Symbolic Math Toolbox    
mfunlist
List special functions for use with mfun.

Syntax

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

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

Runtime depends on the specific function and its parameters. In general, calculations are slower than standard MATLAB calculations.

See Also
mfun, mhelp

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

Table Conventions
The following conventions are used in Table 2-1, MFUN Special Functions, unless otherwise indicated in the Arguments column.

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

Table 2-1: 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.
 LegendreKc(k)
LegendreEc(k)
LegendrePic(a,k)
 a is real

k is real
Complete Elliptic Integrals with Complementary Modulus
 Associated complete elliptic integrals of the first, second, and third kind using complementary modulus.
 LegendreKc1(k)
LegendreEc1(k)

LegendrePic1(a,k)

 a is real

k is real
Complementary Error Function and Its Iterated Integrals







erfc(z)
erfc(n,z)

Dawson's Integral



dawson(x)
Digamma Function



Psi(x)
Dilogarithm Integral



dilog(x)
Error Function



erf(z)
Euler Numbers and Polynomials
 Generating function for Euler numbers:



 euler(n)
euler(n,z)


Exponential Integrals





Ei(n,z)
Ei(x)

Fresnel Sine and Cosine Integrals





FresnelC(x)
FresnelS(x)
Gamma Function



GAMMA(z)
Harmonic Function



harmonic(n)
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 non-negative.
Incomplete Elliptic Integrals
 Legendre's incomplete elliptic integrals of the first, second, and third kind.
 LegendreF(x,k)

LegendreE(x,k)

LegendrePi(x,a,k)

a is real

k is real
Incomplete Gamma Function



GAMMA(z1,z2)

Logarithm of the Gamma Function

lnGAMMA(z)

Logarithmic Integral



Li(x)
Polygamma Function



where is the Digamma function.

 Psi(n,z)
Shifted Sine Integral



Ssi(z)

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

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

Table 2-2: 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