[comment {-*- tcl -*- doctools manpage}]
[manpage_begin math::special n 0.3]
[keywords {Bessel functions}]
[keywords {error function}]
[keywords math]
[keywords {special functions}]
[copyright {2004 Arjen Markus <[email protected]>}]
[moddesc {Tcl Math Library}]
[titledesc {Special mathematical functions}]
[category Mathematics]
[require Tcl [opt 8.3]]
[require math::special [opt 0.3]]
[description]
[para]
This package implements several so-called special functions, like
the Gamma function, the Bessel functions and such.
[para]
Each function is implemented by a procedure that bears its name (well,
in close approximation):
[list_begin itemized]
[item]
J0 for the zeroth-order Bessel function of the first kind
[item]
J1 for the first-order Bessel function of the first kind
[item]
Jn for the nth-order Bessel function of the first kind
[item]
J1/2 for the half-order Bessel function of the first kind
[item]
J-1/2 for the minus-half-order Bessel function of the first kind
[item]
I_n for the modified Bessel function of the first kind of order n
[item]
Gamma for the Gamma function, erf and erfc for the error function and
the complementary error function
[item]
fresnel_C and fresnel_S for the Fresnel integrals
[item]
elliptic_K and elliptic_E (complete elliptic integrals)
[item]
exponent_Ei and other functions related to the so-called exponential
integrals
[item]
legendre, hermite: some of the classical orthogonal polynomials.
[list_end]
[section 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.
[example {
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, ... | |
| 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
}]
[emph 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.
[para]
The following well-known functions are currently missing from the package:
[list_begin itemized]
[item]
Bessel functions of the second kind (Y_n, K_n)
[item]
Bessel functions of arbitrary order (and hence the Airy functions)
[item]
Chebyshev polynomials of the second kind (U_n)
[item]
The digamma function (psi)
[item]
The incomplete gamma and beta functions
[list_end]
[section "PROCEDURES"]
The package defines the following public procedures:
[list_begin definitions]
[call [cmd ::math::special::Beta] [arg x] [arg y]]
Compute the Beta function for arguments "x" and "y"
[list_begin arguments]
[arg_def float x] First argument for the Beta function
[arg_def float y] Second argument for the Beta function
[list_end]
[para]
[call [cmd ::math::special::Gamma] [arg x]]
Compute the Gamma function for argument "x"
[list_begin arguments]
[arg_def float x] Argument for the Gamma function
[list_end]
[para]
[call [cmd ::math::special::erf] [arg x]]
Compute the error function for argument "x"
[list_begin arguments]
[arg_def float x] Argument for the error function
[list_end]
[para]
[call [cmd ::math::special::erfc] [arg x]]
Compute the complementary error function for argument "x"
[list_begin arguments]
[arg_def float x] Argument for the complementary error function
[list_end]
[para]
[call [cmd ::math::special::invnorm] [arg p]]
Compute the inverse of the normal distribution function for argument "p"
[list_begin arguments]
[arg_def float p] Argument for the inverse normal distribution function
(p must be greater than 0 and lower than 1)
[list_end]
[para]
[call [cmd ::math::special::J0] [arg x]]
Compute the zeroth-order Bessel function of the first kind for the
argument "x"
[list_begin arguments]
[arg_def float x] Argument for the Bessel function
[list_end]
[call [cmd ::math::special::J1] [arg x]]
Compute the first-order Bessel function of the first kind for the
argument "x"
[list_begin arguments]
[arg_def float x] Argument for the Bessel function
[list_end]
[call [cmd ::math::special::Jn] [arg n] [arg x]]
Compute the nth-order Bessel function of the first kind for the
argument "x"
[list_begin arguments]
[arg_def integer n] Order of the Bessel function
[arg_def float x] Argument for the Bessel function
[list_end]
[call [cmd ::math::special::J1/2] [arg x]]
Compute the half-order Bessel function of the first kind for the
argument "x"
[list_begin arguments]
[arg_def float x] Argument for the Bessel function
[list_end]
[call [cmd ::math::special::J-1/2] [arg x]]
Compute the minus-half-order Bessel function of the first kind for the
argument "x"
[list_begin arguments]
[arg_def float x] Argument for the Bessel function
[list_end]
[call [cmd ::math::special::I_n] [arg x]]
Compute the modified Bessel function of the first kind of order n for
the argument "x"
[list_begin arguments]
[arg_def int x] Positive integer order of the function
[arg_def float x] Argument for the function
[list_end]
[call [cmd ::math::special::cn] [arg u] [arg k]]
Compute the elliptic function [emph cn] for the argument "u" and
parameter "k".
[list_begin arguments]
[arg_def float u] Argument for the function
[arg_def float k] Parameter
[list_end]
[call [cmd ::math::special::dn] [arg u] [arg k]]
Compute the elliptic function [emph dn] for the argument "u" and
parameter "k".
[list_begin arguments]
[arg_def float u] Argument for the function
[arg_def float k] Parameter
[list_end]
[call [cmd ::math::special::sn] [arg u] [arg k]]
Compute the elliptic function [emph sn] for the argument "u" and
parameter "k".
[list_begin arguments]
[arg_def float u] Argument for the function
[arg_def float k] Parameter
[list_end]
[call [cmd ::math::special::elliptic_K] [arg k]]
Compute the complete elliptic integral of the first kind
for the argument "k"
[list_begin arguments]
[arg_def float k] Argument for the function
[list_end]
[call [cmd ::math::special::elliptic_E] [arg k]]
Compute the complete elliptic integral of the second kind
for the argument "k"
[list_begin arguments]
[arg_def float k] Argument for the function
[list_end]
[call [cmd ::math::special::exponential_Ei] [arg x]]
Compute the exponential integral of the second kind
for the argument "x"
[list_begin arguments]
[arg_def float x] Argument for the function (x != 0)
[list_end]
[call [cmd ::math::special::exponential_En] [arg n] [arg x]]
Compute the exponential integral of the first kind
for the argument "x" and order n
[list_begin arguments]
[arg_def int n] Order of the integral (n >= 0)
[arg_def float x] Argument for the function (x >= 0)
[list_end]
[call [cmd ::math::special::exponential_li] [arg x]]
Compute the logarithmic integral for the argument "x"
[list_begin arguments]
[arg_def float x] Argument for the function (x > 0)
[list_end]
[call [cmd ::math::special::exponential_Ci] [arg x]]
Compute the cosine integral for the argument "x"
[list_begin arguments]
[arg_def float x] Argument for the function (x > 0)
[list_end]
[call [cmd ::math::special::exponential_Si] [arg x]]
Compute the sine integral for the argument "x"
[list_begin arguments]
[arg_def float x] Argument for the function (x > 0)
[list_end]
[call [cmd ::math::special::exponential_Chi] [arg x]]
Compute the hyperbolic cosine integral for the argument "x"
[list_begin arguments]
[arg_def float x] Argument for the function (x > 0)
[list_end]
[call [cmd ::math::special::exponential_Shi] [arg x]]
Compute the hyperbolic sine integral for the argument "x"
[list_begin arguments]
[arg_def float x] Argument for the function (x > 0)
[list_end]
[call [cmd ::math::special::fresnel_C] [arg x]]
Compute the Fresnel cosine integral for real argument x
[list_begin arguments]
[arg_def float x] Argument for the function
[list_end]
[call [cmd ::math::special::fresnel_S] [arg x]]
Compute the Fresnel sine integral for real argument x
[list_begin arguments]
[arg_def float x] Argument for the function
[list_end]
[call [cmd ::math::special::sinc] [arg x]]
Compute the sinc function for real argument x
[list_begin arguments]
[arg_def float x] Argument for the function
[list_end]
[call [cmd ::math::special::legendre] [arg n]]
Return the Legendre polynomial of degree n
(see [sectref "THE ORTHOGONAL POLYNOMIALS"])
[list_begin arguments]
[arg_def int n] Degree of the polynomial
[list_end]
[para]
[call [cmd ::math::special::chebyshev] [arg n]]
Return the Chebyshev polynomial of degree n (of the first kind)
[list_begin arguments]
[arg_def int n] Degree of the polynomial
[list_end]
[para]
[call [cmd ::math::special::laguerre] [arg alpha] [arg n]]
Return the Laguerre polynomial of degree n with parameter alpha
[list_begin arguments]
[arg_def float alpha] Parameter of the Laguerre polynomial
[arg_def int n] Degree of the polynomial
[list_end]
[para]
[call [cmd ::math::special::hermite] [arg n]]
Return the Hermite polynomial of degree n
[list_begin arguments]
[arg_def int n] Degree of the polynomial
[list_end]
[para]
[list_end]
[section "THE ORTHOGONAL POLYNOMIALS"]
For dealing with the classical families of orthogonal polynomials, the
package relies on the [emph math::polynomials] package. To evaluate the
polynomial at some coordinate, use the [emph evalPolyn] command:
[example {
set leg2 [::math::special::legendre 2]
puts "Value at x=$x: [::math::polynomials::evalPolyn $leg2 $x]"
}]
[para]
The return value from the [emph legendre] and other commands is actually
the definition of the corresponding polynomial as used in that package.
[section "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.
[para]
The implementation of J1/2 does not suffer from this: this function can
be expressed exactly in terms of elementary functions.
[para]
The functions J0 and J1 are the ones you will encounter most frequently
in practice.
[para]
The computation of I_n is based on Miller's algorithm for computing the
minimal function from recurrence relations.
[para]
The computation of the Gamma and Beta functions relies on the
combinatorics package, whereas that of the error functions relies on the
statistics package.
[para]
The computation of the complete elliptic integrals uses the AGM
algorithm.
[para]
Much information about these functions can be found in:
[para]
Abramowitz and Stegun: [emph "Handbook of Mathematical Functions"]
(Dover, ISBN 486-61272-4)
[vset CATEGORY {math :: special}]
[include ../doctools2base/include/feedback.inc]
[manpage_end]
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