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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, 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)
 n 0
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
-Inf < a < Inf

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.
 LegendreKc1(k)
LegendreEc1(k)

LegendrePic1(a,k)

 a is real
-Inf < a < Inf

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)

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 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)

 0 < x Inf

a is real
-Inf < a < Inf

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 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 1-1: 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)


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