GNU Octave Manual Version 3 by John W. Eaton, David Bateman, Søren Hauberg Paperback (6"x9"), 568 pages ISBN 095461206X RRP £24.95 ($39.95) 
17.5 Special Functions
 Loadable Function: [j, ierr] = besselj (alpha, x, opt)
 Loadable Function: [y, ierr] = bessely (alpha, x, opt)
 Loadable Function: [i, ierr] = besseli (alpha, x, opt)
 Loadable Function: [k, ierr] = besselk (alpha, x, opt)
 Loadable Function: [h, ierr] = besselh (alpha, k, x, opt)
 Compute Bessel or Hankel functions of various kinds:
besselj

Bessel functions of the first kind. If the argument opt is supplied,
the result is multiplied by
exp(abs(imag(x)))
. bessely

Bessel functions of the second kind. If the argument opt is supplied,
the result is multiplied by
exp(abs(imag(x)))
. besseli

Modified Bessel functions of the first kind. If the argument opt is supplied,
the result is multiplied by
exp(abs(real(x)))
. besselk

Modified Bessel functions of the second kind. If the argument opt is supplied,
the result is multiplied by
exp(x)
. besselh

Compute Hankel functions of the first (k = 1) or second (k
= 2) kind. If the argument opt is supplied, the result is multiplied by
exp (I*x)
for k = 1 orexp (I*x)
for k = 2.
If alpha is a scalar, the result is the same size as x. If x is a scalar, the result is the same size as alpha. If alpha is a row vector and x is a column vector, the result is a matrix with
length (x)
rows andlength (alpha)
columns. Otherwise, alpha and x must conform and the result will be the same size.The value of alpha must be real. The value of x may be complex.
If requested, ierr contains the following status information and is the same size as the result.
 Normal return.

Input error, return
NaN
. 
Overflow, return
Inf
.  Loss of significance by argument reduction results in less than half of machine accuracy.

Complete loss of significance by argument reduction, return
NaN
. 
Errorno computation, algorithm termination condition not met,
return
NaN
.
 Loadable Function: [a, ierr] = airy (k, z, opt)
 Compute Airy functions of the first and second kind, and their
derivatives. If the optional argument opt is nonzero the value is multiplied by the scale factor given in the table below.
K Function Scale factor (if 'opt' is supplied)    0 Ai (Z) exp ((2/3) * Z * sqrt (Z)) 1 dAi(Z)/dZ exp ((2/3) * Z * sqrt (Z)) 2 Bi (Z) exp (abs (real ((2/3) * Z *sqrt (Z)))) 3 dBi(Z)/dZ exp (abs (real ((2/3) * Z *sqrt (Z))))
The function call
airy (z)
is equivalent toairy (0, z)
.The result is the same size as z.
If requested, ierr contains the following status information and is the same size as the result.
 Normal return.

Input error, return
NaN
. 
Overflow, return
Inf
.  Loss of significance by argument reduction results in less than half of machine accuracy.

Complete loss of significance by argument reduction, return
NaN
. 
Errorno computation, algorithm termination condition not met,
return
NaN
.
 Mapping Function: beta (a, b)
 Return the Beta function,
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
 Mapping Function: betainc (x, a, b)
 Return the incomplete Beta function,
x / betainc (x, a, b) = beta (a, b)^(1)  t^(a1) (1t)^(b1) dt. / t=0
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: betaln (a, b)
 Return the log of the Beta function,
betaln (a, b) = gammaln (a) + gammaln (b)  gammaln (a + b)
See also beta, betai, gammaln
 Mapping Function: bincoeff (n, k)
 Return the binomial coefficient of n and k, defined as
/ \  n  n (n1) (n2) ... (nk+1)   =   k  k! \ /
For example,
bincoeff (5, 2) => 10
 Mapping Function: erf (z)
 Computes the error function,
z / erf (z) = (2/sqrt (pi))  e^(t^2) dt / t=0
See also erfc, erfinv
 Mapping Function: erfc (z)
 Computes the complementary error function,
1  erf (z)
.See also erf, erfinv
 Mapping Function: erfinv (z)
 Computes the inverse of the error function.
See also erf, erfc
 Mapping Function: gamma (z)
 Computes the Gamma function,
infinity / gamma (z) =  t^(z1) exp (t) dt. / t=0
See also gammai, lgamma
 Mapping Function: gammainc (x, a)
 Compute the normalized incomplete gamma function,
x 1 / gammainc (x, a) =   exp (t) t^(a1) dt gamma (a) / t=0
with the limiting value of 1 as x approaches infinity. The standard notation is P(a,x), e.g. Abramowitz and Stegun (6.5.1).
If a is scalar, then
gammainc (x, a)
is returned for each element of x and vice versa.If neither x nor a is scalar, the sizes of x and a must agree, and gammainc is applied elementbyelement.
See also gamma, lgamma
 Function File: L = legendre (n, X)

Computes the Legendre Function of degree n and order m where all values for m = 0 ... n are returned. n must be a scalar in the range [0 ... 255]. The return value has one dimension more than x.
The Legendre Function of degree n and order m is defined as
m m 2 m/2 d^m P(x) = (1) * (1x ) *  P (x) n dx^m n
where P_n(x) is the Legendre polynomial of degree n
1 d^n 2 n P (x) =  [(x  1) ] n 2^n n! dx^n
For example,
legendre(3,[1.0 0.9 0.8])
returns a matrix of results corresponding to the following values,x  1.0  0.9  0.8  m=0  1.00000  0.47250  0.08000 m=1  0.00000  1.99420  1.98000 m=2  0.00000  2.56500  4.32000 m=3  0.00000  1.24229  3.24000
 Mapping Function: lgamma (x)
 Mapping Function: gammaln (x)
 Return the natural logarithm of the absolute value of the gamma
function of x.
See also gamma, gammai
 Function File: cross (x, y, dim)
 Computes the vector cross product of the two 3dimensional vectors
x and y.
cross ([1,1,0], [0,1,1]) => [ 1; 1; 1 ]
If x and y are matrices, the cross product is applied along the first dimension with 3 elements. The optional argument dim is used to force the cross product to be calculated along the dimension defined by dim.
 Function File: commutation_matrix (m, n)
 Return the commutation matrix
K(m,n)
which is the unique m*n by m*n
matrix such that K(m,n) * vec(A) = vec(A')
for all m by n
matrices A.
If only one argument m is given, K(m,m)
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
Dn
which is the unique n^2 by n*(n+1)/2
matrix such that Dn vech (A) = vec (A)
for all symmetric n by n
matrices A.
See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.
ISBN 095461206X  GNU Octave Manual Version 3  See the print edition 