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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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