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21: 6.12 Asymptotic Expansions

…

22: 9.5 Integral Representations

… ►

9.5.3

\operatorname{Bi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-{% \tfrac{1}{3}}t^{3}+xt\right)\,\mathrm{d}t+\frac{1}{\pi}\int_{0}^{\infty}\sin% \left(\tfrac{1}{3}t^{3}+xt\right)\,\mathrm{d}t.

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\sin\NVar{z}$ : sine function and $x$ : real variable
Proof sketch:: Use (9.5.5) with the substitions: for the paths $(-\infty,0]$ use $t=-\tau$ , for the paths $[0,\infty{\mathrm{e}}^{\pm\pi\mathrm{i}/3})$ use $t={\mathrm{e}}^{\pm\pi\mathrm{i}/3}\tau$ . For the resulting integrals use (4.14.1) and (4.14.2).
A&S Ref:: 10.4.33
Referenced by:: (9.12.19)
Permalink:: http://dlmf.nist.gov/9.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §9.5(i), §9.5 and Ch.9

…

23: 3.7 Ordinary Differential Equations

… ►Assume that we wish to integrate (3.7.1) along a finite path

\mathscr{P}

from

z=a

z=b

in a domain

D

. The path is partitioned at

P+1

points labeled successively

z_{0},z_{1},\dots,z_{P}

, with

z_{0}=a

z_{P}=b

. … ►Now suppose the path

\mathscr{P}

is such that the rate of growth of

w(z)

along

\mathscr{P}

is intermediate to that of two other solutions. …

24: 15.19 Methods of Computation

… ►As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions. …

25: 11.5 Integral Representations

… ►In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at

s=0,1,2,\dots

from those at

s=-1,-2,-3,\dots

. …

26: 31.11 Expansions in Series of Hypergeometric Functions

… ►

§31.11(v) Doubly-Infinite Series

►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …

27: 8.21 Generalized Sine and Cosine Integrals

… ►(obtained from (5.2.1) by rotation of the integration path) is also needed. … ►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. …

28: 3.5 Quadrature

… ►For these cases the integration path may need to be deformed; see §3.5(ix). … ►

§3.5(ix) Other Contour Integrals

… ►For example, steepest descent paths can be used; see §2.4(iv). … ►with saddle point at

t=1

, and when

c=1

the vertical path intersects the real axis at the saddle point. … ►A special case is the rule for Hilbert transforms (§1.14(v)): …

29: 1.10 Functions of a Complex Variable

… ►The function

f_{1}(z)

D_{1}

is said to be analytically continued along the path

z(t)

a\leq t\leq b

, if there is a chain

(f_{1},D_{1})

(f_{2},D_{2}),\dots,(f_{n},D_{n})

. … ►Here and elsewhere in this subsection the path

C

is described in the positive sense. … ►(b) By specifying the value of

F(z)

at a point

z_{0}

(not a branch point), and requiring

F(z)

to be continuous on any path that begins at

z_{0}

and does not pass through any branch points or other singularities of

F(z)

. ►If the path circles a branch point at

z=a

k

times in the positive sense, and returns to

z_{0}

without encircling any other branch point, then its value is denoted conventionally as

F((z_{0}-a){\mathrm{e}}^{2k\pi\mathrm{i}}+a)

. … ►Thus if

F(z)

is continued along a path that circles

z=1

m

times in the positive sense and returns to

z_{0}

without circling

z=-1

, then

F((z_{0}-1){\mathrm{e}}^{2m\pi\mathrm{i}}+1)={\mathrm{e}}^{\alpha\ln\left(1-z_% {0}\right)}{\mathrm{e}}^{\beta\ln\left(1+z_{0}\right)}{\mathrm{e}}^{2\pi% \mathrm{i}m\alpha}

. …

30: 4.2 Definitions

… ►where the integration path does not intersect the origin. … ►where the path does not intersect

(-\infty,0]

; see Figure 4.2.1. … ►where

k

is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense. …