1.. index:: Airy functions 2.. index:: Ai(x) 3.. index:: Bi(x) 4 5The Airy functions :math:`Ai(x)` and :math:`Bi(x)` are defined by the 6integral representations, 7 8.. only:: not texinfo 9 10 .. math:: 11 12 Ai(x) & = {1\over\pi} \int_0^\infty \cos(t^3/3 + xt ) \,dt \\ 13 Bi(x) & = {1\over\pi} \int_0^\infty (e^{-t^3/3 + xt} + \sin(t^3/3 + xt)) \,dt 14 15.. only:: texinfo 16 17 | Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt 18 | Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3 + xt) + \sin((1/3) t^3 + xt)) dt 19 20For further information see Abramowitz & Stegun, Section 10.4. The Airy 21functions are defined in the header file :file:`gsl_sf_airy.h`. 22 23Airy Functions 24-------------- 25 26.. function:: double gsl_sf_airy_Ai (double x, gsl_mode_t mode) 27 int gsl_sf_airy_Ai_e (double x, gsl_mode_t mode, gsl_sf_result * result) 28 29 These routines compute the Airy function :math:`Ai(x)` with an accuracy 30 specified by :data:`mode`. 31 32.. function:: double gsl_sf_airy_Bi (double x, gsl_mode_t mode) 33 int gsl_sf_airy_Bi_e (double x, gsl_mode_t mode, gsl_sf_result * result) 34 35 These routines compute the Airy function :math:`Bi(x)` with an accuracy 36 specified by :data:`mode`. 37 38.. function:: double gsl_sf_airy_Ai_scaled (double x, gsl_mode_t mode) 39 int gsl_sf_airy_Ai_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result) 40 41 These routines compute a scaled version of the Airy function 42 :math:`S_A(x) Ai(x)`. For :math:`x > 0` the scaling factor :math:`S_A(x)` is 43 :math:`\exp(+(2/3) x^{3/2})`, 44 and is 1 for :math:`x < 0`. 45 46.. function:: double gsl_sf_airy_Bi_scaled (double x, gsl_mode_t mode) 47 int gsl_sf_airy_Bi_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result) 48 49 These routines compute a scaled version of the Airy function 50 :math:`S_B(x) Bi(x)`. For :math:`x > 0` the scaling factor :math:`S_B(x)` is 51 :math:`exp(-(2/3) x^{3/2})`, and is 1 for :math:`x < 0`. 52 53 54Derivatives of Airy Functions 55----------------------------- 56 57.. function:: double gsl_sf_airy_Ai_deriv (double x, gsl_mode_t mode) 58 int gsl_sf_airy_Ai_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result) 59 60 These routines compute the Airy function derivative :math:`Ai'(x)` with 61 an accuracy specified by :data:`mode`. 62 63.. function:: double gsl_sf_airy_Bi_deriv (double x, gsl_mode_t mode) 64 int gsl_sf_airy_Bi_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result) 65 66 These routines compute the Airy function derivative :math:`Bi'(x)` with 67 an accuracy specified by :data:`mode`. 68 69.. function:: double gsl_sf_airy_Ai_deriv_scaled (double x, gsl_mode_t mode) 70 int gsl_sf_airy_Ai_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result) 71 72 These routines compute the scaled Airy function derivative 73 :math:`S_A(x) Ai'(x)`. 74 For :math:`x > 0` the scaling factor :math:`S_A(x)` is 75 :math:`\exp(+(2/3) x^{3/2})`, and is 1 for :math:`x < 0`. 76 77.. function:: double gsl_sf_airy_Bi_deriv_scaled (double x, gsl_mode_t mode) 78 int gsl_sf_airy_Bi_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result) 79 80 These routines compute the scaled Airy function derivative 81 :math:`S_B(x) Bi'(x)`. 82 For :math:`x > 0` the scaling factor :math:`S_B(x)` is 83 :math:`exp(-(2/3) x^{3/2})`, and is 1 for :math:`x < 0`. 84 85Zeros of Airy Functions 86----------------------- 87 88.. function:: double gsl_sf_airy_zero_Ai (unsigned int s) 89 int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl_sf_result * result) 90 91 These routines compute the location of the :data:`s`-th zero of the Airy 92 function :math:`Ai(x)`. 93 94.. function:: double gsl_sf_airy_zero_Bi (unsigned int s) 95 int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl_sf_result * result) 96 97 These routines compute the location of the :data:`s`-th zero of the Airy 98 function :math:`Bi(x)`. 99 100Zeros of Derivatives of Airy Functions 101-------------------------------------- 102 103.. function:: double gsl_sf_airy_zero_Ai_deriv (unsigned int s) 104 int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s, gsl_sf_result * result) 105 106 These routines compute the location of the :data:`s`-th zero of the Airy 107 function derivative :math:`Ai'(x)`. 108 109.. function:: double gsl_sf_airy_zero_Bi_deriv (unsigned int s) 110 int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s, gsl_sf_result * result) 111 112 These routines compute the location of the :data:`s`-th zero of the Airy 113 function derivative :math:`Bi'(x)`. 114