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Compute Bessel or Hankel functions of various kinds:
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.
besselh- Compute Hankel functions of the first (k = 1) or second (k = 2) kind.
If the argument opt is supplied, the result is scaled by the
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.- Error—no computation, algorithm termination condition not met, return
NaN.
Compute Airy functions of the first and second kind, and their derivatives.
K Function Scale factor (if a third argument 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.- Error—no computation, algorithm termination condition not met, return
NaN
Return the Beta function,
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
Return the incomplete Beta function,
x / betainc (x, a, b) = beta (a, b)^(-1) | t^(a-1) (1-t)^(b-1) dt. / t=0If 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.
Return the binomial coefficient of n and k, defined as
/ \ | n | n (n-1) (n-2) ... (n-k+1) | | = ------------------------- | k | k! \ /For example,
bincoeff (5, 2) => 10
Computes the error function,
z / erf (z) = (2/sqrt (pi)) | e^(-t^2) dt / t=0
Computes the Gamma function,
infinity / gamma (z) = | t^(z-1) exp (-t) dt. / t=0
Computes the incomplete gamma function,
x 1 / gammainc (x, a) = --------- | exp (-t) t^(a-1) dt gamma (a) / t=0If 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 element-by-element.
Return the natural logarithm of the gamma function.
Computes the vector cross product of the two 3-dimensional 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 defiend by dim.
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.