Learn more about Symbolic Math Toolbox

Special Functions of Applied Mathematics

On this page…
Numerical Evaluation of Special Functions Using mfun Syntax and Definitions of mfun Special Functions Diffraction Example

Numerical Evaluation of Special Functions Using mfun

Over 50 of the special functions of classical applied mathematics are available in the toolbox. These functions are accessed with the mfun function, which numerically evaluates special functions for the specified parameters. This allows you to evaluate functions that are not available in standard MATLAB software, such as the Fresnel cosine integral. In addition, you can evaluate several MATLAB special functions in the complex plane, such as the error function erf.

For example, suppose you want to evaluate the hyperbolic cosine integral at the points 2 + i, 0, and 4.5. Look in the tables in Syntax and Definitions of mfun Special Functions to find the available functions and their syntax. You can also enter the command

mfunlist

to see the list of functions available for mfun. This list provides a brief mathematical description of each function, its mfun name, and the parameters it needs. From the tables or list, you can see that the hyperbolic cosine integral is called Chi, and it takes one complex argument.

Type

z = [2 + i 0 4.5];
w = mfun('Chi', z)

which returns

w = 
   2.0303 + 1.7227i     NaN       13.9658

mfun returns NaNs where the function has a singularity. The hyperbolic cosine integral has a singularity at z = 0.

Note   mfun functions perform numerical, not symbolic, calculations. The input parameters should be scalars, vectors, or matrices of type double, or complex doubles, not symbolic variables.

Back to Top

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)

Back to Top

Diffraction Example

This example is from diffraction theory in classical electrodynamics. (J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, 1962.)

Suppose you have a plane wave of intensity I₀ and wave number k. Assume that the plane wave is parallel to the xy-plane and travels along the z-axis as shown below. This plane wave is called the incident wave. A perfectly conducting flat diffraction screen occupies half of the xy-plane, that is x < 0. The plane wave strikes the diffraction screen, and you observe the diffracted wave from the line whose coordinates are (x, 0, z₀), where z₀ > 0.

The intensity of the diffracted wave is given by

where

and and are the Fresnel cosine and sine integrals:

How does the intensity of the diffracted wave behave along the line of observation? Since k and z₀ are constants independent of x, you set

and assume an initial intensity of I₀ = 1 for simplicity.

The following code generates a plot of intensity as a function of x:

x = -50:50;
C = mfun('FresnelC',x);
S = mfun('FresnelS',x);
I0 = 1;
T = (C+1/2).^2 + (S+1/2).^2;
I = (I0/2)*T;
plot(x,I);
xlabel('x');
ylabel('I(x)');
title('Intensity of Diffracted Wave');

You see from the graph that the diffraction effect is most prominent near the edge of the diffraction screen (x = 0), as you expect.

Note that values of x that are large and positive correspond to observation points far away from the screen. Here, you would expect the screen to have no effect on the incident wave. That is, the intensity of the diffracted wave should be the same as that of the incident wave. Similarly, x values that are large and negative correspond to observation points under the screen that are far away from the screen edge. Here, you would expect the diffracted wave to have zero intensity. These results can be verified by setting

x = [Inf -Inf] 

in the code to calculate I.

Back to Top

Integral Transforms and Z-Transforms

Generating Code from Symbolic Expressions

© 1984-2009- The MathWorks, Inc.    -   Site Help   -   Patents   -   Trademarks   -   Privacy Policy   -   Preventing Piracy   -   RSS