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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 1 or true,
the result is multiplied by exp (-abs (imag (x)))
.
bessely
Bessel functions of the second kind. If the argument opt is 1 or true,
the result is multiplied by exp (-abs (imag (x)))
.
besseli
Modified Bessel functions of the first kind. If the argument opt is 1
or true, the result is multiplied by exp (-abs (real (x)))
.
besselk
Modified Bessel functions of the second kind. If the argument opt is 1
or true, 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 1 or true, 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
.
Compute the Beta function for real inputs a and b.
The Beta function definition is
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
The Beta function can grow quite large and it is often more useful to work with the logarithm of the output rather than the function directly. See betaln, for computing the logarithm of the Beta function in an efficient manner.
See also: betaln, betainc, betaincinv.
Compute the regularized incomplete Beta function.
The regularized incomplete Beta function is defined by
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.
See also: betaincinv, beta, betaln.
Compute the inverse of the incomplete Beta function.
The inverse is the value x such that
y == betainc (x, a, b)
Compute the natural logarithm of the Beta function for real inputs a and b.
betaln
is defined as
betaln (a, b) = log (beta (a, b))
and is calculated in a way to reduce the occurrence of underflow.
The Beta function can grow quite large and it is often more useful to work with the logarithm of the output rather than the function directly.
See also: beta, betainc, betaincinv, gammaln.
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 Dawson (scaled imaginary error) function.
The Dawson function is defined as
(sqrt (pi) / 2) * exp (-z^2) * erfi (z)
Compute the Jacobi elliptic functions sn, cn, and dn of complex argument u and real parameter m.
If m is a scalar, the results are the same size as u.
If u is a scalar, the results are the same size as m.
If u is a column vector and m is a row vector, the
results are matrices with length (u)
rows and
length (m)
columns. Otherwise, u and
m must conform in size and the results will be the same size as the
inputs.
The value of u may be complex. The value of m must be 0 ≤ m ≤ 1.
The optional input tol is currently ignored (MATLAB uses this to allow faster, less accurate approximation).
If requested, err contains the following status information and is the same size as the result.
NaN
.
Reference: Milton Abramowitz and Irene A Stegun, Handbook of Mathematical Functions, Chapter 16 (Sections 16.4, 16.13, and 16.15), Dover, 1965.
See also: ellipke.
Compute complete elliptic integrals of the first K(m) and second E(m) kind.
m must be a scalar or real array with -Inf ≤ m ≤ 1.
The optional input tol controls the stopping tolerance of the
algorithm and defaults to eps (class (m))
. The tolerance can
be increased to compute a faster, less accurate approximation.
When called with one output only elliptic integrals of the first kind are returned.
Mathematical Note:
Elliptic integrals of the first kind are defined as
1 / dt K (m) = | ------------------------------ / sqrt ((1 - t^2)*(1 - m^2*t^2)) 0
Elliptic integrals of the second kind are defined as
1 / sqrt (1 - m^2*t^2) E (m) = | ------------------ dt / sqrt (1 - t^2) 0
Reference: Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Chapter 17, Dover, 1965.
See also: ellipj.
Compute the error function.
The error function is defined as
z 2 / erf (z) = --------- * | e^(-t^2) dt sqrt (pi) / t=0
Compute the complementary error function.
The complementary error function is defined as
1 - erf (z)
.
Compute the scaled complementary error function.
The scaled complementary error function is defined as
exp (z^2) * erfc (z)
Compute the imaginary error function.
The imaginary error function is defined as
-i * erf (i*z)
Compute the inverse error function.
The inverse error function is defined such that
erf (y) == x
Compute the inverse complementary error function.
The inverse complementary error function is defined such that
erfc (y) == x
Compute the exponential integral:
infinity / E_1 (x) = | exp (-t)/t dt / x
Note: For compatibility, this functions uses the MATLAB definition of the exponential integral. Most other sources refer to this particular value as E_1 (x), and the exponential integral as
infinity / Ei (x) = - | exp (-t)/t dt / -x
The two definitions are related, for positive real values of x, by
E_1 (-x) = -Ei (x) - i*pi
.
Compute the Gamma function.
The Gamma function is defined as
infinity / gamma (z) = | t^(z-1) exp (-t) dt. / t=0
Programming Note: The gamma function can grow quite large even for small
input values. In many cases it may be preferable to use the natural
logarithm of the gamma function (gammaln
) in calculations to minimize
loss of precision. The final result is then
exp (result_using_gammaln).
Compute the normalized incomplete gamma function.
This is defined as
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 value n must be a real non-negative integer.
x is a vector with real-valued elements in the range [-1, 1].
The optional argument normalization may be one of "unnorm"
,
"sch"
, or "norm"
. The default if no normalization is given
is "unnorm"
.
When the optional argument normalization 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
When 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)!
When 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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