§2.3 Integrals of a Real Variable

Permalink:: http://dlmf.nist.gov/2.3

§2.3(i) Integration by Parts
§2.3(ii) Watson’s Lemma
§2.3(iii) Laplace’s Method
§2.3(iv) Method of Stationary Phase
§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method

§2.3(i) Integration by Parts

Notes:: See Olver (1997b, pp. 66–70, 75–76).
Keywords:: Fourier integral, Laplace transform, asymptotic approximations of integrals, error term, variational operator
Referenced by:: §10.40(iii), §2.10(ii), §2.11(i), ¶ ‣ §2.7(iii)
Permalink:: http://dlmf.nist.gov/2.3.i

Assume that the Laplace transform

2.3.1 $\int _{{0}}^{{\infty}}e^{{-xt}}q(t)dt$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $q(t)$ : infinitely differentiable function
Referenced by:: §2.3(ii)
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converges for all sufficiently large $x$ , and $q(t)$ is infinitely differentiable in a neighborhood of the origin. Then

2.3.2 $\int _{{0}}^{{\infty}}e^{{-xt}}q(t)dt\sim\sum _{{s=0}}^{{\infty}}\frac{q^{{(s)}}(0)}{x^{{s+1}}},$ $x\to+\infty$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\sim$ : asymptotic equality and $q(t)$ : infinitely differentiable function
Referenced by:: §2.3(i)
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If, in addition, $q(t)$ is infinitely differentiable on $[0,\infty)$ and

2.3.3 $\sigma _{n}=\sup _{{(0,\infty)}}(t^{{-1}}\mathop{\ln\/}\nolimits|q^{{(n)}}(t)/q^{{(n)}}(0)|)$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b)$ : open interval, $\sup$ : least upper bound (supremum), $q(t)$ : infinitely differentiable function, $\sigma _{n}$ and $n$ : nonnegative integer
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is finite and bounded for $n=0,1,2,\dots$ , 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

$\int _{a}^{b}e^{{ixt}}q(t)dt$

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

2.3.4 $\int _{a}^{b}e^{{ixt}}q(t)dt\sim e^{{iax}}\sum _{{s=0}}^{{\infty}}q^{{(s)}}(a)\left(\frac{i}{x}\right)^{{s+1}}-e^{{ibx}}\sum _{{s=0}}^{{\infty}}q^{{(s)}}(b)\left(\frac{i}{x}\right)^{{s+1}},$ $x\to+\infty$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\sim$ : asymptotic equality, $q(t)$ : infinitely differentiable function, $a$ : left endpoint and $b$ : right endpoint
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Alternatively, assume $b=\infty$ , $q(t)$ is infinitely differentiable on $[a,\infty)$ , and each of the integrals $\int e^{{ixt}}q^{{(s)}}(t)dt$ , $s=0,1,2,\dots$ , converges as $t\to\infty$ uniformly for all sufficiently large $x$ . Then

2.3.5 $\int _{{a}}^{{\infty}}e^{{ixt}}q(t)dt\sim e^{{iax}}\sum _{{s=0}}^{{\infty}}q^{{(s)}}(a)\left(\frac{i}{x}\right)^{{s+1}},$ $x\to+\infty$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\sim$ : asymptotic equality, $q(t)$ : infinitely differentiable function and $a$ : left endpoint
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In both cases the $n$ th error term is bounded in absolute value by $x^{{-n}}\mathop{\mathcal{V}_{{a,b}}\/}\nolimits\!\left(q^{{(n-1)}}(t)\right)$ , where the variational operator $\mathop{\mathcal{V}_{{a,b}}\/}\nolimits$ is defined by

2.3.6 $\mathop{\mathcal{V}_{{a,b}}\/}\nolimits\!\left(f(t)\right)=\int _{a}^{b}|f^{{\prime}}(t)dt|;$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathcal{V}_{{a,b}}\/}\nolimits\!\left(f\right)$ : total variation, $a$ : left endpoint and $b$ : right endpoint
Referenced by:: §10.17(iv)
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see §1.4(v). For other examples, see Wong (1989, Chapter 1).

§2.3(ii) Watson’s Lemma

Notes:: See Olver (1997b, pp. 71–72). For (2.3.9) see Wong (1989, §2.2). For (2.3.12) use termwise integration in an analogous manner to that used to prove Watson’s lemma Olver (1997b, pp. 71–72).
Keywords:: Laplace transform, Watson’s lemma, asymptotic approximations of integrals
Referenced by:: §2.3(iv), §2.4(i), §2.6(i)
Permalink:: http://dlmf.nist.gov/2.3.ii

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

2.3.7 $q(t)\sim\sum _{{s=0}}^{{\infty}}a_{s}t^{{(s+\lambda-\mu)/\mu}},$ $t\to 0+$ ,

Symbols:: $\sim$ : asymptotic equality, $a$ : positive constant, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
Referenced by:: §2.3(ii), §2.3(ii), §2.3(iv), §2.4(i), §2.4(ii)
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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:

2.3.8 $\int _{{0}}^{{\infty}}e^{{-xt}}q(t)dt\sim\sum _{{s=0}}^{{\infty}}\mathop{\Gamma\/}\nolimits\!\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{{(s+\lambda)/\mu}}},$ $x\to+\infty$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\sim$ : asymptotic equality, $a$ : positive constant, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
Referenced by:: §2.3(ii), §2.4(i)
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For the function $\mathop{\Gamma\/}\nolimits$ 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.

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

2.3.9 $\int _{{0}}^{{\infty}}e^{{-xt}}q(t)\mathop{\ln\/}\nolimits tdt\sim\sum _{{s=0}}^{{\infty}}{\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{{(s+\lambda)/\mu}}}-(\mathop{\ln\/}\nolimits x)\sum _{{s=0}}^{{\infty}}\mathop{\Gamma\/}\nolimits\!\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{{(s+\lambda)/\mu}}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $a$ : positive constant, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
Referenced by:: §2.3(ii), §2.3(ii), §9.12(viii)
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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 $(0,\infty)$ , and

2.3.10 $|f(t)|\leq A\mathop{\exp\/}\nolimits\!\left(-at^{{\kappa}}\right),$ $0\leq t<\infty$ ,

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $f(x)$ : function, $A$ : positive constant, $a$ : positive constant and $\kappa$ : positive constant
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2.3.11 $q(t)=\mathop{O\/}\nolimits\!\left(\mathop{\exp\/}\nolimits\!\left(bt^{{\kappa}}\right)\right),$ $t\to+\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\exp\/}\nolimits z$ : exponential function, $b$ : positive constant, $\kappa$ : positive constant and $q(t)$ : function
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where $A,a,b,\kappa$ are positive constants. Then

2.3.12 $\int _{{0}}^{{\infty}}f(xt)q(t)dt\sim\sum _{{s=0}}^{{\infty}}\mathop{\mathscr{M}\/}\nolimits\left(f;\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{{(s+\lambda)/\mu}}},$ $x\to+\infty$ ,

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $\sim$ : asymptotic equality, $f(x)$ : function, $a$ : positive constant, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
Referenced by:: §2.3(ii), §2.3(ii)
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where $\mathop{\mathscr{M}\/}\nolimits\left(f;\alpha\right)$ 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.

§2.3(iii) Laplace’s Method

Notes:: See Olver (1997b, pp. 80–88). (2.3.18) follows from (1.10.15) and (1.10.17) by setting $f(t)=p(t)$ , $g(t)=\ifrac{q(t)}{\left(p^{{\prime}}(t)(p(t)-p(a))^{{(\lambda/\mu)-1}}\right)}$ , using Cauchy’s integral formula (1.9.30) for the residue, and integrating by parts. See also Cicuta and Montaldi (1975).
Keywords:: Laplace’s method for asymptotic expansions of integrals, asymptotic approximations of integrals
Referenced by:: ¶ ‣ §2.10(iii), §2.3(iv), §2.3(v), §2.4(iii), §2.4(iv), §35.7(iv), §7.19(iii)
Permalink:: http://dlmf.nist.gov/2.3.iii

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

2.3.13 $I(x)=\int _{a}^{b}e^{{-xp(t)}}q(t)dt$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $a$ : left endpoint, $b$ : right endpoint, $p(t)$ : real function, $q(t)$ : function and $I(x)$ : integral
Referenced by:: §2.3(iii)
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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:

(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 $a$ .
(b)

As $t\to a+$

2.3.14
$p(t)\sim p(a)+\sum _{{s=0}}^{{\infty}}p_{s}(t-a)^{{s+\mu}},$

$q(t)\sim\sum _{{s=0}}^{{\infty}}q_{s}(t-a)^{{s+\lambda-1}},$

Symbols:

$\sim$ : asymptotic equality, $a$ : left endpoint, $p(t)$ : real function, $p_{s}$ : coefficients, $q(t)$ : function, $q_{s}$ : coefficients, $\lambda$ : positive constant and $\mu$ : positive constant

Referenced by:

§2.3(iv)

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and the expansion for $p(t)$ is differentiable. Again $\lambda$ and $\mu$ are positive constants. Also $p_{0}>0$ (consistent with (a)).
(c)
The integral (2.3.13) converges absolutely for all sufficiently large $x$ .

Then

2.3.15 $\int _{a}^{b}e^{{-xp(t)}}q(t)dt\sim e^{{-xp(a)}}\sum _{{s=0}}^{{\infty}}\mathop{\Gamma\/}\nolimits\!\left(\frac{s+\lambda}{\mu}\right)\frac{b_{s}}{x^{{(s+\lambda)/\mu}}},$ $x\to+\infty$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\sim$ : asymptotic equality, $a$ : left endpoint, $b$ : right endpoint, $p(t)$ : real function, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
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where the coefficients $b_{s}$ are defined by the expansion

2.3.16 $\frac{q(t)}{p^{{\prime}}(t)}\sim\sum _{{s=0}}^{{\infty}}b_{s}v^{{(s+\lambda-\mu)/\mu}},$ $v\to 0+$ ,

Symbols:: $\sim$ : asymptotic equality, $b$ : right endpoint, $p(t)$ : real function, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
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in which $v=p(t)-p(a)$ . For example,

2.3.17

$b_{0}=\frac{q_{0}}{\mu p_{0}^{{\lambda/\mu}}},$

$b_{1}=\left(\frac{q_{1}}{\mu}-\frac{(\lambda+1)p_{1}q_{0}}{\mu^{2}p_{0}}\right)\frac{1}{p_{0}^{{(\lambda+1)/\mu}}},$

$b_{2}=\left(\frac{q_{2}}{\mu}-\frac{(\lambda+2)(p_{1}q_{1}+p_{2}q_{0})}{\mu^{2}p_{0}}+\frac{(\lambda+2)(\lambda+\mu+2)p_{1}^{2}q_{0}}{2\mu^{3}p_{0}^{2}}\right)\frac{1}{p_{0}^{{(\lambda+2)/\mu}}}.$

Symbols:: $b$ : right endpoint, $p_{s}$ : coefficients, $q_{s}$ : coefficients, $\lambda$ : positive constant and $\mu$ : positive constant
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In general

2.3.18 $b_{s}=\frac{1}{\mu}\Residue _{{t=a}}\left[\frac{q(t)}{(p(t)-p(a))^{{(\lambda+s)/\mu}}}\right],$ $s=0,1,2,\dots$ .

Symbols:: $\Residue$ : residue, $a$ : left endpoint, $b$ : right endpoint, $p(t)$ : real function, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
Referenced by:: §2.3(iii), §2.4(iv)
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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.

§2.3(iv) Method of Stationary Phase

Keywords:: Fourier integral, asymptotic approximations of integrals, method of stationary phase
Referenced by:: §36.11, §36.7(ii)
Permalink:: http://dlmf.nist.gov/2.3.iv

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

2.3.19 $I(x)=\int _{a}^{b}e^{{ixp(t)}}q(t)dt$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $p(t)$ : real function, $a$ : left endpoint, $b$ : right endpoint, $I(x)$ : integral and $q(t)$ : function
Referenced by:: §2.3(iv)
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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 $(0,\infty)$ , and each of the integrals $\int e^{{ixt}}q^{{(s)}}(t)dt$ , $s=0,1,2,\dots$ , converges at $t=\infty$ , uniformly for all sufficiently large $x$ . Then

2.3.20 $\int _{{0}}^{{\infty}}e^{{ixt}}q(t)dt\sim\sum _{{s=0}}^{{\infty}}\mathop{\exp\/}\nolimits\!\left(\frac{(s+\lambda)\pi i}{2\mu}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{{(s+\lambda)/\mu}}},$ $x\to+\infty$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $\int$ : integral, $\sim$ : asymptotic equality, $q(t)$ : function, $a_{s}$ : coefficients, $\lambda$ : positive constant and $\mu$ : positive constant
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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:

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

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

2.3.21 $P_{s}(t)=\left(\frac{1}{p^{{\prime}}(t)}\frac{d}{dt}\right)^{s}\frac{q(t)}{p^{{\prime}}(t)},$ $s=0,1,2,\dots$ ,

Symbols:

$\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $p(t)$ : real function, $P_{s}(t)$ : function and $q(t)$ : function

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tends to a finite limit $P_{s}(b)$ .
(d)

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

2.3.22 $\int e^{{ixp(t)}}P_{s}(t)p^{{\prime}}(t)dt,$ $s=0,1,2,\dots$ ,

Symbols:

$dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $p(t)$ : real function and $P_{s}(t)$ : function

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

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

2.3.23 $\int _{a}^{b}e^{{ixp(t)}}q(t)dt\sim e^{{ixp(a)}}\sum _{{s=0}}^{{\infty}}\mathop{\exp\/}\nolimits\!\left(\frac{(s+\lambda)\pi i}{2\mu}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{s+\lambda}{\mu}\right)\frac{b_{s}}{x^{{(s+\lambda)/\mu}}}-e^{{ixp(b)}}\sum _{{s=0}}^{{\infty}}P_{s}(b)\left(\frac{i}{x}\right)^{{s+1}},$ $x\to+\infty$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $\int$ : integral, $\sim$ : asymptotic equality, $p(t)$ : real function, $a$ : left endpoint, $b$ : right endpoint, $P_{s}(t)$ : function, $q(t)$ : function, $\lambda$ : positive constant and $\mu$ : positive constant
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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 logarithmic singularities see Wong and Lin (1978).

§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method

Keywords:: asymptotic approximations of integrals
Referenced by:: §2.4(v), §2.4(vi), §2.8(vi)
Permalink:: http://dlmf.nist.gov/2.3.v

In the integral

2.3.24 $I(\alpha,x)=\int _{0}^{k}e^{{-xp(\alpha,t)}}q(\alpha,t)t^{{\lambda-1}}dt$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $I(\alpha,x)$ : integral, $p(\alpha,t)$ : function, $q(\alpha,t)$ : function and $\lambda$ : positive constant
Referenced by:: §2.3(v)
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$k$ ( $\leq\infty$ ) and $\lambda$ are positive constants, $\alpha$ is a variable parameter in an interval $\alpha _{1}\leq\alpha\leq\alpha _{2}$ with $\alpha _{1}\leq 0$ and $0<\alpha _{2}\leq k$ , and $x$ is a large positive parameter. Assume also that $\ifrac{{\partial}^{2}p(\alpha,t)}{{\partial t}^{2}}$ 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 $\ifrac{\partial p(\alpha,t)}{\partial t}$ vanishes, but both $\ifrac{{\partial}^{2}p(\alpha,t)}{{\partial t}^{2}}$ and $q(\alpha,t)$ are nonzero. When $x\to+\infty$ 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:

2.3.25 $p(\alpha,t)=\tfrac{1}{2}w^{2}-aw+b,$

Symbols:: $p(\alpha,t)$ : function, $w$ : change of variable, $a(\alpha)$ and $b(\alpha)$
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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

2.3.26

$a=(2p(\alpha,0)-2p(\alpha,\alpha))^{{1/2}},$

$b=p(\alpha,0),$

Symbols:: $p(\alpha,t)$ : function, $a(\alpha)$ and $b(\alpha)$
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2.3.27 $w=(2p(\alpha,0)-2p(\alpha,\alpha))^{{1/2}}\pm(2p(\alpha,t)-2p(\alpha,\alpha))^{{1/2}},$

Symbols:: $p(\alpha,t)$ : function and $w$ : change of variable
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the upper or lower sign being taken according as $t\gtrless\alpha$ . The relationship between $t$ and $w$ is one-to-one, and because

2.3.28 $\frac{dw}{dt}=\pm\frac{1}{(2p(\alpha,t)-2p(\alpha,\alpha))^{{1/2}}}\frac{\partial p(\alpha,t)}{\partial t}$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $p(\alpha,t)$ : function and $w$ : change of variable
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it is free from singularity at $t=\alpha$ .

The integral (2.3.24) transforms into

2.3.29 $I(\alpha,x)=e^{{-xp(\alpha,0)}}\*\int _{{0}}^{{\kappa}}\mathop{\exp\/}\nolimits\!\left(-x\left(\tfrac{1}{2}w^{2}-aw\right)\right)f(\alpha,w)w^{{\lambda-1}}dw,$

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $\int$ : integral, $I(\alpha,x)$ : integral, $p(\alpha,t)$ : function, $w$ : change of variable, $a(\alpha)$ , $f(\alpha,w)$ : function, $\kappa=w(k)$ and $\lambda$ : positive constant
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where

2.3.30 $f(\alpha,w)=q(\alpha,t)\left(\frac{t}{w}\right)^{{\lambda-1}}\frac{dt}{dw},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $q(\alpha,t)$ : function, $w$ : change of variable, $f(\alpha,w)$ : function and $\lambda$ : positive constant
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$\kappa=\kappa(\alpha)$ 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:

2.3.31 $f(\alpha,w)=\sum _{{s=0}}^{{\infty}}\phi _{s}(\alpha)(w-a)^{s},$

Symbols:: $w$ : change of variable, $a(\alpha)$ , $f(\alpha,w)$ : function and $\phi _{s}(\alpha)$ : coefficients
Permalink:: http://dlmf.nist.gov/2.3.E31
Encodings:: TeX, pMML, png

with the coefficients $\phi _{s}(\alpha)$ 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 $\infty$ and integrate term-by-term:

2.3.32 $I(\alpha,x)\sim\frac{e^{{-xp(\alpha,0)}}}{x^{{\lambda/2}}}\sum _{{s=0}}^{{\infty}}\phi _{s}(\alpha)\frac{F_{s}(a\sqrt{x})}{x^{{s/2}}},$ $x\to\infty$ ,

Symbols:: $e$ : base of exponential function, $\sim$ : asymptotic equality, $I(\alpha,x)$ : integral, $p(\alpha,t)$ : function, $a(\alpha)$ , $\phi _{s}(\alpha)$ : coefficients, $F_{s}(y)$ : function and $\lambda$ : positive constant
Permalink:: http://dlmf.nist.gov/2.3.E32
Encodings:: TeX, pMML, png