# §2.3 Integrals of a Real Variable

## §2.3(i) Integration by Parts

Assume that the Laplace transform

 2.3.1 $\int_{0}^{\infty}e^{-xt}q(t)\,\mathrm{d}t$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function Keywords: Laplace transform Referenced by: §2.3(ii) Permalink: http://dlmf.nist.gov/2.3.E1 Encodings: TeX, pMML, png See also: Annotations for §2.3(i), §2.3 and Ch.2

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

 2.3.2 $\int_{0}^{\infty}e^{-xt}q(t)\,\mathrm{d}t\sim\sum_{s=0}^{\infty}\frac{q^{(s)}(% 0)}{x^{s+1}},$ $x\to+\infty$. ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function Keywords: Laplace transform Referenced by: §2.3(i) Permalink: http://dlmf.nist.gov/2.3.E2 Encodings: TeX, pMML, png See also: Annotations for §2.3(i), §2.3 and Ch.2

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

 2.3.3 $\sigma_{n}=\sup_{(0,\infty)}(t^{-1}\ln|q^{(n)}(t)/q^{(n)}(0)|)$ ⓘ Defines: $\sigma_{n}$ (locally) Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $(\NVar{a},\NVar{b})$: open interval, $\sup$: least upper bound (supremum), $q(t)$: infinitely differentiable function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/2.3.E3 Encodings: TeX, pMML, png See also: Annotations for §2.3(i), §2.3 and Ch.2

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

 $\int_{a}^{b}e^{ixt}q(t)\,\mathrm{d}t$

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

 2.3.4 $\int_{a}^{b}e^{ixt}q(t)\,\mathrm{d}t\sim e^{iax}\sum_{s=0}^{\infty}q^{(s)}(a)% \left(\frac{i}{x}\right)^{s+1}-e^{ibx}\sum_{s=0}^{\infty}q^{(s)}(b)\left(\frac% {i}{x}\right)^{s+1},$ $x\to+\infty$.

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

 2.3.5 $\int_{a}^{\infty}e^{ixt}q(t)\,\mathrm{d}t\sim e^{iax}\sum_{s=0}^{\infty}q^{(s)% }(a)\left(\frac{i}{x}\right)^{s+1},$ $x\to+\infty$.

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

 2.3.6 $\mathcal{V}_{a,b}\left(f(t)\right)=\int_{a}^{b}\left|f^{\prime}(t)\right|\,% \mathrm{d}t;$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$: total variation, $a$: left endpoint and $b$: right endpoint Referenced by: §10.17(iv), Erratum (V1.1.1) for Equation (2.3.6) Permalink: http://dlmf.nist.gov/2.3.E6 Encodings: TeX, pMML, png Correction (effective with 1.1.1): The integrand was corrected so that the absolute value does not include the differential. Suggested 2021-02-08 by Juan Luis Varona See also: Annotations for §2.3(i), §2.3 and Ch.2

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

## §2.3(ii) Watson’s Lemma

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

 2.3.7 $q(t)\sim\sum_{s=0}^{\infty}a_{s}t^{(s+\lambda-\mu)/\mu},$ $t\to 0+$, ⓘ Defines: $q(t)$: function (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $a$: positive constant, $\lambda$: positive constant and $\mu$: positive constant Referenced by: §2.3(ii), §2.3(ii), §2.3(iv), §2.4(i), §2.4(ii) Permalink: http://dlmf.nist.gov/2.3.E7 Encodings: TeX, pMML, png See also: Annotations for §2.3(ii), §2.3 and Ch.2

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 $\int_{0}^{\infty}e^{-xt}q(t)\,\mathrm{d}t\sim\sum_{s=0}^{\infty}\Gamma\left(% \frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+\lambda)/\mu}},$ $x\to+\infty$.

For the function $\Gamma$ 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,

 2.3.9 $\int_{0}^{\infty}e^{-xt}q(t)\ln t\,\mathrm{d}t\sim\sum_{s=0}^{\infty}\Gamma'% \left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+\lambda)/\mu}}-(\ln x)% \sum_{s=0}^{\infty}\Gamma\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+% \lambda)/\mu}},$

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 $|f(t)|\leq A\exp\left(-at^{\kappa}\right),$ $0\leq t<\infty$,
 2.3.11 $q(t)=O\left(\exp\left(bt^{\kappa}\right)\right),$ $t\to+\infty$,

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

 2.3.12 $\int_{0}^{\infty}f(xt)q(t)\,\mathrm{d}t\sim\sum_{s=0}^{\infty}\mathscr{M}% \mskip-3.0muf\mskip 3.0mu\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+% \lambda)/\mu}},$ $x\to+\infty$, ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\sim$: Poincaré asymptotic expansion, $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $f(x)$: function, $a$: positive constant, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant Referenced by: §2.3(ii), §2.3(ii) Permalink: http://dlmf.nist.gov/2.3.E12 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §2.3(ii), §2.3 and Ch.2

where $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(\alpha\right)$ 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.

## §2.3(iii) Laplace’s Method

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

 2.3.13 $I(x)=\int_{a}^{b}e^{-xp(t)}q(t)\,\mathrm{d}t$ ⓘ Defines: $I(x)$: integral (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: real function and $q(t)$: function Referenced by: item (c) Permalink: http://dlmf.nist.gov/2.3.E13 Encodings: TeX, pMML, png See also: Annotations for §2.3(iii), §2.3 and Ch.2

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 $\displaystyle p(t)$ $\displaystyle\sim p(a)+\sum_{s=0}^{\infty}p_{s}(t-a)^{s+\mu},$ $\displaystyle q(t)$ $\displaystyle\sim\sum_{s=0}^{\infty}q_{s}(t-a)^{s+\lambda-1},$

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

 2.3.15 $\int_{a}^{b}e^{-xp(t)}q(t)\,\mathrm{d}t\sim e^{-xp(a)}\sum_{s=0}^{\infty}% \Gamma\left(\frac{s+\lambda}{\mu}\right)\frac{b_{s}}{x^{(s+\lambda)/\mu}},$ $x\to+\infty$,

where the coefficients $b_{s}$ are defined by the expansion

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

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

 2.3.17 $\displaystyle b_{0}$ $\displaystyle=\frac{q_{0}}{\mu p_{0}^{\lambda/\mu}},$ $\displaystyle b_{1}$ $\displaystyle=\left(\frac{q_{1}}{\mu}-\frac{(\lambda+1)p_{1}q_{0}}{\mu^{2}p_{0% }}\right)\frac{1}{p_{0}^{(\lambda+1)/\mu}},$ $\displaystyle b_{2}$ $\displaystyle=\left(\frac{q_{2}}{\mu}-\frac{(\lambda+2)(p_{1}q_{1}+p_{2}q_{0})% }{\mu^{2}p_{0}}+\frac{(\lambda+2)(\lambda+\mu+2)p_{1}^{2}q_{0}}{2\mu^{3}p_{0}^% {2}}\right)\frac{1}{p_{0}^{(\lambda+2)/\mu}}.$ ⓘ Symbols: $b$: right endpoint, $p_{s}$: coefficients, $q_{s}$: coefficients, $\lambda$: positive constant and $\mu$: positive constant Permalink: http://dlmf.nist.gov/2.3.E17 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §2.3(iii), §2.3 and Ch.2

In general

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

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

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

 2.3.19 $I(x)=\int_{a}^{b}e^{ixp(t)}q(t)\,\mathrm{d}t$ ⓘ Defines: $I(x)$: integral (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $p(t)$: real function, $a$: left endpoint, $b$: right endpoint and $q(t)$: function Referenced by: §2.3(iv) Permalink: http://dlmf.nist.gov/2.3.E19 Encodings: TeX, pMML, png See also: Annotations for §2.3(iv), §2.3 and Ch.2

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 $\int e^{ixt}q^{(s)}(t)\,\mathrm{d}t$, $s=0,1,2,\dots$, converges at $t=\infty$, uniformly for all sufficiently large $x$. Then

 2.3.20 $\int_{0}^{\infty}e^{ixt}q(t)\,\mathrm{d}t\sim\sum_{s=0}^{\infty}\exp\left(% \frac{(s+\lambda)\pi i}{2\mu}\right)\Gamma\left(\frac{s+\lambda}{\mu}\right)% \frac{a_{s}}{x^{(s+\lambda)/\mu}},$ $x\to+\infty$,

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 $P_{s}(t)=\left(\frac{1}{p^{\prime}(t)}\frac{\mathrm{d}}{\mathrm{d}t}\right)^{s% }\frac{q(t)}{p^{\prime}(t)},$ $s=0,1,2,\dots$, ⓘ Defines: $P_{s}(t)$: function (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $p(t)$: 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 $p(b)=\infty$, then $P_{0}(b)=0$ and each of the integrals

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

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

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

 2.3.23 $\int_{a}^{b}e^{ixp(t)}q(t)\,\mathrm{d}t\sim e^{ixp(a)}\sum_{s=0}^{\infty}\exp% \left(\frac{(s+\lambda)\pi i}{2\mu}\right)\Gamma\left(\frac{s+\lambda}{\mu}% \right)\frac{b_{s}}{x^{(s+\lambda)/\mu}}-e^{ixp(b)}\sum_{s=0}^{\infty}P_{s}(b)% \left(\frac{i}{x}\right)^{s+1},$ $x\to+\infty$.

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).

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

In the integral

 2.3.24 $I(\alpha,x)=\int_{0}^{k}e^{-xp(\alpha,t)}q(\alpha,t)t^{\lambda-1}\,\mathrm{d}t$ ⓘ Defines: $I(\alpha,x)$: integral (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $p(\alpha,t)$: function, $q(\alpha,t)$: function and $\lambda$: positive constant Referenced by: §2.3(v) Permalink: http://dlmf.nist.gov/2.3.E24 Encodings: TeX, pMML, png See also: Annotations for §2.3(v), §2.3 and Ch.2

$k$ ($\leq\infty$) 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 $b(\alpha)$ Permalink: http://dlmf.nist.gov/2.3.E25 Encodings: TeX, pMML, png See also: Annotations for §2.3(v), §2.3 and Ch.2

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 $\displaystyle a$ $\displaystyle=(2p(\alpha,0)-2p(\alpha,\alpha))^{1/2},$ $\displaystyle b$ $\displaystyle=p(\alpha,0),$ ⓘ Symbols: $p(\alpha,t)$: function, $a(\alpha)$ and $b(\alpha)$ Permalink: http://dlmf.nist.gov/2.3.E26 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §2.3(v), §2.3 and Ch.2
 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 Permalink: http://dlmf.nist.gov/2.3.E27 Encodings: TeX, pMML, png See also: Annotations for §2.3(v), §2.3 and Ch.2

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 $\frac{\mathrm{d}w}{\mathrm{d}t}=\pm\frac{1}{(2p(\alpha,t)-2p(\alpha,\alpha))^{% 1/2}}\frac{\partial p(\alpha,t)}{\partial t}$

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

The integral (2.3.24) transforms into

 2.3.29 $I(\alpha,x)=e^{-xp(\alpha,0)}\*\int_{0}^{\kappa}\exp\left(-x\left(\tfrac{1}{2}% w^{2}-aw\right)\right)f(\alpha,w)w^{\lambda-1}\,\mathrm{d}w,$

where

 2.3.30 $f(\alpha,w)=q(\alpha,t)\left(\frac{t}{w}\right)^{\lambda-1}\frac{\mathrm{d}t}{% \mathrm{d}w},$ ⓘ Defines: $f(\alpha,w)$: function (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $q(\alpha,t)$: function, $w$: change of variable and $\lambda$: positive constant Permalink: http://dlmf.nist.gov/2.3.E30 Encodings: TeX, pMML, png See also: Annotations for §2.3(v), §2.3 and Ch.2

$\kappa=\kappa(\alpha)$ 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 See also: Annotations for §2.3(v), §2.3 and Ch.2

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 $I(\alpha,x)\sim\frac{e^{-xp(\alpha,0)}}{x^{\lambda/2}}\sum_{s=0}^{\infty}\phi_% {s}(\alpha)\frac{F_{s}(a\sqrt{x})}{x^{s/2}},$ $x\to\infty$,

where

 2.3.33 $F_{s}(y)=\int_{0}^{\infty}\exp\left(-\tfrac{1}{2}\tau^{2}+y\tau\right)(\tau-y)% ^{s}\tau^{\lambda-1}\,\mathrm{d}\tau.$ ⓘ Defines: $F_{s}(y)$: function (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\exp\NVar{z}$: exponential function, $\int$: integral and $\lambda$: positive constant Permalink: http://dlmf.nist.gov/2.3.E33 Encodings: TeX, pMML, png See also: Annotations for §2.3(v), §2.3 and Ch.2

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

## §2.3(vi) Asymptotics of Mellin Transforms

For the asymptotics of the Mellin transform $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\int^{\infty}_{0}t^{z-1}f(t% )\,\mathrm{d}t$ as $z\to\infty$ see Frenzen (1987b), Sidi (1985, 2011).