§2.3 Integrals of a Real Variable

Assume that the Laplace transform

converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. Then

If, in addition, $q(t)$ is infinitely differentiable on $[0,\mathrm{\infty })$ and

is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$.

For the Fourier integral

assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. Then

Alternatively, assume $b=\mathrm{\infty }$, $q(t)$ is infinitely differentiable on $[a,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges as $t\to \mathrm{\infty }$ uniformly for all sufficiently large $x$. Then

In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by

see §1.4(v). For other examples, see Wong (1989, Chapter 1).

Assume again that the integral (2.3.1) converges for all sufficiently large $x$, but now

where $\lambda$ and $\mu$ are positive constants. Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion:

For the function $\mathrm{\Gamma }$ see §5.2(i).

This result is probably the most frequently used method for deriving asymptotic expansions of special functions. Since $q(t)$ need not be continuous (as long as the integral converges), the case of a finite integration range is included. For an extension with more general $t$-powers see Bleistein and Handelsman (1975, §4.1).

Other types of singular behavior in the integrand can be treated in an analogous manner. For example,

provided that the integral on the left-hand side of (2.3.9) converges for all sufficiently large values of $x$. (In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.)

Another extension is to more general factors than the exponential function. In addition to (2.3.7) assume that $f(t)$ and $q(t)$ are piecewise continuous (§1.4(ii)) on $(0,\mathrm{\infty })$, and

where $A,a,b,\kappa$ are positive constants. Then

where $ℳf\left(\alpha \right)$ is the Mellin transform of $f(t)$ (§2.5(i)).

For a more detailed treatment of the integral (2.3.12) see §§2.5, 2.6.

When $p(t)$ is real and $x$ is a large positive parameter, the main contribution to the integral

derives from the neighborhood of the minimum of $p(t)$ in the integration range. Without loss of generality, we assume that this minimum is at the left endpoint $a$. Furthermore:

1. (a)

   $p^{\prime }(t)$ and $q(t)$ are continuous in a neighborhood of $a$, save possibly at $a$, and the minimum of $p(t)$ in $[a,b)$ is approached only at $a$.

2. (b)

   As $t\to a+$

   2.3.14		$p(t)$	$\sim p(a)+{\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{p}_s{(t-a)}^{s+\mu },$
   		$q(t)$	$\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{q}_s{(t-a)}^{s+\lambda -1},$
   ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $a$: left endpoint, $p(t)$: real function, $p_s$: coefficients, $q(t)$: function, $q_s$: coefficients, $\lambda$: positive constant and $\mu$: positive constant Referenced by: item (b) Permalink: http://dlmf.nist.gov/2.3.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §2.3(iii), §2.3 and Ch.2

   and the expansion for $p(t)$ is differentiable. Again $\lambda$ and $\mu$ are positive constants. Also $p_0>0$ (consistent with (a)).

3. (c)

   The integral (2.3.13) converges absolutely for all sufficiently large $x$.

Then

where the coefficients $b_s$ are defined by the expansion

in which $v=p(t)-p(a)$. For example,

In general

Watson’s lemma can be regarded as a special case of this result.

For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). These references and Wong (1989, Chapter 2) also contain examples.

When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in

oscillate rapidly and cancel themselves over most of the range. However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of $p^{\prime }(t)$ because $p(t)$ changes relatively slowly at these stationary points.

The first result is the analog of Watson’s lemma (§2.3(ii)). Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges at $t=\mathrm{\infty }$, uniformly for all sufficiently large $x$. Then

where the coefficients $a_s$ are given by (2.3.7).

For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. Furthermore:

1. (a)

   On $(a,b)$, $p(t)$ and $q(t)$ are infinitely differentiable and $p^{\prime }(t)>0$.

2. (b)

   As $t\to a+$ the asymptotic expansions (2.3.14) apply, and each is infinitely differentiable. Again $\lambda$, $\mu$, and $p_0$ are positive.

3. (c)

   If the limit $p(b)$ of $p(t)$ as $t\to b-$ is finite, then each of the functions

   2.3.21		$$P_s(t)={\left(\frac{1}{p^{\prime }(t)}\frac{d}{dt}\right)}^s\frac{q(t)}{p^{\prime }(t)},$$
   $s=0,1,2,\mathrm{\dots }$,
   ⓘ Defines: $P_s(t)$: function (locally) Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $p(t)$: real function and $q(t)$: function Permalink: http://dlmf.nist.gov/2.3.E21 Encodings: TeX, pMML, png See also: Annotations for §2.3(iv), §2.3 and Ch.2

   tends to a finite limit $P_s(b)$.

4. (d)

   If $p(b)=\mathrm{\infty }$, then $P_0(b)=0$ and each of the integrals

   2.3.22		$$\int {\mathrm{e}}^{\mathrm{i}xp(t)}{P}_s(t)p^{\prime }(t)dt,$$
   $s=0,1,2,\mathrm{\dots }$,
   ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $p(t)$: real function and $P_s(t)$: function Permalink: http://dlmf.nist.gov/2.3.E22 Encodings: TeX, pMML, png See also: Annotations for §2.3(iv), §2.3 and Ch.2

   converges at $t=b$ uniformly for all sufficiently large $x$.

If $p(b)$ is finite, then both endpoints contribute:

But if (d) applies, then the second sum is absent. The coefficients $b_s$ are defined as in §2.3(iii).

For proofs of the results of this subsection, error bounds, and an example, see Olver (1974). For other estimates of the error term see Lyness (1971). For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010).

In the integral

$k$ ($\le \mathrm{\infty }$) and $\lambda$ are positive constants, $\alpha$ is a variable parameter in an interval ${\alpha }_1\le \alpha \le {\alpha }_2$ with ${\alpha }_1\le 0$ and , and $x$ is a large positive parameter. Assume also that ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha ,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are nonzero. When $x\to +\mathrm{\infty }$ Laplace’s method (§2.3(iii)) applies, but the form of the resulting approximation is discontinuous at $\alpha =0$. In consequence, the approximation is nonuniform with respect to $\alpha$ and deteriorates severely as $\alpha \to 0$.

A uniform approximation can be constructed by quadratic change of integration variable:

where $a$ and $b$ are functions of $\alpha$ chosen in such a way that $t=0$ corresponds to $w=0$, and the stationary points $t=\alpha$ and $w=a$ correspond. Thus

the upper or lower sign being taken according as $t\gtrless \alpha$. The relationship between $t$ and $w$ is one-to-one, and because

it is free from singularity at $t=\alpha$.

The integral (2.3.24) transforms into

where

$\kappa =\kappa (\alpha )$ being the value of $w$ at $t=k$. We now expand $f(\alpha ,w)$ in a Taylor series centered at the peak value $w=a$ of the exponential factor in the integrand:

with the coefficients ${\varphi }_s(\alpha )$ continuous at $\alpha =0$. The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. We replace the limit $\kappa$ by $\mathrm{\infty }$ and integrate term-by-term:

where

For examples and proofs see Olver (1997b, Chapter 9), Bleistein (1966), Bleistein and Handelsman (1975, Chapter 9), and Wong (1989, Chapter 7).

For the asymptotics of the Mellin transform $ℳf\left(z\right)={\int }_0^{\mathrm{\infty }}{t}^{z-1}f(t)dt$ as $z\to \mathrm{\infty }$ see Frenzen (1987b), Sidi (1985, 2011).