| 1 | #***************************************************************************** |
| 2 | # Copyright (C) 2010 Oscar Gerardo Lazo Arjona algebraicamente@gmail.com |
| 3 | # |
| 4 | # Distributed under the terms of the GNU General Public License (GPL) |
| 5 | # |
| 6 | # but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 7 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| 8 | # General Public License for more details. |
| 9 | # |
| 10 | # The full text of the GPL is available at: |
| 11 | # |
| 12 | # http://www.gnu.org/licenses/ |
| 13 | #***************************************************************************** |
| 14 | |
| 15 | from sage.symbolic.function import BuiltinFunction |
| 16 | from sage.symbolic.expression import Expression |
| 17 | from sage.symbolic.function import is_inexact |
| 18 | from sage.structure.coerce import parent as sage_structure_coerce_parent |
| 19 | |
| 20 | from sage.functions.other import gamma |
| 21 | from sage.rings.integer_ring import ZZ |
| 22 | from sage.rings.real_double import RDF |
| 23 | from sage.functions.special import meval |
| 24 | from sage.calculus.var import var |
| 25 | from sage.calculus.functional import derivative |
| 26 | |
| 27 | # from sage.rings.real_mpfr import RR |
| 28 | from sage.rings.rational import Rational as R |
| 29 | |
| 30 | class FunctionAiryAiGeneral(BuiltinFunction): |
| 31 | def __init__(self): |
| 32 | r""" |
| 33 | The generalized derivative of the Airy Ai function |
| 34 | |
| 35 | INPUT: |
| 36 | |
| 37 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to `z`. |
| 38 | For `\alpha = n = 1,2,3,\ldots` this gives the derivative `\operatorname{Ai}^{(n)}(z)`, |
| 39 | and for `\alpha = -n = -1,-2,-3,\ldots` this gives the `n`-fold iterated integral |
| 40 | |
| 41 | .. math :: |
| 42 | |
| 43 | f_0(z) = \operatorname{Ai}(z) |
| 44 | |
| 45 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 46 | |
| 47 | - ``x``.- The argument of the function. |
| 48 | |
| 49 | - ``hold``.- Whether or not to stop from returning higher derivatives in terms of |
| 50 | `\operatorname{Ai}(x)` and `\operatorname{Ai}'(x)`. |
| 51 | |
| 52 | EXAMPLES:: |
| 53 | |
| 54 | sage: x,n=var('x n') |
| 55 | sage: airy_ai(-2,x) |
| 56 | airy_ai(-2, x) |
| 57 | sage: derivative(airy_ai(-2,x),x) |
| 58 | airy_ai(-1, x) |
| 59 | sage: airy_ai(n,x) |
| 60 | airy_ai(n, x) |
| 61 | sage: derivative(airy_ai(n,x),x) |
| 62 | airy_ai(n + 1, x) |
| 63 | sage: airy_ai(2,x,True) |
| 64 | airy_ai(2, x) |
| 65 | sage: derivative(airy_ai(2,x,hold_derivative=True),x) |
| 66 | airy_ai(3, x) |
| 67 | """ |
| 68 | |
| 69 | BuiltinFunction.__init__(self, "airy_ai", nargs=2, |
| 70 | latex_name=r"\operatorname{Ai}") |
| 71 | |
| 72 | def _derivative_(self, alpha, *args, **kwds): |
| 73 | """ |
| 74 | EXAMPLES:: |
| 75 | |
| 76 | sage: x,n=var('x n') |
| 77 | sage: derivative(airy_ai(n,x),x) # indirect doctest |
| 78 | airy_ai(n + 1, x) |
| 79 | """ |
| 80 | |
| 81 | x=args[0] |
| 82 | return airy_ai_general(alpha+1,x) |
| 83 | |
| 84 | def _eval_(self, alpha, *args): |
| 85 | """ |
| 86 | EXAMPLES:: |
| 87 | |
| 88 | sage: x,n=var('x n') |
| 89 | sage: airy_ai(-2,1.0) # indirect doctest |
| 90 | 0.136645379421096 |
| 91 | sage: airy_ai(n,1.0) # indirect doctest |
| 92 | airy_ai(n, 1.00000000000000) |
| 93 | """ |
| 94 | |
| 95 | x=args[0] |
| 96 | if not isinstance(x,Expression) and not isinstance(alpha,Expression): |
| 97 | if is_inexact(x): |
| 98 | return self._evalf_(alpha,x) |
| 99 | else: |
| 100 | return None |
| 101 | |
| 102 | def _evalf_(self, alpha, x, **kwargs): |
| 103 | """ |
| 104 | EXAMPLES:: |
| 105 | |
| 106 | sage: airy_ai(-2,1.0) # indirect doctest |
| 107 | 0.136645379421096 |
| 108 | """ |
| 109 | algorithm = kwargs.get('algorithm', None) or 'mpmath' |
| 110 | parent = kwargs.get('parent', None) or sage_structure_coerce_parent(x) |
| 111 | prec = parent.prec() if hasattr(parent, 'prec') else 53 |
| 112 | if algorithm == 'mpmath': |
| 113 | import mpmath |
| 114 | from sage.libs.mpmath import utils as mpmath_utils |
| 115 | return mpmath_utils.call(mpmath.airyai, x, derivative=alpha, parent=parent) |
| 116 | elif algorithm == 'maxima': |
| 117 | raise NotImplementedError("general case not available in maxima") |
| 118 | else: |
| 119 | raise ValueError("unknown algorithm") |
| 120 | |
| 121 | class FunctionAiryAiSimple(BuiltinFunction): |
| 122 | def __init__(self): |
| 123 | """ |
| 124 | The class for the Airy Ai function. |
| 125 | |
| 126 | EXAMPLES:: |
| 127 | |
| 128 | sage: from sage.functions.airy import FunctionAiryAiSimple |
| 129 | sage: airy_ai_simple= FunctionAiryAiSimple() |
| 130 | sage: f=airy_ai_simple(x); f |
| 131 | airy_ai(x) |
| 132 | """ |
| 133 | |
| 134 | BuiltinFunction.__init__(self, "airy_ai", latex_name=r'\operatorname{Ai}', |
| 135 | conversions=dict(mathematica='AiryAi', maxima='airy_ai')) |
| 136 | |
| 137 | def _derivative_(self, x, diff_param=None): |
| 138 | """ |
| 139 | EXAMPLES:: |
| 140 | |
| 141 | sage: derivative(airy_ai(x),x) # indirect doctest |
| 142 | airy_ai_prime(x) |
| 143 | """ |
| 144 | return airy_ai_prime(x) |
| 145 | |
| 146 | def _eval_(self, x): |
| 147 | """ |
| 148 | EXAMPLES:: |
| 149 | |
| 150 | sage: airy_ai(0) # indirect doctest |
| 151 | 1/3*3^(1/3)/gamma(2/3) |
| 152 | sage: airy_ai(0.0) # indirect doctest |
| 153 | 0.355028053887817 |
| 154 | sage: airy_ai(I) # indirect doctest |
| 155 | airy_ai(I) |
| 156 | sage: airy_ai(1.0*I) # indirect doctest |
| 157 | 0.331493305432141 - 0.317449858968444*I |
| 158 | """ |
| 159 | |
| 160 | if not isinstance(x,Expression): |
| 161 | if is_inexact(x): |
| 162 | return self._evalf_(x) |
| 163 | elif x==0: |
| 164 | r=ZZ(2)/3 |
| 165 | return 1/(3**(r)*gamma(r)) |
| 166 | else: |
| 167 | return None |
| 168 | |
| 169 | def _evalf_(self, x, **kwargs): |
| 170 | """ |
| 171 | EXAMPLES:: |
| 172 | |
| 173 | sage: airy_ai(0.0) # indirect doctest |
| 174 | 0.355028053887817 |
| 175 | sage: airy_ai(1.0*I) # indirect doctest |
| 176 | 0.331493305432141 - 0.317449858968444*I |
| 177 | |
| 178 | We can use several methods for numerical evaluation:: |
| 179 | |
| 180 | sage: airy_ai(3).n(algorithm='maxima') |
| 181 | 0.00659113935746 |
| 182 | sage: airy_ai(3).n(algorithm='mpmath') |
| 183 | 0.00659113935746072 |
| 184 | sage: airy_ai(3).n(algorithm='mpmath',prec=100) |
| 185 | 0.0065911393574607191442574484080 |
| 186 | |
| 187 | TESTS:: |
| 188 | |
| 189 | sage: airy_ai(3).n(algorithm='maxima', prec=70) |
| 190 | Traceback (most recent call last): |
| 191 | ... |
| 192 | ValueError: for the maxima algorithm the precision must be 53 |
| 193 | |
| 194 | """ |
| 195 | algorithm = kwargs.get('algorithm', None) or 'mpmath' |
| 196 | parent = kwargs.get('parent', None) or sage_structure_coerce_parent(x) |
| 197 | prec = parent.prec() if hasattr(parent, 'prec') else 53 |
| 198 | if algorithm == 'mpmath': |
| 199 | import mpmath |
| 200 | from sage.libs.mpmath import utils as mpmath_utils |
| 201 | return mpmath_utils.call(mpmath.airyai, x, parent=parent) |
| 202 | elif algorithm == 'maxima': |
| 203 | if prec != 53: |
| 204 | raise ValueError, "for the maxima algorithm the precision must be 53" |
| 205 | return RDF(meval("airy_ai(%s)" % RDF(x))) |
| 206 | else: |
| 207 | raise ValueError("unknown algorithm") |
| 208 | |
| 209 | class FunctionAiryAiPrime(BuiltinFunction): |
| 210 | def __init__(self): |
| 211 | """ |
| 212 | The derivative of the Airy Ai function; see :func:`airy_ai` |
| 213 | for the full documentation. |
| 214 | |
| 215 | EXAMPLES:: |
| 216 | |
| 217 | sage: x,n=var('x n') |
| 218 | sage: airy_ai_prime(x) |
| 219 | airy_ai_prime(x) |
| 220 | sage: airy_ai_prime(0) |
| 221 | -1/3*3^(2/3)/gamma(1/3) |
| 222 | """ |
| 223 | |
| 224 | BuiltinFunction.__init__(self, "airy_ai_prime", |
| 225 | latex_name=r"\operatorname{Ai}'", |
| 226 | conversions=dict(mathematica='AiryAiPrime', maxima='airy_dai')) |
| 227 | |
| 228 | def _derivative_(self, x, diff_param=None): |
| 229 | """ |
| 230 | EXAMPLES:: |
| 231 | |
| 232 | sage: derivative(airy_ai_prime(x),x) # indirect doctest |
| 233 | x*airy_ai(x) |
| 234 | """ |
| 235 | return x*airy_ai_simple(x) |
| 236 | |
| 237 | def _eval_(self, x): |
| 238 | """ |
| 239 | EXAMPLES:: |
| 240 | |
| 241 | sage: airy_ai(1,0) # indirect doctest |
| 242 | -1/3*3^(2/3)/gamma(1/3) |
| 243 | sage: airy_ai(1,0.0) # indirect doctest |
| 244 | -0.258819403792807 |
| 245 | """ |
| 246 | if not isinstance(x,Expression): |
| 247 | if is_inexact(x): |
| 248 | return self._evalf_(x) |
| 249 | elif x==0: |
| 250 | r=ZZ(1)/3 |
| 251 | return -1/(3**(r)*gamma(r)) |
| 252 | else: |
| 253 | return None |
| 254 | |
| 255 | def _evalf_(self, x, **kwargs): |
| 256 | """ |
| 257 | EXAMPLES:: |
| 258 | |
| 259 | sage: airy_ai(1,0.0) # indirect doctest |
| 260 | -0.258819403792807 |
| 261 | |
| 262 | We can use several methods for numerical evaluation:: |
| 263 | |
| 264 | sage: airy_ai(1,4).n(algorithm='maxima') |
| 265 | -0.0019586409502 |
| 266 | sage: airy_ai(1,4).n(algorithm='mpmath') |
| 267 | -0.00195864095020418 |
| 268 | sage: airy_ai(1,4).n(algorithm='mpmath', prec=100) |
| 269 | -0.0019586409502041789001381409184 |
| 270 | |
| 271 | TESTS:: |
| 272 | |
| 273 | sage: airy_ai(1, 4).n(algorithm='maxima', prec=70) |
| 274 | Traceback (most recent call last): |
| 275 | ... |
| 276 | ValueError: for the maxima algorithm the precision must be 53 |
| 277 | |
| 278 | |
| 279 | """ |
| 280 | algorithm = kwargs.get('algorithm', None) or 'mpmath' |
| 281 | parent = kwargs.get('parent', None) or sage_structure_coerce_parent(x) |
| 282 | prec = parent.prec() if hasattr(parent, 'prec') else 53 |
| 283 | if algorithm == 'mpmath': |
| 284 | import mpmath |
| 285 | from sage.libs.mpmath import utils as mpmath_utils |
| 286 | return mpmath_utils.call(mpmath.airyai, x, derivative=1, parent=parent) |
| 287 | elif algorithm == 'maxima': |
| 288 | if prec != 53: |
| 289 | raise ValueError, "for the maxima algorithm the precision must be 53" |
| 290 | return RDF(meval("airy_dai(%s)" % RDF(x))) |
| 291 | else: |
| 292 | raise ValueError("unknown algorithm") |
| 293 | |
| 294 | |
| 295 | airy_ai_general=FunctionAiryAiGeneral() |
| 296 | airy_ai_simple= FunctionAiryAiSimple() |
| 297 | airy_ai_prime= FunctionAiryAiPrime() |
| 298 | |
| 299 | |
| 300 | def airy_ai(alpha,x=None, hold_derivative=False, *args, **kwds): |
| 301 | r""" |
| 302 | The Airy Ai function `\operatorname{Ai}(x)` is one of the two |
| 303 | linearly independent solutions to the Airy differental equation |
| 304 | `f''(z) +f(z)x=0`, defined by the initial conditions: |
| 305 | |
| 306 | .. math :: |
| 307 | \operatorname{Ai}(0)=\frac{1}{2^{2/3} \Gamma(\frac{2}{3})}, |
| 308 | |
| 309 | \operatorname{Ai}'(0)=-\frac{1}{2^{1/3} \Gamma(\frac{1}{3})}. |
| 310 | |
| 311 | Another way to define the Airy Ai function is: |
| 312 | |
| 313 | .. math:: |
| 314 | \operatorname{Ai}(x)=\frac{1}{\pi}\int_0^\infty |
| 315 | \cos\left(\frac{1}{3}t^3+xt\right) dt. |
| 316 | |
| 317 | INPUT: |
| 318 | |
| 319 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to `z`. |
| 320 | For `\alpha = n = 1,2,3,\ldots` this gives the derivative `\operatorname{Ai}^{(n)}(z)`, |
| 321 | and for `\alpha = -n = -1,-2,-3,\ldots` this gives the `n`-fold iterated integral |
| 322 | |
| 323 | .. math :: |
| 324 | |
| 325 | f_0(z) = \operatorname{Ai}(z) |
| 326 | |
| 327 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 328 | |
| 329 | - ``x``.- The argument of the function. |
| 330 | |
| 331 | - ``hold_derivative``.- Whether or not to stop from returning higher derivatives |
| 332 | in terms of `\operatorname{Ai}(x)` and `\operatorname{Ai}'(x)`. |
| 333 | |
| 334 | |
| 335 | EXAMPLES:: |
| 336 | |
| 337 | sage: n,x=var('n x') |
| 338 | sage: airy_ai(x) |
| 339 | airy_ai(x) |
| 340 | |
| 341 | It can return derivatives or integrals:: |
| 342 | |
| 343 | sage: airy_ai(1,x) |
| 344 | airy_ai_prime(x) |
| 345 | sage: airy_ai(2,x) |
| 346 | x*airy_ai(x) |
| 347 | sage: airy_ai(2,x,True) |
| 348 | airy_ai(2, x) |
| 349 | sage: airy_ai(-2,x) |
| 350 | airy_ai(-2, x) |
| 351 | sage: airy_ai(n, x) |
| 352 | airy_ai(n, x) |
| 353 | |
| 354 | It can be evaluated symbolically or numerically for real or complex values:: |
| 355 | |
| 356 | sage: airy_ai(0) |
| 357 | 1/3*3^(1/3)/gamma(2/3) |
| 358 | sage: airy_ai(0.0) |
| 359 | 0.355028053887817 |
| 360 | sage: airy_ai(I) |
| 361 | airy_ai(I) |
| 362 | sage: airy_ai(1.0*I) |
| 363 | 0.331493305432141 - 0.317449858968444*I |
| 364 | |
| 365 | The functions can be evaluated numerically using either mpmath (the default) |
| 366 | or maxima, where mpmath can compute the values to arbitrary precision:: |
| 367 | |
| 368 | sage: airy_ai(2).n(prec=100) |
| 369 | 0.034924130423274379135322080792 |
| 370 | sage: airy_ai(2).n(algorithm='mpmath',prec=100) |
| 371 | 0.034924130423274379135322080792 |
| 372 | sage: airy_ai(2).n(algorithm='maxima') |
| 373 | 0.0349241304233 |
| 374 | |
| 375 | And the derivatives can be evaluated:: |
| 376 | |
| 377 | sage: airy_ai(1,0) |
| 378 | -1/3*3^(2/3)/gamma(1/3) |
| 379 | sage: airy_ai(1,0.0) |
| 380 | -0.258819403792807 |
| 381 | |
| 382 | Plots:: |
| 383 | |
| 384 | sage: plot(airy_ai(x),(x,-10,5))+plot(airy_ai_prime(x),(x,-10,5),color='red') |
| 385 | |
| 386 | **References** |
| 387 | |
| 388 | - Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10" |
| 389 | |
| 390 | - http://en.wikipedia.org/wiki/Airy_function |
| 391 | """ |
| 392 | |
| 393 | #We catch the case with no alpha |
| 394 | if x==None: |
| 395 | x=alpha |
| 396 | return airy_ai_simple(x, **kwds) |
| 397 | #We raise an error if there are too many arguments |
| 398 | if len(args) > 0: |
| 399 | raise TypeError("Symbolic function airy_ai takes at most 3 arguments (%s given)" % |
| 400 | (len(args)+3)) |
| 401 | |
| 402 | #We take care of all other cases. |
| 403 | if hold_derivative: |
| 404 | return airy_ai_general(alpha,x,**kwds) |
| 405 | elif alpha==0: |
| 406 | return airy_ai_simple(x, **kwds) |
| 407 | elif alpha==1: |
| 408 | return airy_ai_prime(x, **kwds) |
| 409 | elif alpha>1: |
| 410 | #we use a different variable here because if x is a |
| 411 | #particular value, we would be differentiating a constant |
| 412 | #which would return 0. What we want is the value of |
| 413 | #the derivative at the value and not the derivative of |
| 414 | #a particular value of the function. |
| 415 | v=var('v') |
| 416 | return derivative(airy_ai_simple(v,**kwds),v,alpha).subs(v=x) |
| 417 | else: |
| 418 | return airy_ai_general(alpha,x,**kwds) |
| 419 | |
| 420 | ######################################################################## |
| 421 | ######################################################################## |
| 422 | |
| 423 | class FunctionAiryBiGeneral(BuiltinFunction): |
| 424 | def __init__(self): |
| 425 | r""" |
| 426 | The generalized derivative of the Airy Bi function |
| 427 | |
| 428 | INPUT: |
| 429 | |
| 430 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to |
| 431 | `z`. For `\alpha = n = 1,2,3,\ldots` this gives the derivative |
| 432 | `\operatorname{Bi}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots` this gives |
| 433 | the `n`-fold iterated integral |
| 434 | |
| 435 | .. math :: |
| 436 | |
| 437 | f_0(z) = \operatorname{Bi}(z) |
| 438 | |
| 439 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 440 | |
| 441 | - ``x``.- The argument of the function. |
| 442 | |
| 443 | - ``hold``.- Whether or not to stop from returning higher derivatives in terms of |
| 444 | `\operatorname{Bi}(x)` and `\operatorname{Bi}'(x)`. |
| 445 | |
| 446 | EXAMPLES:: |
| 447 | |
| 448 | sage: x,n=var('x n') |
| 449 | sage: airy_bi(-2,x) |
| 450 | airy_bi(-2, x) |
| 451 | sage: derivative(airy_bi(-2,x),x) |
| 452 | airy_bi(-1, x) |
| 453 | sage: airy_bi(n,x) |
| 454 | airy_bi(n, x) |
| 455 | sage: derivative(airy_bi(n,x),x) |
| 456 | airy_bi(n + 1, x) |
| 457 | sage: airy_bi(2,x,True) |
| 458 | airy_bi(2, x) |
| 459 | sage: derivative(airy_bi(2,x,hold_derivative=True),x) |
| 460 | airy_bi(3, x) |
| 461 | """ |
| 462 | |
| 463 | BuiltinFunction.__init__(self, "airy_bi", nargs=2, |
| 464 | latex_name=r"\operatorname{Bi}") |
| 465 | |
| 466 | def _derivative_(self, alpha, *args, **kwds): |
| 467 | """ |
| 468 | EXAMPLES:: |
| 469 | |
| 470 | sage: x,n=var('x n') |
| 471 | sage: derivative(airy_bi(n,x),x) # indirect doctest |
| 472 | airy_bi(n + 1, x) |
| 473 | """ |
| 474 | |
| 475 | x=args[0] |
| 476 | return airy_bi_general(alpha+1,x) |
| 477 | |
| 478 | def _eval_(self, alpha, *args): |
| 479 | """ |
| 480 | EXAMPLES:: |
| 481 | |
| 482 | sage: x,n=var('x n') |
| 483 | sage: airy_bi(-2,1.0) # indirect doctest |
| 484 | 0.388621540699059 |
| 485 | sage: airy_bi(n,1.0) # indirect doctest |
| 486 | airy_bi(n, 1.00000000000000) |
| 487 | """ |
| 488 | |
| 489 | |
| 490 | x=args[0] |
| 491 | if not isinstance(x,Expression) and not isinstance(alpha,Expression): |
| 492 | if is_inexact(x): |
| 493 | return self._evalf_(alpha,x) |
| 494 | else: |
| 495 | return None |
| 496 | |
| 497 | def _evalf_(self, alpha, x, **kwargs): |
| 498 | """ |
| 499 | EXAMPLES:: |
| 500 | |
| 501 | sage: airy_bi(-2,1.0) # indirect doctest |
| 502 | 0.388621540699059 |
| 503 | |
| 504 | """ |
| 505 | algorithm = kwargs.get('algorithm', None) or 'mpmath' |
| 506 | parent = kwargs.get('parent', None) or sage_structure_coerce_parent(x) |
| 507 | prec = parent.prec() if hasattr(parent, 'prec') else 53 |
| 508 | if algorithm == 'mpmath': |
| 509 | import mpmath |
| 510 | from sage.libs.mpmath import utils as mpmath_utils |
| 511 | return mpmath_utils.call(mpmath.airybi, x, derivative=alpha, parent=parent) |
| 512 | elif algorithm == 'maxima': |
| 513 | raise NotImplementedError("general case not available in maxima") |
| 514 | else: |
| 515 | raise ValueError("unknown algorithm") |
| 516 | |
| 517 | class FunctionAiryBiSimple(BuiltinFunction): |
| 518 | def __init__(self): |
| 519 | """ |
| 520 | The class for the Airy Bi function. |
| 521 | |
| 522 | EXAMPLES:: |
| 523 | |
| 524 | sage: from sage.functions.airy import FunctionAiryBiSimple |
| 525 | sage: airy_bi_simple= FunctionAiryBiSimple() |
| 526 | sage: f=airy_bi_simple(x); f |
| 527 | airy_bi(x) |
| 528 | """ |
| 529 | |
| 530 | BuiltinFunction.__init__(self, "airy_bi", latex_name=r'\operatorname{Bi}', |
| 531 | conversions=dict(mathematica='AiryBi', maxima='airy_bi')) |
| 532 | |
| 533 | def _derivative_(self, x, diff_param=None): |
| 534 | """ |
| 535 | EXAMPLES:: |
| 536 | |
| 537 | sage: derivative(airy_bi(x),x) # indirect doctest |
| 538 | airy_bi_prime(x) |
| 539 | """ |
| 540 | return airy_bi_prime(x) |
| 541 | |
| 542 | def _eval_(self, x): |
| 543 | """ |
| 544 | EXAMPLES:: |
| 545 | |
| 546 | sage: airy_bi(0) # indirect doctest |
| 547 | 1/3*3^(5/6)/gamma(1/3) |
| 548 | sage: airy_bi(0.0) # indirect doctest |
| 549 | 0.614926627446001 |
| 550 | sage: airy_bi(I) # indirect doctest |
| 551 | airy_bi(I) |
| 552 | sage: airy_bi(1.0*I) # indirect doctest |
| 553 | 0.648858208330395 + 0.344958634768048*I |
| 554 | """ |
| 555 | |
| 556 | if not isinstance(x,Expression): |
| 557 | if is_inexact(x): |
| 558 | return self._evalf_(x) |
| 559 | elif x==0: |
| 560 | one_sixth = ZZ(1)/6 |
| 561 | return 1/(3**(one_sixth)*gamma(2*one_sixth)) |
| 562 | else: |
| 563 | return None |
| 564 | |
| 565 | def _evalf_(self, x, **kwargs): |
| 566 | """ |
| 567 | EXAMPLES:: |
| 568 | |
| 569 | sage: airy_bi(0.0) # indirect doctest |
| 570 | 0.614926627446001 |
| 571 | sage: airy_bi(1.0*I) # indirect doctest |
| 572 | 0.648858208330395 + 0.344958634768048*I |
| 573 | |
| 574 | We can use several methods for numerical evaluation:: |
| 575 | |
| 576 | sage: airy_bi(3).n(algorithm='maxima') |
| 577 | 14.0373289637 |
| 578 | sage: airy_bi(3).n(algorithm='mpmath') |
| 579 | 14.0373289637302 |
| 580 | sage: airy_bi(3).n(algorithm='mpmath', prec=100) |
| 581 | 14.037328963730232031740267314 |
| 582 | |
| 583 | TESTS:: |
| 584 | |
| 585 | sage: airy_bi(3).n(algorithm='maxima', prec=100) |
| 586 | Traceback (most recent call last): |
| 587 | ... |
| 588 | ValueError: for the maxima algorithm the precision must be 53 |
| 589 | |
| 590 | """ |
| 591 | algorithm = kwargs.get('algorithm', None) or 'mpmath' |
| 592 | parent = kwargs.get('parent', None) or sage_structure_coerce_parent(x) |
| 593 | prec = parent.prec() if hasattr(parent, 'prec') else 53 |
| 594 | if algorithm == 'mpmath': |
| 595 | import mpmath |
| 596 | from sage.libs.mpmath import utils as mpmath_utils |
| 597 | return mpmath_utils.call(mpmath.airybi, x, parent=parent) |
| 598 | elif algorithm == 'maxima': |
| 599 | if prec != 53: |
| 600 | raise ValueError, "for the maxima algorithm the precision must be 53" |
| 601 | return RDF(meval("airy_bi(%s)" % RDF(x))) |
| 602 | else: |
| 603 | raise ValueError("unknown algorithm") |
| 604 | |
| 605 | class FunctionAiryBiPrime(BuiltinFunction): |
| 606 | def __init__(self): |
| 607 | """ |
| 608 | The derivative of the Airy Bi function; see :func:`airy_bi` |
| 609 | for the full documentation. |
| 610 | |
| 611 | EXAMPLES:: |
| 612 | |
| 613 | sage: x,n=var('x n') |
| 614 | sage: airy_bi_prime(x) # indirect doctest |
| 615 | airy_bi_prime(x) |
| 616 | sage: airy_bi_prime(0) # indirect doctest |
| 617 | 3^(1/6)/gamma(1/3) |
| 618 | """ |
| 619 | |
| 620 | BuiltinFunction.__init__(self, "airy_bi_prime", |
| 621 | latex_name=r"\operatorname{Bi}'", |
| 622 | conversions=dict(mathematica='AiryBiPrime', maxima='airy_dbi')) |
| 623 | |
| 624 | def _derivative_(self, x, diff_param=None): |
| 625 | """ |
| 626 | EXAMPLES:: |
| 627 | |
| 628 | sage: derivative(airy_bi_prime(x),x) # indirect doctest |
| 629 | x*airy_bi(x) |
| 630 | """ |
| 631 | return x*airy_bi_simple(x) |
| 632 | |
| 633 | def _eval_(self, x): |
| 634 | """ |
| 635 | EXAMPLES:: |
| 636 | |
| 637 | sage: airy_bi(1,0) # indirect doctest |
| 638 | 3^(1/6)/gamma(1/3) |
| 639 | sage: airy_bi(1,0.0) # indirect doctest |
| 640 | 0.448288357353826 |
| 641 | """ |
| 642 | if not isinstance(x,Expression): |
| 643 | if is_inexact(x): |
| 644 | return self._evalf_(x) |
| 645 | elif x==0: |
| 646 | one_sixth = ZZ(1)/6 |
| 647 | return 3**(one_sixth)/gamma(2*one_sixth) |
| 648 | else: |
| 649 | return None |
| 650 | |
| 651 | def _evalf_(self, x, **kwargs): |
| 652 | """ |
| 653 | EXAMPLES:: |
| 654 | |
| 655 | sage: airy_bi(1,0.0) # indirect doctest |
| 656 | 0.448288357353826 |
| 657 | |
| 658 | We can use several methods for numerical evaluation:: |
| 659 | |
| 660 | sage: airy_bi(1,4).n(algorithm='maxima') |
| 661 | 161.926683505 |
| 662 | sage: airy_bi(1,4).n(algorithm='mpmath') |
| 663 | 161.926683504613 |
| 664 | sage: airy_bi(1,4).n(algorithm='mpmath', prec=100) |
| 665 | 161.92668350461340184309492429 |
| 666 | |
| 667 | TESTS:: |
| 668 | |
| 669 | sage: airy_bi(1,4).n(algorithm='maxima', prec=70) |
| 670 | Traceback (most recent call last): |
| 671 | ... |
| 672 | ValueError: for the maxima algorithm the precision must be 53 |
| 673 | |
| 674 | """ |
| 675 | algorithm = kwargs.get('algorithm', None) or 'mpmath' |
| 676 | parent = kwargs.get('parent', None) or sage_structure_coerce_parent(x) |
| 677 | prec = parent.prec() if hasattr(parent, 'prec') else 53 |
| 678 | if algorithm == 'mpmath': |
| 679 | import mpmath |
| 680 | from sage.libs.mpmath import utils as mpmath_utils |
| 681 | return mpmath_utils.call(mpmath.airybi, x, derivative=1, parent=parent) |
| 682 | elif algorithm == 'maxima': |
| 683 | if prec != 53: |
| 684 | raise ValueError, "for the maxima algorithm the precision must be 53" |
| 685 | return RDF(meval("airy_dbi(%s)" % RDF(x))) |
| 686 | else: |
| 687 | raise ValueError("unknown algorithm") |
| 688 | |
| 689 | airy_bi_general=FunctionAiryBiGeneral() |
| 690 | airy_bi_simple= FunctionAiryBiSimple() |
| 691 | airy_bi_prime= FunctionAiryBiPrime() |
| 692 | |
| 693 | def airy_bi(alpha,x=None, hold_derivative=False, *args, **kwds): |
| 694 | r""" |
| 695 | The Airy Bi function `\operatorname{Bi}(x)` is one of the two |
| 696 | linearly independent solutions to the Airy differental equation |
| 697 | `f''(z) +f(z)x=0`, defined by the initial conditions: |
| 698 | |
| 699 | .. math :: |
| 700 | \operatorname{Bi}(0)=\frac{1}{3^{1/6} \Gamma(\frac{2}{3})}, |
| 701 | |
| 702 | \operatorname{Bi}'(0)=\frac{3^{1/6}}{ \Gamma(\frac{1}{3})}. |
| 703 | |
| 704 | Another way to define the Airy Bi function is: |
| 705 | |
| 706 | .. math:: |
| 707 | \operatorname{Bi}(x)=\frac{1}{\pi}\int_0^\infty |
| 708 | \left[ \exp\left( xt -\frac{t^3}{3} \right) |
| 709 | +\sin\left(xt + \frac{1}{3}t^3\right) \right ] dt. |
| 710 | |
| 711 | INPUT: |
| 712 | |
| 713 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to `z`. |
| 714 | For `\alpha = n = 1,2,3,\ldots` this gives the derivative `\operatorname{Bi}^{(n)}(z)`, |
| 715 | and for `\alpha = -n = -1,-2,-3,\ldots` this gives the `n`-fold iterated integral |
| 716 | |
| 717 | .. math :: |
| 718 | |
| 719 | f_0(z) = \operatorname{Bi}(z) |
| 720 | |
| 721 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 722 | |
| 723 | - ``x``.- The argument of the function. |
| 724 | |
| 725 | - ``hold_derivative``.- Whether or not to stop from returning higher derivatives in |
| 726 | terms of `\operatorname{Bi}(x)` and `\operatorname{Bi}'(x)`. |
| 727 | |
| 728 | |
| 729 | EXAMPLES:: |
| 730 | |
| 731 | sage: n,x=var('n x') |
| 732 | sage: airy_bi(x) |
| 733 | airy_bi(x) |
| 734 | |
| 735 | It can return derivatives or integrals:: |
| 736 | |
| 737 | sage: airy_bi(1,x) |
| 738 | airy_bi_prime(x) |
| 739 | sage: airy_bi(2,x) |
| 740 | x*airy_bi(x) |
| 741 | sage: airy_bi(2,x,True) |
| 742 | airy_bi(2, x) |
| 743 | sage: airy_bi(-2,x) |
| 744 | airy_bi(-2, x) |
| 745 | sage: airy_bi(n, x) |
| 746 | airy_bi(n, x) |
| 747 | |
| 748 | It can be evaluated symbolically or numerically for real or complex values:: |
| 749 | |
| 750 | sage: airy_bi(0) |
| 751 | 1/3*3^(5/6)/gamma(1/3) |
| 752 | sage: airy_bi(0.0) |
| 753 | 0.614926627446001 |
| 754 | sage: airy_bi(I) |
| 755 | airy_bi(I) |
| 756 | sage: airy_bi(1.0*I) |
| 757 | 0.648858208330395 + 0.344958634768048*I |
| 758 | |
| 759 | The functions can be evaluated numerically using either mpmath (the default) |
| 760 | or maxima, where mpmath can compute the values to arbitrary precision:: |
| 761 | |
| 762 | sage: airy_bi(2).n(prec=100) |
| 763 | 3.2980949999782147102806044252 |
| 764 | sage: airy_bi(2).n(algorithm='mpmath',prec=100) |
| 765 | 3.2980949999782147102806044252 |
| 766 | sage: airy_bi(2).n(algorithm='maxima') |
| 767 | 3.29809499998 |
| 768 | |
| 769 | And the derivatives can be evaluated:: |
| 770 | |
| 771 | sage: airy_bi(1,0) |
| 772 | 3^(1/6)/gamma(1/3) |
| 773 | sage: airy_bi(1,0.0) |
| 774 | 0.448288357353826 |
| 775 | |
| 776 | Plots:: |
| 777 | |
| 778 | sage: plot(airy_bi(x),(x,-10,5))+plot(airy_bi_prime(x),(x,-10,5),color='red') |
| 779 | |
| 780 | **References** |
| 781 | |
| 782 | - Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10" |
| 783 | |
| 784 | - http://en.wikipedia.org/wiki/Airy_function |
| 785 | """ |
| 786 | |
| 787 | #We catch the case with no alpha |
| 788 | if x==None: |
| 789 | x=alpha |
| 790 | return airy_bi_simple(x, **kwds) |
| 791 | #We raise an error if there are too many arguments |
| 792 | if len(args) > 0: |
| 793 | raise TypeError("Symbolic function airy_ai takes at most 3 arguments (%s given)" % |
| 794 | (len(args)+3)) |
| 795 | |
| 796 | #We take care of all other cases. |
| 797 | if hold_derivative: |
| 798 | return airy_bi_general(alpha,x,**kwds) |
| 799 | elif alpha==0: |
| 800 | return airy_bi_simple(x, **kwds) |
| 801 | elif alpha==1: |
| 802 | return airy_bi_prime(x, **kwds) |
| 803 | elif alpha>1: |
| 804 | #we use a different variable here because if x is a |
| 805 | #particular value, we would be differentiating a constant |
| 806 | #which would return 0. What we want is the value of |
| 807 | #the derivative at the value and not the derivative of |
| 808 | #a particular value of the function. |
| 809 | v=var('v') |
| 810 | return derivative(airy_bi_simple(v,**kwds),v,alpha).subs(v=x) |
| 811 | else: |
| 812 | return airy_bi_general(alpha,x,**kwds) |
| 813 | |