Hypergeometric function \(\,_0F_1\) and related

Text explaining this chapter

• Hypergeometric Limit Function \(\,_0F_1\)
  • Confluent Hypergeometric Limit Function \({}_0F_1(b,x)\)
    • ctx.hyperg_0f1()
  • Regularized Confluent Hypergeometric Limit Function \({}_0\widetilde{F}_1(b;x)\)
    • ctx.hyperg_0f1r()
• Bessel functions
  • Bessel function of the 1st kind, \(J_{\nu}(x)\)
    • ctx.bessel_jv()
  • Bessel function of the 2nd kind, \(Y_{\nu}(x)\)
    • ctx.bessel_yv()
  • First derivative of the Bessel function of the first kind, \(J'_{\nu}(x)\)
    • ctx.bessel_jv_prime()
  • First derivative of the Bessel function of the second kind, \(Y'_{\nu}(x)\)
    • ctx.bessel_yv_prime()
  • Boost: Zeros \(j_{\nu, m}\) of the Bessel function of the first kind: \(J_{\nu}(j_{\nu, m})=0\)
    • ctx.bessel_jv_zero()
  • Boost: Zeros \(y_{\nu, m}\) of the Bessel function of the first kind: \(Y_{\nu}(y_{\nu, m})=0\)
    • ctx.bessel_yv_zero()
• Modified Bessel functions
  • Modified Bessel function of the 1st kind, \(I_{\nu}(x)\)
    • ctx.bessel_iv()
  • Modified Bessel function of the 2nd kind, \(K_{\nu}(x)\)
    • ctx.bessel_kv()
  • First derivative of the modified Bessel function of the first kind \(I'_{\nu}(x)\)
    • ctx.bessel_iv_prime()
  • First derivative of the modified Bessel function of the second kind \(K'_{\nu}(x)\)
    • ctx.bessel_kv_prime()
  • Marcum Q function, \(Q_m(a,b)\) (up to octuple precision)
    • math53.marcum_q()
• Spherical Bessel functions
  • Spherical Bessel function of the first kind, \(j_n(x)\)
    • ctx.sph_bessel_jn()
  • Spherical Bessel function of the second kind, \(y_n(x)\)
    • ctx.sph_bessel_yn()
  • First derivative of the spherical Bessel function of the first kind, \(j'_n(x)\)
    • ctx.sph_bessel_jn_prime()
  • First derivative of the spherical Bessel function of the second kind, \(y'_n(x)\)
    • ctx.sph_bessel_yn_prime()
  • Boost: Zeros \(a_{n, m}\) of the spherical Bessel function of the first kind: \(j_{n}(a_{n, m})=0\)
    • ctx.sph_bessel_jn_zero()
  • Boost: Zeros \(b_{n, m}\) of the spherical Bessel function of the second kind: \(y_{n}(b_{n, m})=0\)
    • ctx.sph_bessel_yn_zero()
• Modified spherical Bessel functions
  • Modified spherical Bessel function of the first kind, \(i_n(x)\)
    • math53.sph_bessel_in()
  • Modified Spherical Bessel function of the second kind, \(k_n(x)\)
    • math53.sph_bessel_kn()
  • First derivative of the modified spherical Bessel function of the first kind, \(i'_n(x)\)
    • math53.sph_bessel_in_prime()
  • First derivative of the modified spherical Bessel function of the second kind, \(k'_n(x)\)
    • math53.sph_bessel_kn_prime()
  • Bessel polynomials \(y_n(x)\)
    • math53.besselpoly()
  • Reverse Bessel polynomials \(\theta_n(x)\)
    • math53.besseltheta()
• Hankel functions
  • Hankel function of the first kind \(H^{(1)}_{\nu}(x)\)
    • ctx.hankel_h1()
  • Hankel function of the second kind \(H^{(2)}_{\nu}(x)\)
    • ctx.hankel_h2()
  • Spherical Hankel function of the first kind, \(h^{(1)}_{\nu}(x)\)
    • ctx.sph_hankel_h1()
  • Spherical Hankel function of the second kind, \(h^{(2)}_{\nu}(x)\)
    • ctx.sph_hankel_h2()
• Airy functions
  • Airy function \(\mathrm{Ai}(x)\)
    • ctx.airy_ai()
  • Airy function \(\mathrm{Bi}(x)\)
    • ctx.airy_bi()
  • First derivative of the Airy function \(\mathrm{Ai}\), \(\mathrm{Ai}'(x)\)
    • ctx.airy_ai_prime()
  • First derivative of the Airy function \(\mathrm{Bi}\), \(\mathrm{Bi}'(x)\)
    • ctx.airy_bi_prime()
  • Real zeros \(a_k\) of the Airy function \(\mathrm{Ai}\), \(\mathrm{Ai}(a_k)=0\)
    • ctx.airy_ai_zero()
  • Real zeros \(b_k\) of the Airy function \(\mathrm{Bi}\), \(\mathrm{Bi}(b_k)=0\)
    • ctx.airy_bi_zero()
• Kelvin functions
  • Kelvin function \(\mathrm{ber}(\nu, x)\)
    • ctx.kelvin_ber()
  • Kelvin function \(\mathrm{bei}(\nu, x)\)
    • ctx.kelvin_bei()
  • Kelvin function \(\mathrm{ker}(\nu, x)\)
    • ctx.kelvin_ker()
  • Kelvin function \(\mathrm{kei}(\nu, x)\)
    • ctx.kelvin_kei()
  • First derivative of the Kelvin function \(\mathrm{ber}(\nu, x)\), \(\mathrm{ber}'(\nu, x)\)
    • ctx.kelvin_ber_prime()
  • First derivative of the Kelvin function \(\mathrm{bei}(\nu, x)\), \(\mathrm{bei}'(\nu, x)\)
    • ctx.kelvin_bei_prime()
  • First derivative of the Kelvin function \(\mathrm{ker}(\nu, x)\), \(\mathrm{ker}'(\nu, x)\)
    • ctx.kelvin_ker_prime()
  • First derivative of the Kelvin function \(\mathrm{kei}(\nu, x)\), \(\mathrm{kei}'(\nu, x)\)
    • ctx.kelvin_kei_prime()