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### 15.1 Introduction to Special Functions

Special function notation follows:

    bessel_j (index, expr)         Bessel function, 1st kind
    bessel_y (index, expr)         Bessel function, 2nd kind
    bessel_i (index, expr)         Modified Bessel function, 1st kind
    bessel_k (index, expr)         Modified Bessel function, 2nd kind

    hankel_1 (v,z)                 Hankel function of the 1st kind
    hankel_2 (v,z)                 Hankel function of the 2nd kind
    struve_h (v,z)                 Struve H function
    struve_l (v,z)                 Struve L function

    assoc_legendre_p[v,u] (z)      Legendre function of degree v and order u
    assoc_legendre_q[v,u] (z)      Legendre function, 2nd kind

    %f[p,q] ([], [], expr)         Generalized Hypergeometric function
    gamma (z)                      Gamma function
    gamma_incomplete_lower (a,z)   Lower incomplete gamma function
    gamma_incomplete (a,z)         Tail of incomplete gamma function
    hypergeometric (l1, l2, z)     Hypergeometric function
    %s[u,v] (z)                    Lommel "small" s function
    slommel[u,v] (z)               Lommel "big" S function
    %m[u,k] (z)                    Whittaker function, 1st kind
    %w[u,k] (z)                    Whittaker function, 2nd kind
    erfc (z)                       Complement of the erf function

    expintegral_e (v,z)            Exponential integral E
    expintegral_e1 (z)             Exponential integral E1
    expintegral_ei (z)             Exponential integral Ei
    expintegral_li (z)             Logarithmic integral Li
    expintegral_si (z)             Exponential integral Si
    expintegral_ci (z)             Exponential integral Ci
    expintegral_shi (z)            Exponential integral Shi
    expintegral_chi (z)            Exponential integral Chi

    kelliptic (z)                  Complete elliptic integral of the first
                                   kind (K)
    parabolic_cylinder_d (v,z)     Parabolic cylinder D function

Categories: Bessel functions · Airy functions · Special functions ·

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