _a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right) q(t)\mathrm{d}t, \end{aligned}$$

in which z is a large real or complex parameter; p(t), q(t), and r(t) are analytic functions of t; and the positive constants $\mu$ and $\nu$ are related to the local behavior of the functions p(t) and r(t) near the endpoint a. Our main theorem includes as special cases several important asymptotic methods for integrals such as those of Laplace, Watson, Erdélyi, and Olver. Asymptotic expansions similar to ours were derived earlier by Dingle using formal, nonrigorous methods. The results of the paper also serve to place Dingle’s investigations on a rigorous mathematical foundation. The new results have potential applications in the asymptotic theory of special functions in transition regions, and we illustrate this by two examples.

Appendices

Appendix A. Basic Properties of Faxén’s Integral

In this appendix, we discuss some of the basic properties of the Faxén integral defined by Eq. (4). It is assumed throughout that $0 \le \alpha < 1$ and $\mathfrak {R}(\beta ) > 0$.

By uniform convergence, we may expand the factor $\exp \left( xt^\alpha \right)$ in ascending powers of $x t^\alpha$ and integrate (4) term by term. In this way we obtain the following expansion, valid for all x:

$$\begin{aligned} {\text {Fi}}(\alpha ,\beta ;x) = \sum \limits _{n = 0}^\infty \Gamma (\alpha n + \beta )\frac{x^n}{n!} . \end{aligned}$$

This expansion is useful for computing ${\text {Fi}}(\alpha ,\beta ;x)$ when x is small or moderate in size. For large values of |x|, asymptotic expansions should be used instead. Faxén’s integral is a particular case of the so-called Wright function since ${\text {Fi}}(\alpha ,\beta ;x) = {}_1\Psi _1 \left( {\alpha ,\beta ;0,1;x} \right)$. Accordingly, the large-x asymptotic behavior of ${\text {Fi}}(\alpha ,\beta ;x)$ can be determined by employing the well-established asymptotic theory of the Wright function [17]. In particular, to leading order,

$$\begin{aligned} {\text {Fi}}(\alpha ,\beta ;x) \sim \left( {\alpha x} \right) ^{(2\beta - 1)/(2 - 2\alpha )} \exp \left( {\left( {1 - \alpha } \right) \left( {\alpha ^\alpha x} \right) ^{1/(1 - \alpha )} } \right) \sqrt{\frac{{2\pi }}{{1 - \alpha }}} \end{aligned}$$

and

$$\begin{aligned} {\text {Fi}}(\alpha ,\beta ;-x) \sim \Gamma \left( {\frac{\beta }{\alpha }} \right) \frac{1}{{\alpha x^{\beta /\alpha } }} \end{aligned}$$

as $x\rightarrow +\infty$. For further representations, we refer the reader to [4].

In the remaining part of this appendix, we show that for certain specific values of the parameters $\alpha$ and $\beta$, the function ${\text {Fi}}(\alpha ,\beta ;x)$ is expressible in terms of well-known special functions. These representations lead to simplified forms of the asymptotic expansions derived in Sect. 4. When $\alpha =\frac{1}{2}$ and $\beta$ is any complex number with positive real part, Faxén’s integral is essentially a parabolic cylinder function [14, §12.2]. Indeed, substituting $t=\frac{1}{2} s^2$, we obtain

$$\begin{aligned} {\text {Fi}}\left( \frac{1}{2},\beta ;x \right)&= \int _0^{ + \infty } \exp \left( - t + xt^{1/2} \right) t^{\beta - 1} \mathrm{d}t \nonumber \\&= 2^{1 - \beta } \int _0^{ + \infty } \exp \left( - \frac{1}{2}s^2 + \frac{x}{2^{1/2}}s \right) s^{2\beta - 1} \mathrm{d}s \nonumber \\&= 2^{1 - \beta } \Gamma (2\beta )\exp \left( \frac{x^2}{8} \right) U\left( 2\beta - \frac{1}{2}, - 2^{ - 1/2} x \right) ; \end{aligned}$$

(54)

compare [14, eq. 12.5.1]. When $\alpha =\beta = \frac{1}{3}$, the substitution $t = \frac{1}{3}s^3$ produces

$$\begin{aligned} {\text {Fi}}\left( \frac{1}{3},\frac{1}{3};x \right)&= \int _0^{ + \infty } \exp \left( - t + xt^{1/3} \right) t^{ - 2/3} \mathrm{d}t \nonumber \\&= 3^{2/3} \int _0^{ + \infty } \exp \left( - \frac{1}{3}s^3 + \frac{x}{3^{1/3}}s \right) \mathrm{d}s = 3^{2/3} \pi {\text {Hi}}( 3^{ - 1/3} x ) \end{aligned}$$

(55)

(cf. [14, eq. 9.12.20]), where ${\text {Hi}}(x)$ is Scorer’s function [14, §9.12]. The function ${\text {Hi}}(x)$ is known to satisfy the inhomogeneous Airy equation

$$\begin{aligned} {\text {Hi}}''(x) - x{\text {Hi}}(x) = \frac{1}{\pi }. \end{aligned}$$

(56)

For other special cases, see [13, p. 332].

Appendix B

In this appendix, we prove the estimate (23). With the notation of Sect. 2, we define

$$\begin{aligned} F(w,y) := v^{1 - \lambda /\mu } f(v,y/v) \end{aligned}$$

for sufficiently small |w| and $\left| \arg y\right| <\frac{\pi }{2}$. Thus, by (20), F(w, y) is an analytic function of w, and it has the power series expansion

$$\begin{aligned} F(w,y) = \sum \limits _{n = 0}^{\infty } \left( \sum \limits _{m = 0}^n f_{n,m} y^{m\nu /\mu } \right) w^n, \end{aligned}$$

valid within some circle $\left| w\right| \le \rho$, where $\rho$ is independent of y. The branch of $y^{\nu /\mu }$ has phase $\nu \arg y/\mu$. We truncate this power series after $N\ge 0$ terms and write

$$\begin{aligned} F(w,y) = \sum \limits _{n = 0}^{N - 1} \left( \sum \limits _{m = 0}^n f_{n,m} y^{m\nu /\mu } \right) w^n + w^N F_N (w,y), \end{aligned}$$

with the remainder

$$\begin{aligned} F_N (w,y) = f_N (v,y/v) \end{aligned}$$

(57)

(compare (22)). By choosing k sufficiently close to a, we can assume that $\left| p(k)-p(a)\right| ^{1/\mu }<\rho$. Then

$$\begin{aligned} \left| F_N (w,y) \right| \le \frac{\rho ^{ - N} }{2\pi }\frac{M(\rho )}{\rho - \left| w \right| },\qquad M(\rho ) = \mathop {\max }\limits _{\left| \zeta \right| = \rho } \left| F(\zeta ,y) \right| , \end{aligned}$$

(58)

for any w satisfying $w \le \left| p(k)-p(a)\right| ^{1/\mu }$ (see, for example, [7, p. 70]). It is readily seen from (13) that

$$\begin{aligned} F(\zeta ,y) = \mathcal {O}(1)\exp \left( C_k y^{\nu /\mu } \right) ,\quad \left| \zeta \right| =\rho , \end{aligned}$$

(59)

where the implied constant depends only on k and $\rho$, and $C_k$ is an assignable constant that may depend on k. Hence, from (10), (57), (58), and (59), we can infer that

$$\begin{aligned} f_N (v,z) = \mathcal {O}(1)\exp \left( C_k (zv)^{\nu /\mu } \right) , \end{aligned}$$

uniformly with respect to z and v, if v lies on the segment connecting 0 to $\kappa$.

Appendix C

In this appendix, we show that if $0\le \alpha <1$, and $\beta$ and x are arbitrary (fixed) complex numbers, then

$$\begin{aligned} \int _\zeta ^\infty \exp \left( - t + xt^\alpha \right) t^{\beta - 1} \mathrm{d}t = \mathcal {O}(1)\exp \left( - \zeta + x\zeta ^\alpha \right) \zeta ^{\beta - 1} \end{aligned}$$

(60)

uniformly in the region $\left| \arg \zeta \right| \le \frac{\pi }{2}$ as $\left| \zeta \right| \rightarrow +\infty$. To this end, we perform a change of integration variable from t to u by $t = \zeta (u + 1)$ in (60), and deform the contour of integration so that $\arg u = -\arg \zeta$ along the new contour. Thus, the left-hand side of (60) becomes

$$\begin{aligned}&\zeta ^\beta \int _0^{\infty e^{ - i\arg \zeta } } \exp \left( - \zeta \left( u + 1\right) + x\left( \zeta \left( u + 1 \right) \right) ^\alpha \right) \left( u + 1 \right) ^{\beta - 1} \mathrm{d}u \nonumber \\&\quad = \exp \left( - \zeta + x\zeta ^\alpha \right) \zeta ^\beta \int _0^{\infty e^{ - i\arg \zeta } } \exp \left( - \zeta u + x\zeta ^\alpha \left( \left( u + 1 \right) ^\alpha - 1\right) \right) \left( u + 1 \right) ^{\beta - 1} \mathrm{d}u \end{aligned}$$

(61)

for any $\zeta$ in the closed sector $\left| \arg \zeta \right| \le \frac{\pi }{2}$. Since $0\le \alpha <1$, we may assert that

$$\begin{aligned} \mathfrak {R}\left( - \zeta u + x\zeta ^\alpha \left( \left( u + 1 \right) ^\alpha - 1 \right) \right) = - \left| \zeta \right| \left| u \right| \mathfrak {R}\left( 1 - \frac{x}{\zeta ^{1 - \alpha } }\frac{\left( u + 1 \right) ^\alpha - 1}{u} \right) \le - \frac{1}{2}\left| \zeta \right| \left| u \right| \end{aligned}$$

for sufficiently large $\left| \zeta \right|$. Therefore, for large $\left| \zeta \right|$, the integral in the second line of (61) can be estimated as follows:

$$\begin{aligned}&\left| \int _0^{\infty e^{ - i\arg \zeta } } \exp \left( - \zeta u + x\zeta ^\alpha \left( \left( u + 1 \right) ^\alpha - 1 \right) \right) \left( u + 1 \right) ^{\beta - 1} \mathrm{d}u \right| \\&\quad \le \int _0^{\infty e^{ - i\arg \zeta } } e^{ - \frac{1}{2}\left| \zeta \right| \left| u \right| } \left| \left( u + 1 \right) ^{\beta - 1} \right| \left| \mathrm{d}u \right| = \int _0^{ + \infty } e^{ - \frac{1}{2}\left| \zeta \right| s} \left| \left( se^{ - i\arg \zeta } + 1 \right) ^{\beta - 1} \right| \mathrm{d}s, \end{aligned}$$

with $s=\left| u \right|$. Since the factor $\left( se^{ - i\arg \zeta } + 1 \right) ^{\beta - 1}$ grows sub-exponentially in s, the last quantity is $\mathcal {O}(\left| \zeta \right| ^{-1})$, which completes the proof of the estimate (60).