Provided by: tcllib_1.21+dfsg-1_all 
NAME
math::special - Special mathematical functions
SYNOPSIS
package require Tcl ?8.5?
package require math::special ?0.5?
::math::special::eulerNumber index
::math::special::bernoulliNumber index
::math::special::Beta x y
::math::special::incBeta a b x
::math::special::regIncBeta a b x
::math::special::Gamma x
::math::special::digamma x
::math::special::erf x
::math::special::erfc x
::math::special::invnorm p
::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
________________________________________________________________________________________________________________
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):
• J0 for the zeroth-order Bessel function of the first kind
• J1 for the first-order Bessel function of the first kind
• Jn for the nth-order Bessel function of the first kind
• J1/2 for the half-order Bessel function of the first kind
• J-1/2 for the minus-half-order Bessel function of the first kind
• I_n for the modified Bessel function of the first kind of order n
• Gamma for the Gamma function, erf and erfc for the error function and the complementary error
function
• fresnel_C and fresnel_S for the Fresnel integrals
• elliptic_K and elliptic_E (complete elliptic integrals)
• exponent_Ei and other functions related to the so-called exponential integrals
• 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 | | |
| | | |
Inverse | invnorm | 0 < x < 1 | -- | < 1.2e-9
normal | | | |
distribution | | | |
| | | |
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, ... | |
| incBeta | | a, b > 0 | < 1.0e-9
| regIncBeta | | a, b > 0 | < 1.0e-9
| digamma | 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:
• Bessel functions of the second kind (Y_n, K_n)
• Bessel functions of arbitrary order (and hence the Airy functions)
• Chebyshev polynomials of the second kind (U_n)
• The incomplete gamma function
PROCEDURES
The package defines the following public procedures:
::math::special::eulerNumber index
Return the index'th Euler number (note: these are integer values). As the size of these numbers
grows very fast, only a limited number are available.
int index
Index of the number to be returned (should be between 0 and 54)
::math::special::bernoulliNumber index
Return the index'th Bernoulli number. As the size of the numbers grows very fast, only a limited
number are available.
int index
Index of the number to be returned (should be between 0 and 52)
::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::incBeta a b x
Compute the incomplete Beta function for argument "x" with parameters "a" and "b"
float a
First parameter for the incomplete Beta function, a > 0
float b
Second parameter for the incomplete Beta function, b > 0
float x
Argument for the incomplete Beta function
::math::special::regIncBeta a b x
Compute the regularized incomplete Beta function for argument "x" with parameters "a" and "b"
float a
First parameter for the incomplete Beta function, a > 0
float b
Second parameter for the incomplete Beta function, b > 0
float x
Argument for the regularized incomplete Beta function
::math::special::Gamma x
Compute the Gamma function for argument "x"
float x
Argument for the Gamma function
::math::special::digamma x
Compute the digamma function (psi) for argument "x"
float x
Argument for the digamma 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::invnorm p
Compute the inverse of the normal distribution function for argument "p"
float p
Argument for the inverse normal distribution function (p must be greater than 0 and lower
than 1)
::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 Trackers
[http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for enhancements you may have for
either package and/or documentation.
When proposing code changes, please provide unified diffs, i.e the output of diff -u.
Note further that attachments are strongly preferred over inlined patches. Attachments can be made by
going to the Edit form of the ticket immediately after its creation, and then using the left-most button
in the secondary navigation bar.
KEYWORDS
Bessel functions, error function, math, special functions
CATEGORY
Mathematics
COPYRIGHT
Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>