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mfunlist See Also

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

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.

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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