# asymptotic expansions of integrals

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##### 1: 6.12 Asymptotic Expansions
###### §6.12(i) Exponential and Logarithmic Integrals
For the function $\chi$ see §9.7(i). …
##### 2: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …
##### 3: 6.13 Zeros
Values of $c_{1}$ and $c_{2}$ to 30D are given by MacLeod (1996b). …
6.13.2 $c_{k},s_{k}\sim\alpha+\frac{1}{\alpha}-\frac{16}{3}\frac{1}{\alpha^{3}}+\frac{% 1673}{15}\frac{1}{\alpha^{5}}-\frac{5\;07746}{105}\frac{1}{\alpha^{7}}+\cdots,$
##### 4: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$
###### §8.20(i) Large $z$
Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). For an exponentially-improved asymptotic expansion of $E_{p}\left(z\right)$ see §2.11(iii).
##### 5: 7.12 Asymptotic Expansions
###### §7.12(ii) Fresnel Integrals
The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and
7.12.2 $\mathrm{f}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}},$
They are bounded by $|\csc\left(4\operatorname{ph}z\right)|$ times the first neglected terms when $\frac{1}{8}\pi\leq|\operatorname{ph}z|<\frac{1}{4}\pi$. …
##### 6: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 7: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
##### 8: 2.3 Integrals of a Real Variable
For the Fourier integral
###### §2.3(ii) Watson’s Lemma
2.3.12 $\int_{0}^{\infty}f(xt)q(t)\,\mathrm{d}t\sim\sum_{s=0}^{\infty}\mathscr{M}% \mskip-3.0muf\mskip 3.0mu\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+% \lambda)/\mu}},$ $x\to+\infty$,
Then …
##### 9: 2.6 Distributional Methods
2.6.3 $\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\,\mathrm{d}t,$ $s=1,2,3,\dots$.
###### §2.6(iii) Fractional Integrals
If both $f$ and $g$ in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution $f\ast g$; see Li and Wong (1994). … The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …
##### 10: 7.13 Zeros
As $n\to\infty$ the $x_{n}$ and $y_{n}$ corresponding to the zeros of $C\left(z\right)$ satisfy … For an asymptotic expansion of the zeros of $\int_{0}^{z}\exp\left(\tfrac{1}{2}\pi it^{2}\right)\,\mathrm{d}t$ ($=\mathcal{F}\left(0\right)-\mathcal{F}\left(z\right)$ $=C\left(z\right)+iS\left(z\right)$) see Tuẑilin (1971).