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9 Airy and Related FunctionsAiry Functions

§9.5 Integral Representations

Contents

§9.5(i) Real Variable

9.5.1 Ai(x)=1π0cos(13t3+xt)dt.
9.5.2 Ai(-x)=x1/2π-1cos(x3/2(13t3+t2-23))dt,
x>0.
9.5.3 Bi(x)=1π0exp(-13t3+xt)dt+1π0sin(13t3+xt)dt.

See also (9.10.19), (9.11.3), (36.9.2), and Vallée and Soares (2010, §2.1.3).

§9.5(ii) Complex Variable

9.5.4 Ai(z)=12πie-πi/3eπi/3exp(13t3-zt)dt,
9.5.5 Bi(z)=12π-eπi/3exp(13t3-zt)dt+12π-e-πi/3exp(13t3-zt)dt.
9.5.6 Ai(z)=32π0exp(-t33-z33t3)dt.
9.5.7 Ai(z)=e-ζπ0exp(-z1/2t2)cos(13t3)dt,
|phz|<π.
9.5.8 Ai(z)=e-ζζ-1/6π(48)1/6Γ(56)0e-tt-1/6(2+tζ)-1/6dt,
|phz|<23π.

In (9.5.7) and (9.5.8) ζ=23z3/2.

See also (9.10.18) and (9.11.4).