Compute Airy functions of the first and second kind, and their derivatives.
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.- Error—no computation, algorithm termination condition not met, return
NaN.
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.- Error—no computation, algorithm termination condition not met, return
NaN.
For real inputs, 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 natural logarithm of the Beta function,
betaln (a, b) = gammaln (a) + gammaln (b) - gammaln (a + b)
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) ⇒ 10In most cases, the
nchoosekfunction is faster for small scalar integer arguments. It also warns about loss of precision for big arguments.See also: nchoosek.
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.
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.
Compute the error function,
z / erf (z) = (2/sqrt (pi)) | e^(-t^2) dt / t=0
Compute the Gamma function,
infinity / gamma (z) = | t^(z-1) exp (-t) dt. / t=0
Compute the normalized incomplete gamma function,
x 1 / gammainc (x, a) = --------- | exp (-t) t^(a-1) dt gamma (a) / t=0with 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
gammaincis applied element-by-element.By default the incomplete gamma function integrated from 0 to x is computed. If "upper" is given then the complementary function integrated from x to infinity is calculated. It should be noted that
gammainc (x, a) == 1 - gammainc (x, a, "upper")
Compute the Legendre function of degree n and order m = 0 ... N. The optional argument, normalization, may be one of
"unnorm","sch", or"norm". The default is"unnorm". The value of n must be a non-negative scalar integer.If the optional argument normalization is missing or is
"unnorm", compute the Legendre function of degree n and order m and return all values for m = 0 ... n. The return value has one dimension more than x.The Legendre Function of degree n and order m:
m m 2 m/2 d^m P(x) = (-1) * (1-x ) * ---- P (x) n dx^m nwith Legendre polynomial of degree n:
1 d^n 2 n P (x) = ------ [----(x - 1) ] n 2^n n! dx^n
legendre (3, [-1.0, -0.9, -0.8])returns the matrix: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 | -3.24000If the optional argument
normalizationis"sch", compute the Schmidt semi-normalized associated Legendre function. The Schmidt semi-normalized associated Legendre function is related to the unnormalized Legendre functions by the following:For Legendre functions of degree n and order 0:
0 0 SP (x) = P (x) n nFor Legendre functions of degree n and order m:
m m m 2(n-m)! 0.5 SP (x) = P (x) * (-1) * [-------] n n (n+m)!If the optional argument normalization is
"norm", compute the fully normalized associated Legendre function. The fully normalized associated Legendre function is related to the unnormalized Legendre functions by the following:For Legendre functions of degree n and order m
m m m (n+0.5)(n-m)! 0.5 NP (x) = P (x) * (-1) * [-------------] n n (n+m)!