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17.6 Special Functions

Built-in Function: [a, ierr] = airy (k, z, opt)

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.

  1. Normal return.
  2. Input error, return NaN.
  3. Overflow, return Inf.
  4. Loss of significance by argument reduction results in less than half of machine accuracy.
  5. Complete loss of significance by argument reduction, return NaN.
  6. Error—no computation, algorithm termination condition not met, return NaN.
Built-in Function: [j, ierr] = besselj (alpha, x, opt)
Built-in Function: [y, ierr] = bessely (alpha, x, opt)
Built-in Function: [i, ierr] = besseli (alpha, x, opt)
Built-in Function: [k, ierr] = besselk (alpha, x, opt)
Built-in 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 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.

  1. Normal return.
  2. Input error, return NaN.
  3. Overflow, return Inf.
  4. Loss of significance by argument reduction results in less than half of machine accuracy.
  5. Complete loss of significance by argument reduction, return NaN.
  6. Error—no computation, algorithm termination condition not met, return NaN.
Mapping Function: beta (a, b)

For real inputs, return the Beta function, 

beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).

See also: betaln, betainc.

Mapping Function: betainc (x, 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.

See also: betaincinv, beta, betaln.

Mapping Function: betaincinv (y, a, b)

Compute the inverse of the incomplete Beta function, i.e., x such that 

y == betainc (x, a, b)

See also: betainc, beta, betaln.

Mapping Function: betaln (a, b)

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.

See also: beta, betainc, gammaln.

Mapping Function: bincoeff (n, k)

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.

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.

Mapping Function: dawson (z)

Compute the Dawson (scaled imaginary error) function, 

(sqrt (pi) / 2) * exp (-z^2) * erfi (z)

See also: erfc, erf, erfcx, erfi, erfinv, erfcinv.

Built-in Function: [sn, cn, dn, err] = ellipj (u, m)
Built-in Function: [sn, cn, dn, err] = ellipj (u, m, tol)

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.

  1. Normal return.
  2. Error—no computation, algorithm termination condition not met, return 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.

Function File: k = ellipke (m)
Function File: k = ellipke (m, tol)
Function File: [k, e] = 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 is currently ignored (MATLAB uses this to allow a faster, less accurate approximation).

Called with only one output, elliptic integrals of the first kind are returned.

Reference: Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Chapter 17, Dover, 1965.

See also: ellipj.

Mapping Function: erf (z)

Compute the error function, 

                        z
              2        /
erf (z) = --------- *  | e^(-t^2) dt
          sqrt (pi)    /
                    t=0

See also: erfc, erfcx, erfi, dawson, erfinv, erfcinv.

Mapping Function: erfc (z)

Compute the complementary error function, 1 - erf (z).

See also: erfcinv, erfcx, erfi, dawson, erf, erfinv.

Mapping Function: erfcx (z)

Compute the scaled complementary error function, 

exp (z^2) * erfc (z)

See also: erfc, erf, erfi, dawson, erfinv, erfcinv.

Mapping Function: erfi (z)

Compute the imaginary error function, 

-i * erf (i*z)

See also: erfc, erf, erfcx, dawson, erfinv, erfcinv.

Mapping Function: erfinv (x)

Compute the inverse error function, i.e., y such that 

erf (y) == x

See also: erf, erfc, erfcx, erfi, dawson, erfcinv.

Mapping Function: erfcinv (x)

Compute the inverse complementary error function, i.e., y such that 

erfc (y) == x

See also: erfc, erf, erfcx, erfi, dawson, erfinv.

Function File: expint (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 is 

            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.

Mapping Function: gamma (z)

Compute the Gamma function, 

             infinity
            /
gamma (z) = | t^(z-1) exp (-t) dt.
            /
         t=0

See also: gammainc, lgamma.

Mapping Function: gammainc (x, a)
Mapping Function: gammainc (x, a, "lower")
Mapping Function: gammainc (x, a, "upper")

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")

See also: gamma, lgamma.

Function File: l = legendre (n, x)
Function File: l = legendre (n, x, normalization)

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)!
Mapping Function: lgamma (x)
Mapping Function: gammaln (x)

Return the natural logarithm of the gamma function of x.

See also: gamma, gammainc.

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