1math::special(n) Tcl Math Library math::special(n)
2______________________________________________________________________________
6
8 math::special - Special mathematical functions
9
11 package require Tcl ?8.3?
12 package require math::special ?0.2?
14 ::math::special::Beta x y
16 ::math::special::Gamma x y
18 ::math::special::erf x
20 ::math::special::erfc x
22 ::math::special::J0 x
24 ::math::special::J1 x
26 ::math::special::Jn n x
28 ::math::special::J1/2 x
30 ::math::special::J-1/2 x
32 ::math::special::I_n x
34 ::math::special::cn u k
36 ::math::special::dn u k
38 ::math::special::sn u k
40 ::math::special::elliptic_K k
42 ::math::special::elliptic_E k
44 ::math::special::exponential_Ei x
46 ::math::special::exponential_En n x
48 ::math::special::exponential_li x
50 ::math::special::exponential_Ci x
52 ::math::special::exponential_Si x
54 ::math::special::exponential_Chi x
56 ::math::special::exponential_Shi x
58 ::math::special::fresnel_C x
60 ::math::special::fresnel_S x
62 ::math::special::sinc x
64 ::math::special::legendre n
66 ::math::special::chebyshev n
68 ::math::special::laguerre alpha n
70 ::math::special::hermite n
72_________________________________________________________________
74
76 This package implements several so-called special functions, like the
77 Gamma function, the Bessel functions and such.
78 Each function is implemented by a procedure that bears its name (well,
80 in close approximation):
81 · J0 for the zeroth-order Bessel function of the first kind
83 · J1 for the first-order Bessel function of the first kind
85 · Jn for the nth-order Bessel function of the first kind
87 · J1/2 for the half-order Bessel function of the first kind
89 · J-1/2 for the minus-half-order Bessel function of the first kind
91 · I_n for the modified Bessel function of the first kind of order
93 n
94 · Gamma for the Gamma function, erf and erfc for the error func‐
96 tion and the complementary error function
97 · fresnel_C and fresnel_S for the Fresnel integrals
99 · elliptic_K and elliptic_E (complete elliptic integrals)
101 · exponent_Ei and other functions related to the so-called expo‐
103 nential integrals
104 · legendre, hermite: some of the classical orthogonal polynomials.
106
108 In the following table several characteristics of the functions in this
109 package are summarized: the domain for the argument, the values for the
110 parameters and error bounds.
111 Family | Function | Domain x | Parameter | Error bound
113 -------------+-------------+-------------+-------------+--------------
114 Bessel | J0, J1, | all of R | n = integer | < 1.0e-8
115 | Jn | | | (|x|<20, n<20)
116 Bessel | J1/2, J-1/2,| x > 0 | n = integer | exact
117 Bessel | I_n | all of R | n = integer | < 1.0e-6
118 | | | |
119 Elliptic | cn | 0 <= x <= 1 | -- | < 1.0e-10
120 functions | dn | 0 <= x <= 1 | -- | < 1.0e-10
121 | sn | 0 <= x <= 1 | -- | < 1.0e-10
122 Elliptic | K | 0 <= x < 1 | -- | < 1.0e-6
123 integrals | E | 0 <= x < 1 | -- | < 1.0e-6
124 | | | |
125 Error | erf | | -- |
126 functions | erfc | | |
127 | ierfc_n | | |
128 | | | |
129 Exponential | Ei | x != 0 | -- | < 1.0e-10 (relative)
130 integrals | En | x > 0 | -- | as Ei
131 | li | x > 0 | -- | as Ei
132 | Chi | x > 0 | -- | < 1.0e-8
133 | Shi | x > 0 | -- | < 1.0e-8
134 | Ci | x > 0 | -- | < 2.0e-4
135 | Si | x > 0 | -- | < 2.0e-4
136 | | | |
137 Fresnel | C | all of R | -- | < 2.0e-3
138 integrals | S | all of R | -- | < 2.0e-3
139 | | | |
140 general | Beta | (see Gamma) | -- | < 1.0e-9
141 | Gamma | x != 0,-1, | -- | < 1.0e-9
142 | | -2, ... | |
143 | sinc | all of R | -- | exact
144 | | | |
145 orthogonal | Legendre | all of R | n = 0,1,... | exact
146 polynomials | Chebyshev | all of R | n = 0,1,... | exact
147 | Laguerre | all of R | n = 0,1,... | exact
148 | | | alpha el. R |
149 | Hermite | all of R | n = 0,1,... | exact
150 Note: Some of the error bounds are estimated, as no "formal" bounds
152 were available with the implemented approximation method, others hold
153 for the auxiliary functions used for estimating the primary functions.
154 The following well-known functions are currently missing from the pack‐
156 age:
157 · Bessel functions of the second kind (Y_n, K_n)
159 · Bessel functions of arbitrary order (and hence the Airy func‐
161 tions)
162 · Chebyshev polynomials of the second kind (U_n)
164 · The digamma function (psi)
166 · The incomplete gamma and beta functions
168
170 The package defines the following public procedures:
171 ::math::special::Beta x y
173 Compute the Beta function for arguments "x" and "y"
174 x float First argument for the Beta function
176 y float Second argument for the Beta function
178 ::math::special::Gamma x y
181 Compute the Gamma function for argument "x"
182 x float Argument for the Gamma function
184 ::math::special::erf x
187 Compute the error function for argument "x"
188 x float Argument for the error function
190 ::math::special::erfc x
193 Compute the complementary error function for argument "x"
194 x float Argument for the complementary error function
196 ::math::special::J0 x
199 Compute the zeroth-order Bessel function of the first kind for
200 the argument "x"
201 x float Argument for the Bessel function
203 ::math::special::J1 x
205 Compute the first-order Bessel function of the first kind for
206 the argument "x"
207 x float Argument for the Bessel function
209 ::math::special::Jn n x
211 Compute the nth-order Bessel function of the first kind for the
212 argument "x"
213 n integer Order of the Bessel function
215 x float Argument for the Bessel function
217 ::math::special::J1/2 x
219 Compute the half-order Bessel function of the first kind for the
220 argument "x"
221 x float Argument for the Bessel function
223 ::math::special::J-1/2 x
225 Compute the minus-half-order Bessel function of the first kind
226 for the argument "x"
227 x float Argument for the Bessel function
229 ::math::special::I_n x
231 Compute the modified Bessel function of the first kind of order
232 n for the argument "x"
233 x int Positive integer order of the function
235 x float Argument for the function
237 ::math::special::cn u k
239 Compute the elliptic function cn for the argument "u" and param‐
240 eter "k".
241 u float Argument for the function
243 k float Parameter
245 ::math::special::dn u k
247 Compute the elliptic function dn for the argument "u" and param‐
248 eter "k".
249 u float Argument for the function
251 k float Parameter
253 ::math::special::sn u k
255 Compute the elliptic function sn for the argument "u" and param‐
256 eter "k".
257 u float Argument for the function
259 k float Parameter
261 ::math::special::elliptic_K k
263 Compute the complete elliptic integral of the first kind for the
264 argument "k"
265 k float Argument for the function
267 ::math::special::elliptic_E k
269 Compute the complete elliptic integral of the second kind for
270 the argument "k"
271 k float Argument for the function
273 ::math::special::exponential_Ei x
275 Compute the exponential integral of the second kind for the
276 argument "x"
277 x float Argument for the function (x != 0)
279 ::math::special::exponential_En n x
281 Compute the exponential integral of the first kind for the argu‐
282 ment "x" and order n
283 n int Order of the integral (n >= 0)
285 x float Argument for the function (x >= 0)
287 ::math::special::exponential_li x
289 Compute the logarithmic integral for the argument "x"
290 x float Argument for the function (x > 0)
292 ::math::special::exponential_Ci x
294 Compute the cosine integral for the argument "x"
295 x float Argument for the function (x > 0)
297 ::math::special::exponential_Si x
299 Compute the sine integral for the argument "x"
300 x float Argument for the function (x > 0)
302 ::math::special::exponential_Chi x
304 Compute the hyperbolic cosine integral for the argument "x"
305 x float Argument for the function (x > 0)
307 ::math::special::exponential_Shi x
309 Compute the hyperbolic sine integral for the argument "x"
310 x float Argument for the function (x > 0)
312 ::math::special::fresnel_C x
314 Compute the Fresnel cosine integral for real argument x
315 x float Argument for the function
317 ::math::special::fresnel_S x
319 Compute the Fresnel sine integral for real argument x
320 x float Argument for the function
322 ::math::special::sinc x
324 Compute the sinc function for real argument x
325 x float Argument for the function
327 ::math::special::legendre n
329 Return the Legendre polynomial of degree n (see THE ORTHOGONAL
330 POLYNOMIALS)
331 n int Degree of the polynomial
333 ::math::special::chebyshev n
336 Return the Chebyshev polynomial of degree n (of the first kind)
337 n int Degree of the polynomial
339 ::math::special::laguerre alpha n
342 Return the Laguerre polynomial of degree n with parameter alpha
343 alpha float Parameter of the Laguerre polynomial
345 n int Degree of the polynomial
347 ::math::special::hermite n
350 Return the Hermite polynomial of degree n
351 n int Degree of the polynomial
353
356 For dealing with the classical families of orthogonal polynomials, the
357 package relies on the math::polynomials package. To evaluate the poly‐
358 nomial at some coordinate, use the evalPolyn command:
359 set leg2 [::math::special::legendre 2]
361 puts "Value at x=$x: [::math::polynomials::evalPolyn $leg2 $x]"
362 The return value from the legendre and other commands is actually the
365 definition of the corresponding polynomial as used in that package.
366
368 It should be noted, that the actual implementation of J0 and J1 depends
369 on straightforward Gaussian quadrature formulas. The (absolute) accu‐
370 racy of the results is of the order 1.0e-4 or better. The main reason
371 to implement them like that was that it was fast to do (the formulas
372 are simple) and the computations are fast too.
373 The implementation of J1/2 does not suffer from this: this function can
375 be expressed exactly in terms of elementary functions.
376 The functions J0 and J1 are the ones you will encounter most frequently
378 in practice.
379 The computation of I_n is based on Miller's algorithm for computing the
381 minimal function from recurrence relations.
382 The computation of the Gamma and Beta functions relies on the combina‐
384 torics package, whereas that of the error functions relies on the sta‐
385 tistics package.
386 The computation of the complete elliptic integrals uses the AGM algo‐
388 rithm.
389 Much information about these functions can be found in:
391 Abramowitz and Stegun: Handbook of Mathematical Functions (Dover, ISBN
393 486-61272-4)
394
396 Bessel functions, error function, math, special functions
397
399 Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
400math 0.2 math::special(n)