§ 9.6. Relations to Other Functions

Permalink:: http://dlmf.nist.gov/9.6

§9.6(i)Airy Functions as Bessel Functions
§9.6(ii)Bessel Functions as Airy Functions
§9.6(iii)Airy Functions as Confluent Hypergeometric Functions

§ 9.6(i). Airy Functions as Bessel Functions

Notes:: These formulas are derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).
Referenced by:: §9.17(iv)
Permalink:: http://dlmf.nist.gov/9.6.SS1

For the notation see §§Ch.10 and Ch.10. With

9.6.1 $\zeta=\tfrac{2}{3}z^{{3/2}},$

Defines:: $\zeta(z)$ : change of variable
Symbols:: $z$ : complex variable
Referenced by:: §9.6(iii), §9.6(iii), §9.7(iv)
Permalink:: http://dlmf.nist.gov/9.6.E1
Encodings:: TeX, pMathML, png

9.6.2 $\mathrm{Ai}\!\left(z\right)=\pi^{{-1}}\sqrt{z/3}K_{{\pm 1/3}}\!\left(\zeta\right)=\tfrac{1}{3}\sqrt{z}\left(I_{{-1/3}}\!\left(\zeta\right)-I_{{1/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}e^{{2\pi i/3}}{H^{{(1)}}_{{1/3}}}\!\left(\zeta e^{{\pi i/2}}\right)=\tfrac{1}{2}\sqrt{z/3}e^{{\pi i/3}}{H^{{(1)}}_{{-1/3}}}\!\left(\zeta e^{{\pi i/2}}\right)=\tfrac{1}{2}\sqrt{z/3}e^{{-2\pi i/3}}{H^{{(2)}}_{{1/3}}}\!\left(\zeta e^{{-\pi i/2}}\right)=\tfrac{1}{2}\sqrt{z/3}e^{{-\pi i/3}}{H^{{(2)}}_{{-1/3}}}\!\left(\zeta e^{{-\pi i/2}}\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.14 (partial)
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.6.E2
Encodings:: TeX, pMathML, png

9.6.3 ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)=-\pi^{{-1}}(z/\sqrt{3})K_{{\pm 2/3}}\!\left(\zeta\right)=(z/3)\left(I_{{2/3}}\!\left(\zeta\right)-I_{{-2/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})e^{{-\pi i/6}}{H^{{(1)}}_{{2/3}}}\!\left(\zeta e^{{\pi i/2}}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{{-5\pi i/6}}{H^{{(1)}}_{{-2/3}}}\!\left(\zeta e^{{\pi i/2}}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{{\pi i/6}}{H^{{(2)}}_{{2/3}}}\!\left(\zeta e^{{-\pi i/2}}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{{5\pi i/6}}{H^{{(2)}}_{{-2/3}}}\!\left(\zeta e^{{-\pi i/2}}\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.16 (with opposite sign!)
Permalink:: http://dlmf.nist.gov/9.6.E3
Encodings:: TeX, pMathML, png

9.6.4 $\mathrm{Bi}\!\left(z\right)=\sqrt{z/3}\left(I_{{1/3}}\!\left(\zeta\right)+I_{{-1/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{{\pi i/6}}{H^{{(1)}}_{{1/3}}}\!\left(\zeta e^{{-\pi i/2}}\right)+e^{{-\pi i/6}}{H^{{(2)}}_{{1/3}}}\!\left(\zeta e^{{\pi i/2}}\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{{-\pi i/6}}{H^{{(1)}}_{{-1/3}}}\!\left(\zeta e^{{-\pi i/2}}\right)+e^{{\pi i/6}}{H^{{(2)}}_{{-1/3}}}\!\left(\zeta e^{{\pi i/2}}\right)\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.18 (partial)
Permalink:: http://dlmf.nist.gov/9.6.E4
Encodings:: TeX, pMathML, png

9.6.5 ${{\mathrm{Bi}}^{{\prime}}}\!\left(z\right)=(z/\sqrt{3})\left(I_{{2/3}}\!\left(\zeta\right)+I_{{-2/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{{\pi i/3}}{H^{{(1)}}_{{2/3}}}\!\left(\zeta e^{{-\pi i/2}}\right)+e^{{-\pi i/3}}{H^{{(2)}}_{{2/3}}}\!\left(\zeta e^{{\pi i/2}}\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{{-\pi i/3}}{H^{{(1)}}_{{-2/3}}}\!\left(\zeta e^{{-\pi i/2}}\right)+e^{{\pi i/3}}{H^{{(2)}}_{{-2/3}}}\!\left(\zeta e^{{\pi i/2}}\right)\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.20 (partial)
Referenced by:: §9.6(iii), §9.7(iv)
Permalink:: http://dlmf.nist.gov/9.6.E5
Encodings:: TeX, pMathML, png

9.6.6 $\mathrm{Ai}\!\left(-z\right)=(\sqrt{z}/3)\left(J_{{1/3}}\!\left(\zeta\right)+J_{{-1/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{{\pi i/6}}{H^{{(1)}}_{{1/3}}}\!\left(\zeta\right)+e^{{-\pi i/6}}{H^{{(2)}}_{{1/3}}}\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{{-\pi i/6}}{H^{{(1)}}_{{-1/3}}}\!\left(\zeta\right)+e^{{\pi i/6}}{H^{{(2)}}_{{-1/3}}}\!\left(\zeta\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.15 (partial)
Permalink:: http://dlmf.nist.gov/9.6.E6
Encodings:: TeX, pMathML, png

9.6.7 ${{\mathrm{Ai}}^{{\prime}}}\!\left(-z\right)=(z/3)\left(J_{{2/3}}\!\left(\zeta\right)-J_{{-2/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{{-\pi i/6}}{H^{{(1)}}_{{2/3}}}\!\left(\zeta\right)+e^{{\pi i/6}}{H^{{(2)}}_{{2/3}}}\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{{-5\pi i/6}}{H^{{(1)}}_{{-2/3}}}\!\left(\zeta\right)+e^{{5\pi i/6}}{H^{{(2)}}_{{-2/3}}}\!\left(\zeta\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.17 (partial)
Permalink:: http://dlmf.nist.gov/9.6.E7
Encodings:: TeX, pMathML, png

9.6.8 $\mathrm{Bi}\!\left(-z\right)=\sqrt{z/3}\left(J_{{-1/3}}\!\left(\zeta\right)-J_{{1/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{{2\pi i/3}}{H^{{(1)}}_{{1/3}}}\!\left(\zeta\right)+e^{{-2\pi i/3}}{H^{{(2)}}_{{1/3}}}\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{{\pi i/3}}{H^{{(1)}}_{{-1/3}}}\!\left(\zeta\right)+e^{{-\pi i/3}}{H^{{(2)}}_{{-1/3}}}\!\left(\zeta\right)\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.19 (in slightly different form)
Permalink:: http://dlmf.nist.gov/9.6.E8
Encodings:: TeX, pMathML, png

9.6.9 ${{\mathrm{Bi}}^{{\prime}}}\!\left(-z\right)=(z/\sqrt{3})\left(J_{{-2/3}}\!\left(\zeta\right)+J_{{2/3}}\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{{\pi i/3}}{H^{{(1)}}_{{2/3}}}\!\left(\zeta\right)+e^{{-\pi i/3}}{H^{{(2)}}_{{2/3}}}\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{{-\pi i/3}}{H^{{(1)}}_{{-2/3}}}\!\left(\zeta\right)+e^{{\pi i/3}}{H^{{(2)}}_{{-2/3}}}\!\left(\zeta\right)\right).$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.21 (partial)
Permalink:: http://dlmf.nist.gov/9.6.E9
Encodings:: TeX, pMathML, png

§ 9.6(ii). Bessel Functions as Airy Functions

Notes:: These formulas are derivable from those given in Olver (1997b, pp. 392–393).
Permalink:: http://dlmf.nist.gov/9.6.SS2

Again, for the notation see §§Ch.10 and Ch.10. With

9.6.10 $z=(\tfrac{3}{2}\zeta)^{{2/3}},$

Defines:: $\zeta(z)$ : change of variable
Symbols:: $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.6.E10
Encodings:: TeX, pMathML, png

9.6.11 $J_{{\pm 1/3}}\!\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}\left(\sqrt{3}\mathrm{Ai}\!\left(-z\right)\mp\mathrm{Bi}\!\left(-z\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.22
Permalink:: http://dlmf.nist.gov/9.6.E11
Encodings:: TeX, pMathML, png

9.6.12 $J_{{\pm 2/3}}\!\left(\zeta\right)=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}{{\mathrm{Ai}}^{{\prime}}}\!\left(-z\right)+{{\mathrm{Bi}}^{{\prime}}}\!\left(-z\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.27
Permalink:: http://dlmf.nist.gov/9.6.E12
Encodings:: TeX, pMathML, png

9.6.13 $I_{{\pm 1/3}}\!\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}\left(\mp\sqrt{3}\mathrm{Ai}\!\left(z\right)+\mathrm{Bi}\!\left(z\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.25
Permalink:: http://dlmf.nist.gov/9.6.E13
Encodings:: TeX, pMathML, png

9.6.14 $I_{{\pm 2/3}}\!\left(\zeta\right)=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}{{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)+{{\mathrm{Bi}}^{{\prime}}}\!\left(z\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.30
Permalink:: http://dlmf.nist.gov/9.6.E14
Encodings:: TeX, pMathML, png

9.6.15 $K_{{\pm 1/3}}\!\left(\zeta\right)=\pi\sqrt{3/z}\mathrm{Ai}\!\left(z\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.26
Permalink:: http://dlmf.nist.gov/9.6.E15
Encodings:: TeX, pMathML, png

9.6.16 $K_{{\pm 2/3}}\!\left(\zeta\right)=-\pi(\sqrt{3}/z){{\mathrm{Ai}}^{{\prime}}}\!\left(z\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.31
Permalink:: http://dlmf.nist.gov/9.6.E16
Encodings:: TeX, pMathML, png

9.6.17 ${H^{{(1)}}_{{1/3}}}\!\left(\zeta\right)=e^{{-\pi i/3}}{H^{{(1)}}_{{-1/3}}}\!\left(\zeta\right)=e^{{-\pi i/6}}\sqrt{3/z}\left(\mathrm{Ai}\!\left(-z\right)-i\mathrm{Bi}\!\left(-z\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.23 (with different sign, different form!)
Referenced by:: §9.8(i)
Permalink:: http://dlmf.nist.gov/9.6.E17
Encodings:: TeX, pMathML, png

9.6.18 ${H^{{(1)}}_{{2/3}}}\!\left(\zeta\right)=e^{{-2\pi i/3}}{H^{{(1)}}_{{-2/3}}}\!\left(\zeta\right)=e^{{\pi i/6}}(\sqrt{3}/z)\left({{\mathrm{Ai}}^{{\prime}}}\!\left(-z\right)-i{{\mathrm{Bi}}^{{\prime}}}\!\left(-z\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.28
Referenced by:: §9.8(i)
Permalink:: http://dlmf.nist.gov/9.6.E18
Encodings:: TeX, pMathML, png

9.6.19 ${H^{{(2)}}_{{1/3}}}\!\left(\zeta\right)=e^{{\pi i/3}}{H^{{(2)}}_{{-1/3}}}\!\left(\zeta\right)=e^{{\pi i/6}}\sqrt{3/z}\left(\mathrm{Ai}\!\left(-z\right)+i\mathrm{Bi}\!\left(-z\right)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.24
Permalink:: http://dlmf.nist.gov/9.6.E19
Encodings:: TeX, pMathML, png

9.6.20 ${H^{{(2)}}_{{2/3}}}\!\left(\zeta\right)=e^{{2\pi i/3}}{H^{{(2)}}_{{-2/3}}}\!\left(\zeta\right)=e^{{-\pi i/6}}(\sqrt{3}/z)\left({{\mathrm{Ai}}^{{\prime}}}\!\left(-z\right)+i{{\mathrm{Bi}}^{{\prime}}}\!\left(-z\right)\right).$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
A&S Ref:: 10.4.29
Permalink:: http://dlmf.nist.gov/9.6.E20
Encodings:: TeX, pMathML, png

§ 9.6(iii). Airy Functions as Confluent Hypergeometric Functions

Notes:: For (9.6.21)–(9.6.24) combine (9.6.1)–(9.6.5) and (Ch.10)–(Ch.10). For (9.6.25), (9.6.26) combine (9.6.23), (9.6.24) with (Ch.13) and refer to §Ch.13.
Permalink:: http://dlmf.nist.gov/9.6.SS3

For the notation see §§Ch.13, Ch.13, and Ch.13. With $\zeta$ as in (9.6.1),

9.6.21 $\mathrm{Ai}\!\left(z\right)=\tfrac{1}{2}\pi^{{-1/2}}z^{{-1/4}}W_{{0,1/3}}\!\left(2\zeta\right)=3^{{-1/6}}\pi^{{-1/2}}\zeta^{{2/3}}e^{{-\zeta}}U\!\left(\tfrac{5}{6},\tfrac{5}{3},2\zeta\right),$

Defines:: $\zeta(z)$ : change of variable
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $z$ : complex variable
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/9.6.E21
Encodings:: TeX, pMathML, png

9.6.22 ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)=-\tfrac{1}{2}\pi^{{-1/2}}z^{{1/4}}W_{{0,2/3}}\!\left(2\zeta\right)=-3^{{1/6}}\pi^{{-1/2}}\zeta^{{4/3}}e^{{-\zeta}}U\!\left(\tfrac{7}{6},\tfrac{7}{3},2\zeta\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $z$ : complex variable and $\zeta(z)$ : change of variable
Permalink:: http://dlmf.nist.gov/9.6.E22
Encodings:: TeX, pMathML, png

9.6.23 $\mathrm{Bi}\!\left(z\right)=\frac{1}{2^{{1/3}}\Gamma\!\left(\tfrac{2}{3}\right)}z^{{-1/4}}M_{{0,-1/3}}\!\left(2\zeta\right)+\frac{3}{2^{{5/3}}\Gamma\!\left(\tfrac{1}{3}\right)}z^{{-1/4}}M_{{0,1/3}}\!\left(2\zeta\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable and $\zeta(z)$ : change of variable
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/9.6.E23
Encodings:: TeX, pMathML, png

9.6.24 ${{\mathrm{Bi}}^{{\prime}}}\!\left(z\right)=\frac{2^{{1/3}}}{\Gamma\!\left(\tfrac{1}{3}\right)}z^{{1/4}}M_{{0,-2/3}}\!\left(2\zeta\right)+\frac{3}{2^{{10/3}}\Gamma\!\left(\tfrac{2}{3}\right)}z^{{1/4}}M_{{0,2/3}}\!\left(2\zeta\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable and $\zeta(z)$ : change of variable
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/9.6.E24
Encodings:: TeX, pMathML, png

9.6.25 $\mathrm{Bi}\!\left(z\right)=\frac{1}{3^{{1/6}}\Gamma\!\left(\tfrac{2}{3}\right)}e^{{-\zeta}}{{}_{{1}}F_{{1}}}\!\left(\tfrac{1}{6};\tfrac{1}{3};2\zeta\right)+\frac{3^{{5/6}}}{2^{{2/3}}\Gamma\!\left(\tfrac{1}{3}\right)}\zeta^{{2/3}}e^{{-\zeta}}{{}_{{1}}F_{{1}}}\!\left(\tfrac{5}{6};\tfrac{5}{3};2\zeta\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable and $\zeta(z)$ : change of variable
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/9.6.E25
Encodings:: TeX, pMathML, png

9.6.26 ${{\mathrm{Bi}}^{{\prime}}}\!\left(z\right)=\frac{3^{{1/6}}}{\Gamma\!\left(\tfrac{1}{3}\right)}e^{{-\zeta}}{{}_{{1}}F_{{1}}}\!\left(-\tfrac{1}{6};-\tfrac{1}{3};2\zeta\right)+\frac{3^{{7/6}}}{2^{{7/3}}\Gamma\!\left(\tfrac{2}{3}\right)}\zeta^{{4/3}}e^{{-\zeta}}{{}_{{1}}F_{{1}}}\!\left(\tfrac{7}{6};\tfrac{7}{3};\zeta\right).$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable and $\zeta(z)$ : change of variable
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/9.6.E26
Encodings:: TeX, pMathML, png

§ 9.6. Relations to Other Functions

Contents

§ 9.6(i). Airy Functions as Bessel Functions

§ 9.6(ii). Bessel Functions as Airy Functions

§ 9.6(iii). Airy Functions as Confluent Hypergeometric Functions