mfunlist - List special functions for use with mfun

x, y

real argument

z, z1, z2

complex argument

m, n

integer argument

Bernoulli numbers and polynomials

Generating functions:

bernoulli(n)

bernoulli(n,t)

Bessel functions

BesselI, BesselJ—Bessel functions of the first kind.
BesselK, BesselY—Bessel functions of the second kind.

BesselJ(v,x)

BesselY(v,x)

BesselI(v,x)

BesselK(v,x)

v is real.

Beta function

Beta(x,y)

Binomial coefficients

binomial(m,n)

Complete elliptic integrals

Legendre's complete elliptic integrals of the first, second, and third kind. This definition uses modulus k. The numerical ellipke function and the MuPAD functions for computing elliptic integrals use the parameter .

EllipticK(k)

EllipticE(k)

EllipticPi(a,k)

a is real, –∞ < a < ∞.

k is real, 0 < k < 1.

Complete elliptic integrals with complementary modulus

Associated complete elliptic integrals of the first, second, and third kind using complementary modulus. This definition uses modulus k. The numerical ellipke function and the MuPAD functions for computing elliptic integrals use the parameter .

EllipticCK(k)

EllipticCE(k)

EllipticCPi(a,k)

a is real, –∞ < a < ∞.

k is real, 0 < k < 1.

Complementary error function and its iterated integrals

erfc(z)

erfc(n,z)

n > 0

Dawson's integral

dawson(x)

Digamma function

Psi(x)

Dilogarithm integral

dilog(x)

x > 1

Error function

erf(z)

Euler numbers and polynomials

Generating function for Euler numbers:

euler(n)

euler(n,z)

n ≥ 0

Exponential integrals

Ei(n,z)

Ei(x)

n ≥ 0

Real(z) > 0

Fresnel sine and cosine integrals

FresnelC(x)

FresnelS(x)

Gamma function

GAMMA(z)

Harmonic function

harmonic(n)

n > 0

Hyperbolic sine and cosine integrals

Shi(z)

Chi(z)

(Generalized) hypergeometric function

where j and m are the number of terms in n and d, respectively.

hypergeom(n,d,x)

where

n = [n1,n2,...]

d = [d1,d2,...]

n1,n2,... are real.

d1,d2,... are real and nonnegative.

Incomplete elliptic integrals

Legendre's incomplete elliptic integrals of the first, second, and third kind. This definition uses modulus k. The numerical ellipke function and the MuPAD functions for computing elliptic integrals use the parameter .

EllipticF(x,k)

EllipticE(x,k)

EllipticPi(x,a,k)

0 < x ≤ ∞.

a is real, –∞ < a < ∞.

k is real, 0 < k < 1.

Incomplete gamma function

GAMMA(z1,z2)

z1 = a
z2 = z

Logarithm of the gamma function

lnGAMMA(z)

Logarithmic integral

Li(x)

x > 1

Polygamma function

where is the Digamma function.

Psi(n,z)

n ≥ 0

Shifted sine integral

Ssi(z)

Chebyshev of the first and second kind

T(n,x)

U(n,x)

Gegenbauer

G(n,a,x)

a is a nonrational algebraic expression or a rational number greater than -1/2.

Hermite

H(n,x)

Jacobi

P(n,a,b,x)

a, b are nonrational algebraic expressions or rational numbers greater than -1.

Laguerre

L(n,x)

Generalized Laguerre

L(n,a,x)

a is a nonrational algebraic expression or a rational number greater than -1.

Legendre

P(n,x)