§9.6 Relations to Other Functions

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

§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions

Notes:: These formulas are derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).
Keywords:: Airy functions, Bessel functions, Hankel functions, modified Bessel functions
Referenced by:: ¶ ‣ §10.16, ¶ ‣ §10.39, ¶ ‣ §10.72(i), §9.17(iv)
Permalink:: http://dlmf.nist.gov/9.6.i

For the notation see §§10.2(ii) and 10.25(ii). With

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

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

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, png

§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions

Notes:: These formulas are derivable from those given in Olver (1997b, pp. 392–393).
Keywords:: Airy functions, Bessel functions, Hankel functions, modified Bessel functions
Referenced by:: ¶ ‣ §10.16, ¶ ‣ §10.39
Permalink:: http://dlmf.nist.gov/9.6.ii

Again, for the notation see §§10.2(ii) and 10.25(ii). With

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

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

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

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

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

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

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential 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, pMML, 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 (10.39.8)–(10.39.10). For (9.6.25), (9.6.26) combine (9.6.23), (9.6.24) with (13.14.2) and refer to §13.1.
Keywords:: Airy functions, confluent hypergeometric functions
Permalink:: http://dlmf.nist.gov/9.6.iii

For the notation see §§13.1, 13.2, and 13.14(i). With $\zeta$ as in (9.6.1),

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $z$ : complex variable and $\zeta(z)$ : change of variable
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/9.6.E21
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $z$ : complex variable and $\zeta(z)$ : change of variable
Permalink:: http://dlmf.nist.gov/9.6.E22
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $e$ : base of exponential 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, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $e$ : base of exponential 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, pMML, png

§9.6 Relations to Other Functions

Contents

§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions

§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions

§9.6(iii) Airy Functions as Confluent Hypergeometric Functions