Symbolic Math Toolbox™

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.

You can access more detailed descriptions by typing

mhelp function

Limitations

In general, the accuracy of a function will be lower near its roots and when its arguments are relatively large.

Run-time depends on the specific function and its parameters. In general, calculations are slower than standard MATLAB^® calculations.

References

[1] Abramowitz, M. and I.A., Stegun, Handbook of Mathematical Functions, Dover Publications, 1965.

Table Conventions

The following conventions are used in the following 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 `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)` `Elliptic`CPi(a,k)	`a` is real `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)`	`a` is real `k` is real 0 < k < 1
Incomplete Gamma Function		`GAMMA(z1,z2)`
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)`

Orthogonal Polynomials

The following functions require the Maple^® Orthogonal Polynomial Package. They are available only with Extended Symbolic Math Toolbox™ software. Before using these functions, you must first initialize the Orthogonal Polynomial Package by typing

maple('with','orthopoly')

Note that in all cases, n is a non-negative integer and x is real.

Orthogonal Polynomials

Polynomial	Maple Name	Arguments
Gegenbauer	`G(n,a,x)`	`a` is a nonrational algebraic expression or a rational number greater than `-1/2.`
Hermite	`H(n,x)`
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)`
Jacobi	`P(n,a,b,x)`	`a`, `b` are nonrational algebraic expressions or rational numbers greater than `-1`.
Chebyshev of the First and Second Kind	`T(n,x)` `U(n,x)`

mfun

mhelp

mfunlist - List special functions for use with mfun