Special Functions

Mapping Function: besseli (alpha, x)

Mapping Function: besselj (alpha, x)

Mapping Function: besselk (alpha, x)

Mapping Function: bessely (alpha, x)

Compute Bessel functions of the following types:

besselj

Bessel functions of the first kind.

bessely

Bessel functions of the second kind.

besseli

Modified Bessel functions of the first kind.

besselk

Modified Bessel functions of the second kind.

The second argument, x, must be a real matrix, vector, or scalar.

The first argument, alpha, must be greater than or equal to zero. If alpha is a range, it must have an increment equal to one.

If alpha is a scalar, the result is the same size as x.

If alpha is a range, x must be a vector or scalar, and the result is a matrix with length(x) rows and length(alpha) columns.

Mapping Function: beta (a, b)

Return the Beta function,

Mapping Function: betai (a, b, x)

Return the incomplete Beta function,

If x has more than one component, both a and b must be scalars. If x is a scalar, a and b must be of compatible dimensions.

Mapping Function: bincoeff (n, k)

Return the binomial coefficient of n and k, defined as

For example,

bincoeff (5, 2)
     => 10

Mapping Function: erf (z)

Computes the error function,

Mapping Function: erfc (z)

Computes the complementary error function,

Mapping Function: erfinv (z)

Computes the inverse of the error function,

Mapping Function: gamma (z)

Computes the Gamma function,

Mapping Function: gammai (a, x)

Computes the incomplete gamma function,

If a is scalar, then gammai (a, x) is returned for each element of x and vice versa.

If neither a nor x is scalar, the sizes of a and x must agree, and gammai is applied element-by-element.

Mapping Function: lgamma (a, x)

Mapping Function: gammaln (a, x)

Return the natural logarithm of the gamma function.

Function File: cross (x, y)

Computes the vector cross product of the two 3-dimensional vectors x and y. For example,

cross ([1,1,0], [0,1,1])
     => [ 1; -1; 1 ]

Function File: commutation_matrix (m, n)

Return the commutation matrix which is the unique matrix such that for all matrices

If only one argument m is given, is returned.

See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

Function File: duplication_matrix (n)

Return the duplication matrix which is the unique matrix such that for all symmetric matrices

See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

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