math::special(n)               Tcl Math Library               math::special(n)



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NAME
       math::special - Special mathematical functions

SYNOPSIS
       package require Tcl  ?8.3?

       package require math::special  ?0.2?

       ::math::special::Beta x y

       ::math::special::Gamma x y

       ::math::special::erf x

       ::math::special::erfc x

       ::math::special::J0 x

       ::math::special::J1 x

       ::math::special::Jn n x

       ::math::special::J1/2 x

       ::math::special::J-1/2 x

       ::math::special::I_n x

       ::math::special::cn u k

       ::math::special::dn u k

       ::math::special::sn u k

       ::math::special::elliptic_K k

       ::math::special::elliptic_E k

       ::math::special::exponential_Ei x

       ::math::special::exponential_En n x

       ::math::special::exponential_li x

       ::math::special::exponential_Ci x

       ::math::special::exponential_Si x

       ::math::special::exponential_Chi x

       ::math::special::exponential_Shi x

       ::math::special::fresnel_C x

       ::math::special::fresnel_S x

       ::math::special::sinc x

       ::math::special::legendre n

       ::math::special::chebyshev n

       ::math::special::laguerre alpha n

       ::math::special::hermite n

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DESCRIPTION
       This package implements several so-called special functions, like the
       Gamma function, the Bessel functions and such.

       Each function is implemented by a procedure that bears its name (well,
       in close approximation):

       o      J0 for the zeroth-order Bessel function of the first kind

       o      J1 for the first-order Bessel function of the first kind

       o      Jn for the nth-order Bessel function of the first kind

       o      J1/2 for the half-order Bessel function of the first kind

       o      J-1/2 for the minus-half-order Bessel function of the first kind

       o      I_n for the modified Bessel function of the first kind of order
              n

       o      Gamma for the Gamma function, erf and erfc for the error
              function and the complementary error function

       o      fresnel_C and fresnel_S for the Fresnel integrals

       o      elliptic_K and elliptic_E (complete elliptic integrals)

       o      exponent_Ei and other functions related to the so-called
              exponential integrals

       o      legendre, hermite: some of the classical orthogonal polynomials.

OVERVIEW
       In the following table several characteristics of the functions in this
       package are summarized: the domain for the argument, the values for the
       parameters and error bounds.


              Family       | Function    | Domain x    | Parameter   | Error bound
              -------------+-------------+-------------+-------------+--------------
              Bessel       | J0, J1,     | all of R    | n = integer |   < 1.0e-8
                           | Jn          |             |             |  (|x|<20, n<20)
              Bessel       | J1/2, J-1/2,|  x > 0      | n = integer |   exact
              Bessel       | I_n         | all of R    | n = integer |   < 1.0e-6
                           |             |             |             |
              Elliptic     | cn          | 0 <= x <= 1 |     --      |   < 1.0e-10
              functions    | dn          | 0 <= x <= 1 |     --      |   < 1.0e-10
                           | sn          | 0 <= x <= 1 |     --      |   < 1.0e-10
              Elliptic     | K           | 0 <= x < 1  |     --      |   < 1.0e-6
              integrals    | E           | 0 <= x < 1  |     --      |   < 1.0e-6
                           |             |             |             |
              Error        | erf         |             |     --      |
              functions    | erfc        |             |             |
                           | ierfc_n     |             |             |
                           |             |             |             |
              Exponential  | Ei          |  x != 0     |     --      |   < 1.0e-10 (relative)
              integrals    | En          |  x >  0     |     --      |   as Ei
                           | li          |  x > 0      |     --      |   as Ei
                           | Chi         |  x > 0      |     --      |   < 1.0e-8
                           | Shi         |  x > 0      |     --      |   < 1.0e-8
                           | Ci          |  x > 0      |     --      |   < 2.0e-4
                           | Si          |  x > 0      |     --      |   < 2.0e-4
                           |             |             |             |
              Fresnel      | C           |  all of R   |     --      |   < 2.0e-3
              integrals    | S           |  all of R   |     --      |   < 2.0e-3
                           |             |             |             |
              general      | Beta        | (see Gamma) |     --      |   < 1.0e-9
                           | Gamma       |  x != 0,-1, |     --      |   < 1.0e-9
                           |             |  -2, ...    |             |
                           | sinc        |  all of R   |     --      |   exact
                           |             |             |             |
              orthogonal   | Legendre    |  all of R   | n = 0,1,... |   exact
              polynomials  | Chebyshev   |  all of R   | n = 0,1,... |   exact
                           | Laguerre    |  all of R   | n = 0,1,... |   exact
                           |             |             | alpha el. R |
                           | Hermite     |  all of R   | n = 0,1,... |   exact

       Note: Some of the error bounds are estimated, as no "formal" bounds
       were available with the implemented approximation method, others hold
       for the auxiliary functions used for estimating the primary functions.

       The following well-known functions are currently missing from the
       package:

       o      Bessel functions of the second kind (Y_n, K_n)

       o      Bessel functions of arbitrary order (and hence the Airy
              functions)

       o      Chebyshev polynomials of the second kind (U_n)

       o      The digamma function (psi)

       o      The incomplete gamma and beta functions

PROCEDURES
       The package defines the following public procedures:

       ::math::special::Beta x y
              Compute the Beta function for arguments "x" and "y"

              float x
                     First argument for the Beta function

              float y
                     Second argument for the Beta function


       ::math::special::Gamma x y
              Compute the Gamma function for argument "x"

              float x
                     Argument for the Gamma function


       ::math::special::erf x
              Compute the error function for argument "x"

              float x
                     Argument for the error function


       ::math::special::erfc x
              Compute the complementary error function for argument "x"

              float x
                     Argument for the complementary error function


       ::math::special::J0 x
              Compute the zeroth-order Bessel function of the first kind for
              the argument "x"

              float x
                     Argument for the Bessel function

       ::math::special::J1 x
              Compute the first-order Bessel function of the first kind for
              the argument "x"

              float x
                     Argument for the Bessel function

       ::math::special::Jn n x
              Compute the nth-order Bessel function of the first kind for the
              argument "x"

              integer n
                     Order of the Bessel function

              float x
                     Argument for the Bessel function

       ::math::special::J1/2 x
              Compute the half-order Bessel function of the first kind for the
              argument "x"

              float x
                     Argument for the Bessel function

       ::math::special::J-1/2 x
              Compute the minus-half-order Bessel function of the first kind
              for the argument "x"

              float x
                     Argument for the Bessel function

       ::math::special::I_n x
              Compute the modified Bessel function of the first kind of order
              n for the argument "x"

              int x  Positive integer order of the function

              float x
                     Argument for the function

       ::math::special::cn u k
              Compute the elliptic function cn for the argument "u" and
              parameter "k".

              float u
                     Argument for the function

              float k
                     Parameter

       ::math::special::dn u k
              Compute the elliptic function dn for the argument "u" and
              parameter "k".

              float u
                     Argument for the function

              float k
                     Parameter

       ::math::special::sn u k
              Compute the elliptic function sn for the argument "u" and
              parameter "k".

              float u
                     Argument for the function

              float k
                     Parameter

       ::math::special::elliptic_K k
              Compute the complete elliptic integral of the first kind for the
              argument "k"

              float k
                     Argument for the function

       ::math::special::elliptic_E k
              Compute the complete elliptic integral of the second kind for
              the argument "k"

              float k
                     Argument for the function

       ::math::special::exponential_Ei x
              Compute the exponential integral of the second kind for the
              argument "x"

              float x
                     Argument for the function (x != 0)

       ::math::special::exponential_En n x
              Compute the exponential integral of the first kind for the
              argument "x" and order n

              int n  Order of the integral (n >= 0)

              float x
                     Argument for the function (x >= 0)

       ::math::special::exponential_li x
              Compute the logarithmic integral for the argument "x"

              float x
                     Argument for the function (x > 0)

       ::math::special::exponential_Ci x
              Compute the cosine integral for the argument "x"

              float x
                     Argument for the function (x > 0)

       ::math::special::exponential_Si x
              Compute the sine integral for the argument "x"

              float x
                     Argument for the function (x > 0)

       ::math::special::exponential_Chi x
              Compute the hyperbolic cosine integral for the argument "x"

              float x
                     Argument for the function (x > 0)

       ::math::special::exponential_Shi x
              Compute the hyperbolic sine integral for the argument "x"

              float x
                     Argument for the function (x > 0)

       ::math::special::fresnel_C x
              Compute the Fresnel cosine integral for real argument x

              float x
                     Argument for the function

       ::math::special::fresnel_S x
              Compute the Fresnel sine integral for real argument x

              float x
                     Argument for the function

       ::math::special::sinc x
              Compute the sinc function for real argument x

              float x
                     Argument for the function

       ::math::special::legendre n
              Return the Legendre polynomial of degree n (see THE ORTHOGONAL
              POLYNOMIALS)

              int n  Degree of the polynomial


       ::math::special::chebyshev n
              Return the Chebyshev polynomial of degree n (of the first kind)

              int n  Degree of the polynomial


       ::math::special::laguerre alpha n
              Return the Laguerre polynomial of degree n with parameter alpha

              float alpha
                     Parameter of the Laguerre polynomial

              int n  Degree of the polynomial


       ::math::special::hermite n
              Return the Hermite polynomial of degree n

              int n  Degree of the polynomial


THE ORTHOGONAL POLYNOMIALS
       For dealing with the classical families of orthogonal polynomials, the
       package relies on the math::polynomials package. To evaluate the
       polynomial at some coordinate, use the evalPolyn command:


                 set leg2 [::math::special::legendre 2]
                 puts "Value at x=$x: [::math::polynomials::evalPolyn $leg2 $x]"


       The return value from the legendre and other commands is actually the
       definition of the corresponding polynomial as used in that package.

REMARKS ON THE IMPLEMENTATION
       It should be noted, that the actual implementation of J0 and J1 depends
       on straightforward Gaussian quadrature formulas. The (absolute)
       accuracy of the results is of the order 1.0e-4 or better. The main
       reason to implement them like that was that it was fast to do (the
       formulas are simple) and the computations are fast too.

       The implementation of J1/2 does not suffer from this: this function can
       be expressed exactly in terms of elementary functions.

       The functions J0 and J1 are the ones you will encounter most frequently
       in practice.

       The computation of I_n is based on Miller's algorithm for computing the
       minimal function from recurrence relations.

       The computation of the Gamma and Beta functions relies on the
       combinatorics package, whereas that of the error functions relies on
       the statistics package.

       The computation of the complete elliptic integrals uses the AGM
       algorithm.

       Much information about these functions can be found in:

       Abramowitz and Stegun: Handbook of Mathematical Functions (Dover, ISBN
       486-61272-4)

BUGS, IDEAS, FEEDBACK
       This document, and the package it describes, will undoubtedly contain
       bugs and other problems.  Please report such in the category math ::
       special of the Tcllib SF Trackers
       [http://sourceforge.net/tracker/?group_id=12883].  Please also report
       any ideas for enhancements you may have for either package and/or
       documentation.

KEYWORDS
       Bessel functions, error function, math, special functions

CATEGORY
       Mathematics

COPYRIGHT
       Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>




math                                  0.2                     math::special(n)
