Procedures

Compute Airy functions and their derivatives

Compute Airy functions and their derivatives

Compute the first NT zeros of Airy functions Ai(x) and Ai'(x), a and a', and the associated values of Ai(a') and Ai'(a); and the first NT zeros of Airy functions Bi(x) and Bi'(x), b and b', and the associated values of Bi(b') and Bi'(b)

Compute Bessel functions Jv(x) and Yv(x), and modified Bessel functions Iv(x) and Kv(x), and their derivatives with v=1/3,2/3

Compute the prolate and oblate spheroidal angular functions of the first kind and their derivatives

Compute the prolate and oblate spheroidal angular functions of the first kind and their derivatives

Compute Bernoulli number Bn

Compute Bernoulli number Bn

Compute the beta function B(p,q)

Compute Bessel functions Jn(x) and their first and second derivatives ( n= 0,1,… )

Compute coefficient Bk's for oblate radial functions with a small argument

Compute confluent hypergeometric function M(a,b,z) with real parameters a, b and a complex argument z

Compute complex Error function erf(z) & erf'(z)

Compute error function erf(z) for a complex argument (z=x+iy)

Evaluate the complex zeros of error function erf(z) using the modified Newton's iteration method

Compute complex Fresnel integral C(z) and C'(z)

Compute complex Fresnel Integral S(z) and S'(z)

Compute the gamma function Г(z) or ln[Г(z)] for a complex argument

Compute Hankel functions of the first and second kinds and their derivatives for a complex argument

Compute confluent hypergeometric function M(a,b,x)

Compute the confluent hypergeometric function U(a,b,x)

Compute confluent hypergeometric function U(a,b,x) with integer b ( b = ±1,±2,... )

Compute hypergeometric function U(a,b,x) by using Gaussian-Legendre integration (n=60)

Compute the confluent hypergeometric function U(a,b,x) for large argument x

Compute confluent hypergeometric function U(a,b,x) for small argument x

Compute modified Bessel functions I0(z), I1(z), K0(z), K1(z), and their derivatives for a complex argument

Compute modified Bessel functions Iv(z) and Kv(z) and their derivatives with a complex argument and a large order

Compute modified Bessel functions In(z), Kn(x) and their derivatives for a complex argument

Compute modified Bessel functions In(z) and Kn(z), and their derivatives for a complex argument

Compute the modified Bessel functions Iv(z), Kv(z) and their derivatives for an arbitrary order and complex argument

Compute the modified Bessel functions Iv(z), Kv(z) and their derivatives for an arbitrary order and complex argument

Compute cosine and sine integrals Si(x) and Ci(x) ( x ≥ 0 )

Compute cosine and sine integrals Si(x) and Ci(x) ( x ≥ 0 )

Compute the expansion coefficients for the asymptotic expansion of Bessel functions with large orders

Compute Bessel functions Jv(z) and Yv(z) and their derivatives with a complex argument and a large order

Compute Bessel functions Jn(z), Yn(z) and their derivatives for a complex argument

Compute Bessel functions Jv(z), Yv(z) and their derivatives for a complex argument

Compute Bessel functions Jv(z), Yv(z) and their derivatives for a complex argument

Compute the associated Legendre functions Pmn(z) and their derivatives Pmn'(z) for a complex argument

Compute Legendre polynomials Pn(z) and their derivatives Pn'(z) for a complex argument

Compute the associated Legendre functions of the second kind, Qmn(z) and Qmn'(z), for a complex argument

Compute the Legendre functions Qn(z) and their derivatives Qn'(z) for a complex argument

Compute complete elliptic integrals K(k) and E(k)

Compute the parabolic cylinder functions Dn(z) and Dn'(z) for a complex argument

Compute complex parabolic cylinder function Dn(z) for large argument

Compute complex parabolic cylinder function Dn(z) for small argument

Compute the psi function for a complex argument

Compute the initial characteristic value of Mathieu functions for m ≤ 12 or q ≤ 300 or q ≥ m*m

Compute a sequence of characteristic values of Mathieu functions

Calculate a specific characteristic value of Mathieu functions

Compute the value of F for characteristic equation of Mathieu functions

Compute the characteristic value of Mathieu functions for q ≥ 3m

Compute the characteristic value of Mathieu functions for q ≤ m*m

Compute complex Bessel functions Y0(z), Y1(z) and their derivatives

Compute the complex zeros of Y0(z), Y1(z) and Y1'(z), and their associated values at the zeros using the modified Newton's iteration method

Compute parabolic cylinder functions Dv(x) for large argument

Compute parabolic cylinder function Dv(x) for small argument

Compute exponential integral E1(x)

Compute exponential integral E1(x)

Compute complex exponential integral E1(z)

Compute exponential integral Ei(x)

Compute exponential integral Ei(x)

Compute complete and incomplete elliptic integrals F(k,phi) and E(k,phi)

Compute the elliptic integral of the third kind using Gauss-Legendre quadrature

Compute exponential integral En(x)

Compute exponential integral En(x)

Compute error function erf(x)

Compute Euler number En

Compute Euler number En

Compute expansion coefficients for Mathieu functions and modified Mathieu functions

Compute Fresnel integrals C(x) and S(x)

Compute the complex zeros of Fresnel integral C(z) or S(z) using modified Newton's iteration method

Compute modified Fresnel integrals F±(x) and K±(x)

Compute gamma function Г(x)

Compute gamma function Г(x)

Compute gamma function Г(x)

Compute gmn(-ic,ix) and its derivative for oblate radial functions with a small argument

Compute the zeros of Hermite polynomial Ln(x) in the interval [-∞,∞], and the corresponding weighting coefficients for Gauss-Hermite integration

Compute hypergeometric function F(a,b,c,x)

Compute the hypergeometric function for a complex argument, F(a,b,c,z)

Compute modified Bessel functions I0(x), I1(1), K0(x) and K1(x), and their derivatives

Compute modified Bessel functions I0(x), I1(1), K0(x) and K1(x), and their derivatives

Compute modified Bessel functions In(x) and Kn(x), and their derivatives

Compute modified Bessel functions In(x) and Kn(x), and their derivatives

Compute modified Bessel functions Iv(x) and Kv(x), and their derivatives

Compute the incomplete beta function Ix(a,b)

Compute the incomplete gamma function r(a,x), Г(a,x) and P(a,x)

Compute the integrals of Airy fnctions with respect to t from 0 and x ( x ≥ 0 )

Integrate modified Bessel functions I0(t) and K0(t) with respect to t from 0 to x

Integrate Bessel functions I0(t) and K0(t) with respect to t from 0 to x

Integrate Bessel functions J0(t) & Y0(t) with respect to t from 0 to x

Integrate Bessel functions J0(t) and Y0(t) with respect to t from 0 to x ( x ≥ 0 )

Evaluate the integral of Struve function H0(t) with respect to t from 0 and x

Evaluate the integral of modified Struve function L0(t) with respect to t from 0 to x

Evaluate the integral H0(t)/t with respect to t from x to infinity

Integrate [I0(t)-1]/t with respect to t from 0 to x, and K0(t)/t with respect to t from x to ∞

Integrate [I0(t)-1]/t with respect to t from 0 to x, and K0(t)/t with respect to t from x to ∞

Integrate [1-J0(t)]/t with respect to t from 0 to x, and Y0(t)/t with respect to t from x to ∞

Integrate [1-J0(t)]/t with respect to t from 0 to x, and Y0(t)/t with respect to t from x to ∞

Compute the zeros of Bessel functions Jn(x) and Jn'(x), and arrange them in the order of their magnitudes

Compute Jacobian elliptic functions sn u, cn u and dn u

Compute Bessel functions J0(x), J1(x), Y0(x), Y1(x), and their derivatives

Compute Bessel functions J0(x), J1(x), Y0(x), Y1(x), and their derivatives

Compute Bessel functions Jn(x) & Yn(x) and their derivatives

Compute Bessel functions Jn(x), Yn(x) and their derivatives

Compute Bessel functions Jn(x), Yn(x)

Compute Bessel functions Jn(x) and Yn(x), and their first and second derivatives

Compute Bessel functions Jv(x) and Yv(x) and their derivatives

Compute the zeros of Bessel functions Jn(x), Yn(x), and their derivatives

Compute Kelvin functions ber x, bei x, ker x and kei x, and their derivatives ( x > 0 )

Compute Kelvin functions ber x, bei x, ker x and kei x, and their derivatives ( x > 0 )

Compute the zeros of Kelvin functions

Compute the expansion coefficients of the prolate and oblate spheroidal functions and joining factors

Compute the zeros of Laguerre polynomial Ln(x) in the interval [0,∞], and the corresponding weighting coefficients for Gauss-Laguerre integration

Compute lambda functions and their derivatives

Compute lambda function with arbitrary order v, and their derivative

Compute the zeros of Legendre polynomial Pn(x) in the interval [-1,1], and the corresponding weighting coefficients for Gauss-Legendre integration

Compute gamma function Г(x) or ln[Г(x)]

Compute the associated Legendre functions Pmn(x) and their derivatives Pmn'(x) for real argument

Compute associated Legendre functions Pmn(x) and Pmn'(x) for a given order

Compute the associated Legendre function Pmv(x) with an integer order and an arbitrary degree v, using recursion for large degrees

Compute the associated Legendre function Pmv(x) with an integer order and an arbitrary nonnegative degree v

Compute Legendre polynomials Pn(x) and their derivatives Pn'(x)

Compute Legendre polynomials Pn(x), Pn'(x) and the integral of Pn(t) from 0 to x

Compute the associated Legendre functions of the second kind, Qmn(x) and Qmn'(x)

Compute associated Legendre functions Qmn(x) and Qmn'(x) for a given order

Compute Legendre functions Qn(x) and Qn'(x)

Compute Legendre functions Qn(x) & Qn'(x)

Determine the starting point for backward recurrence such that the magnitude of Jn(x) at that point is about 10^(-MP)

Determine the starting point for backward recurrence such that all Jn(x) has MP significant digits

Compute Mathieu functions cem(x,q) and sem(x,q) and their derivatives ( q ≥ 0 )

Compute modified Mathieu functions of the first and second kinds, Mcm(1)(2)(x,q) and Msm(1)(2)(x,q), and their derivatives

Compute orthogonal polynomials: Tn(x) or Un(x), or Ln(x) or Hn(x), and their derivatives

Compute parabolic cylinder functions Dv(x) and their derivatives

Compute parabolic cylinder functions Vv(x) and their derivatives

Compute parabolic cylinder functions W(a,±x) and their derivatives

Compute Psi function

Compute Q*mn(-ic) for oblate radial functions with a small argument

Compute Riccati-Bessel functions of the first kind and their derivatives

Compute Riccati-Bessel functions of the second kind and their derivatives

calculate the accurate characteristic value by the secant method

Compute prolate and oblate spheroidal radial functions of the first kind for given m, n, c and x

Compute prolate and oblate spheroidal radial functions of the second kind for given m, n, c and a large cx

Compute oblate radial functions of the second kind with a small argument, Rmn(-ic,ix) & Rmn'(-ic,ix)

Compute prolate spheroidal radial function of the second kind with a small argument

Compute oblate radial functions of the first and second kinds, and their derivatives

Compute prolate spheriodal radial functions of the first and second kinds, and their derivatives

Compute the expansion coefficients of the prolate and oblate spheroidal functions, c2k

Compute the expansion coefficients of the prolate and oblate spheroidal functions

Compute the expansion coefficients of the prolate and oblate spheroidal functions, dk

Compute the characteristic values of spheroidal wave functions

Compute modified spherical Bessel functions of the first kind, in(x) and in'(x)

Compute spherical Bessel functions jn(x) and their derivatives

Compute modified spherical Bessel functions of the second kind, kn(x) and kn'(x)

Compute spherical Bessel functions yn(x) and their derivatives

Compute parabolic cylinder function Vv(x) for large argument

Compute parabolic cylinder function Vv(x) for small argument