mfunlist - List special functions for use with `mfun`

Syntax

mfunlist

Description

mfunlist lists the special mathematical functions for use with the mfun function. The following tables describe these special functions.

Syntax and Definitions of mfun Special Functions

The following conventions are used in the next table, unless otherwise indicated in the Arguments column.

`x`, `y`	real argument
`z`, `z1`, `z2`	complex argument
`m`, `n`	integer argument

MFUN Special Functions

Function Name	Definition	mfun Name	Arguments
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	`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	`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.	`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)`

The following orthogonal polynomials are available using mfun. In all cases, n is a nonnegative integer and x is real.

Orthogonal Polynomials

Polynomial	mfun Name	Arguments
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)`