Symbolic Math Toolbox

Special Mathematical Functions

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

For example, suppose you want to evaluate the hyperbolic cosine integral at the points 2+i, 0, and 4.5. First type

• help mfunlist
  

to see the list of functions available for mfun. This list provides a brief mathematical description of each function, its Maple name, and the parameters it needs. From the list, you can see that the hyperbolic cosine integral is called Chi, and it takes one complex argument. For additional information, you can access Maple help on the hyperbolic cosine integral using

• mhelp Chi
  

Now 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.

These special functions can be used with the mfun function:

• Airy Functions
• Binomial Coefficients
• Riemann Zeta Functions
• Bernoulli Numbers and Polynomials
• Euler Numbers and Polynomials
• Harmonic Function
• Exponential Integrals
• Logarithmic Integral
• Sine and Cosine Integrals
• Hyperbolic Sine and Cosine Integrals
• Shifted Sine Integral
• Fresnel Sine and Cosine Integral
• Dawson's Integral
• Error Function
• Complementary Error Function and its Iterated Integrals
• Gamma Function
• Logarithm of the Gamma Function
• Incomplete Gamma Function
• Digamma Function
• Polygamma Function
• Generalized Hypergeometric Function
• Bessel Functions
• Incomplete Elliptic Integrals
• Complete Elliptic Integrals
• Complete Elliptic Integrals with Complementary Modulus
• Beta Function
• Dilogarithm Integral
• Lambert's W Function
• Dirac Delta Function (distribution)
• Heaviside Function (distribution)

The orthogonal polynomials listed below are available with the Extended Symbolic Math Toolbox:

• Gegenbauer
• Hermite
• Laguerre
• Generalized Laguerre
• Legendre
• Jacobi
• Chebyshev of the First and Second Kind

Several Differential Equations

Diffraction