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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 to
airy (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.
NaN
.
Inf
.
NaN
.
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 or exp (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 and
length (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.
NaN
.
Inf
.
NaN
.
NaN
.
For real inputs, return the Beta function,
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
Return the regularized incomplete Beta function,
x 1 / betainc (x, a, b) = ----------- | t^(a-1) (1-t)^(b-1) dt. beta (a, b) / 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.
Return the natural logarithm of the Beta function,
betaln (a, b) = log (beta (a, b))
calculated in a way to reduce the occurrence of underflow.
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
In most cases, the nchoosek
function 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 complementary error function,
1 - erf (z)
.
Compute the scaled complementary error function,
exp (z^2) * erfc (x)
Compute the inverse error function, i.e., y such that
erf (y) == x
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=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 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 n
with 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 | -3.24000
If the optional argument normalization
is "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 n
For 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)!
Return the natural logarithm of the gamma function of x.
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