| 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 |
| 19 | |
| 20 | from sage.functions.other import gamma |
| 21 | from mpmath import airyai as airy_ai_mpmath |
| 22 | from mpmath import airybi as airy_bi_mpmath |
| 23 | from sage.libs.mpmath import utils as mpmath_utils |
| 24 | |
| 25 | from sage.calculus.var import var |
| 26 | from sage.calculus.functional import derivative |
| 27 | |
| 28 | from sage.rings.real_mpfr import RR |
| 29 | from sage.rings.rational import Rational as R |
| 30 | |
| 31 | class FunctionAiryAiSimple(BuiltinFunction): |
| 32 | """ |
| 33 | The class for the Airy Ai function. |
| 34 | |
| 35 | Examples:: |
| 36 | |
| 37 | sage: airy_ai_simple= FunctionAiryAiSimple() |
| 38 | sage: f=airy_ai_simple(x); f |
| 39 | airy_ai(x) |
| 40 | """ |
| 41 | def __init__(self): |
| 42 | BuiltinFunction.__init__(self, "airy_ai", latex_name=r'\operatorname{Ai}', |
| 43 | conversions=dict(mathematica='AiryAi')) |
| 44 | |
| 45 | def _derivative_(self, x, diff_param=None): |
| 46 | """ |
| 47 | Examples:: |
| 48 | |
| 49 | sage: derivative(airy_ai(x),x) |
| 50 | airy_ai_prime(x) |
| 51 | """ |
| 52 | return airy_ai_prime(x) |
| 53 | |
| 54 | def _eval_(self, x): |
| 55 | """ |
| 56 | Examples:: |
| 57 | |
| 58 | sage: airy_ai(0) |
| 59 | 1/3*3^(1/3)/gamma(2/3) |
| 60 | sage: airy_ai(0.0) |
| 61 | 0.355028053887817 |
| 62 | sage: airy_ai(I) |
| 63 | airy_ai(I) |
| 64 | sage: airy_ai(1.0*I) |
| 65 | 0.331493305432141 - 0.317449858968444*I |
| 66 | """ |
| 67 | |
| 68 | if not isinstance(x,Expression): |
| 69 | if is_inexact(x): |
| 70 | return self._evalf_(x, parent(x)) |
| 71 | elif x==0: |
| 72 | r=R('2/3') |
| 73 | return 1/(3**(r)*gamma(r)) |
| 74 | else: |
| 75 | return None |
| 76 | |
| 77 | def _evalf_(self, x, parent=None): |
| 78 | """ |
| 79 | Examples:: |
| 80 | |
| 81 | sage: airy_ai(0.0) |
| 82 | 0.355028053887817 |
| 83 | sage: airy_ai(1.0*I) |
| 84 | 0.331493305432141 - 0.317449858968444*I |
| 85 | """ |
| 86 | return mpmath_utils.call(airy_ai_mpmath, x, parent=RR) |
| 87 | |
| 88 | class FunctionAiryAiPrime(BuiltinFunction): |
| 89 | """ |
| 90 | The derivative of the Airy Ai function. |
| 91 | |
| 92 | Examples:: |
| 93 | |
| 94 | sage: x,n=var('x n') |
| 95 | sage: airy_ai_prime(x) |
| 96 | airy_ai_prime(x) |
| 97 | sage: airy_ai_prime(0) |
| 98 | -1/3*3^(2/3)/gamma(1/3) |
| 99 | """ |
| 100 | def __init__(self): |
| 101 | BuiltinFunction.__init__(self, "airy_ai_prime", |
| 102 | latex_name=r"\operatorname{Ai}'", |
| 103 | conversions=dict(mathematica='AiryAiPrime')) |
| 104 | |
| 105 | def _derivative_(self, x, diff_param=None): |
| 106 | """ |
| 107 | Examples:: |
| 108 | |
| 109 | sage: derivative(airy_ai_prime(x),x) |
| 110 | x*airy_ai(x) |
| 111 | """ |
| 112 | return x*airy_ai_simple(x) |
| 113 | |
| 114 | def _eval_(self, x): |
| 115 | """ |
| 116 | Examples:: |
| 117 | |
| 118 | sage: airy_ai(1,0) |
| 119 | -1/3*3^(2/3)/gamma(1/3) |
| 120 | sage: airy_ai(1,0.0) |
| 121 | -0.258819403792807 |
| 122 | """ |
| 123 | if not isinstance(x,Expression): |
| 124 | if is_inexact(x): |
| 125 | return self._evalf_(x, parent(x)) |
| 126 | elif x==0: |
| 127 | r=R('1/3') |
| 128 | return -1/(3**(r)*gamma(r)) |
| 129 | else: |
| 130 | return None |
| 131 | |
| 132 | def _evalf_(self, x, parent=None): |
| 133 | """ |
| 134 | Examples:: |
| 135 | |
| 136 | sage: airy_ai(1,0.0) |
| 137 | -0.258819403792807 |
| 138 | """ |
| 139 | return mpmath_utils.call(airy_ai_mpmath, x, derivative=1, parent=RR) |
| 140 | |
| 141 | class FunctionAiryAiGeneral(BuiltinFunction): |
| 142 | """ |
| 143 | The generalized derivative of the Airy Ai function |
| 144 | |
| 145 | INPUT: |
| 146 | |
| 147 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to `z`. For `\alpha = n = 1,2,3,\ldots` this gives the derivative `\operatorname{Ai}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots` this gives the `n`-fold iterated integral |
| 148 | |
| 149 | .. math :: |
| 150 | |
| 151 | f_0(z) = \operatorname{Ai}(z) |
| 152 | |
| 153 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 154 | |
| 155 | - ``x``.- The argument of the function. |
| 156 | |
| 157 | - ``hold``.- Whether or not to stop from returning higher derivatives in terms of `\operatorname{Ai}(x)` and `\operatorname{Ai}'(x)`. |
| 158 | |
| 159 | Examples:: |
| 160 | |
| 161 | sage: x,n=var('x n') |
| 162 | sage: airy_ai(-2,x) |
| 163 | airy_ai(-2, x) |
| 164 | sage: derivative(airy_ai(-2,x),x) |
| 165 | airy_ai(-1, x) |
| 166 | sage: airy_ai(n,x) |
| 167 | airy_ai(n, x) |
| 168 | sage: derivative(airy_ai(n,x),x) |
| 169 | airy_ai(n + 1, x) |
| 170 | sage: airy_ai(2,x,True) |
| 171 | airy_ai(2, x) |
| 172 | sage: derivative(airy_ai(2,x,hold_derivative=True),x) |
| 173 | airy_ai(3, x) |
| 174 | """ |
| 175 | def __init__(self): |
| 176 | BuiltinFunction.__init__(self, "airy_ai", nargs=2, |
| 177 | latex_name=r"\operatorname{Ai}") |
| 178 | |
| 179 | def _derivative_(self, alpha, *args, **kwds): |
| 180 | """ |
| 181 | Examples:: |
| 182 | |
| 183 | sage: x,n=var('x n') |
| 184 | sage: derivative(airy_ai(n,x),x) |
| 185 | airy_ai(n + 1, x) |
| 186 | """ |
| 187 | |
| 188 | x=args[0] |
| 189 | return airy_ai_general(alpha+1,x) |
| 190 | |
| 191 | def _eval_(self, alpha, *args): |
| 192 | """ |
| 193 | Examples:: |
| 194 | |
| 195 | sage: x,n=var('x n') |
| 196 | sage: airy_ai(-2,1.0) |
| 197 | 0.136645379421096 |
| 198 | sage: airy_ai(n,1.0) |
| 199 | airy_ai(n, 1.00000000000000) |
| 200 | """ |
| 201 | |
| 202 | |
| 203 | x=args[0] |
| 204 | if not isinstance(x,Expression) and not isinstance(alpha,Expression): |
| 205 | if is_inexact(x): |
| 206 | return self._evalf_(alpha,x, parent(x)) |
| 207 | else: |
| 208 | return None |
| 209 | |
| 210 | def _evalf_(self, alpha, x, parent=None): |
| 211 | """ |
| 212 | Examples:: |
| 213 | sage: airy_ai(-2,1.0) |
| 214 | 0.136645379421096 |
| 215 | """ |
| 216 | return mpmath_utils.call(airy_ai_mpmath, x, derivative=alpha, parent=RR) |
| 217 | |
| 218 | airy_ai_simple= FunctionAiryAiSimple() |
| 219 | airy_ai_prime= FunctionAiryAiPrime() |
| 220 | airy_ai_general=FunctionAiryAiGeneral() |
| 221 | |
| 222 | def airy_ai(alpha,x=None, hold_derivative=False, *args, **kwds): |
| 223 | r""" |
| 224 | The Airy Ai function `\operatorname{Ai}(x)` is one of the two |
| 225 | linearly independent solutions the Airy differental equation `f''(z) +f(z)x=0`, |
| 226 | defined by the initial conditions: |
| 227 | |
| 228 | .. math :: |
| 229 | \operatorname{Ai}(0)=\frac{1}{2^{2/3} \Gamma(\frac{2}{3})}, |
| 230 | |
| 231 | \operatorname{Ai}'(0)=-\frac{1}{2^{1/3} \Gamma(\frac{1}{3})}. |
| 232 | |
| 233 | Another way to define the Airy Ai function is: |
| 234 | |
| 235 | .. math:: |
| 236 | \operatorname{Ai}(x)=\frac{1}{\pi}\int_0^\infty |
| 237 | \cos\left(\frac{1}{3}t^3+xt\right) dt. |
| 238 | |
| 239 | INPUT: |
| 240 | |
| 241 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to `z`. For `\alpha = n = 1,2,3,\ldots` this gives the derivative `\operatorname{Ai}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots` this gives the `n`-fold iterated integral |
| 242 | |
| 243 | .. math :: |
| 244 | |
| 245 | f_0(z) = \operatorname{Ai}(z) |
| 246 | |
| 247 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 248 | |
| 249 | - ``x``.- The argument of the function. |
| 250 | |
| 251 | - ``hold_derivative``.- Whether or not to stop from returning higher derivatives in terms of `\operatorname{Ai}(x)` and `\operatorname{Ai}'(x)`. |
| 252 | |
| 253 | |
| 254 | EXAMPLES:: |
| 255 | |
| 256 | sage: n,x=var('n x') |
| 257 | sage: airy_ai(x) |
| 258 | airy_ai(x) |
| 259 | |
| 260 | It can return derivatives or integrals:: |
| 261 | |
| 262 | sage: airy_ai(1,x) |
| 263 | airy_ai_prime(x) |
| 264 | sage: airy_ai(2,x) |
| 265 | x*airy_ai(x) |
| 266 | sage: airy_ai(2,x,True) |
| 267 | airy_ai(2, x) |
| 268 | sage: airy_ai(-2,x) |
| 269 | airy_ai(-2, x) |
| 270 | sage: airy_ai(n, x) |
| 271 | airy_ai(n, x) |
| 272 | |
| 273 | It can be evaluated symbolically or numerically for real or complex values:: |
| 274 | |
| 275 | sage: airy_ai(0) |
| 276 | 1/3*3^(1/3)/gamma(2/3) |
| 277 | sage: airy_ai(0.0) |
| 278 | 0.355028053887817 |
| 279 | sage: airy_ai(I) |
| 280 | airy_ai(I) |
| 281 | sage: airy_ai(1.0*I) |
| 282 | 0.331493305432141 - 0.317449858968444*I |
| 283 | |
| 284 | And the derivatives can be evaluated:: |
| 285 | |
| 286 | sage: airy_ai(1,0) |
| 287 | -1/3*3^(2/3)/gamma(1/3) |
| 288 | sage: airy_ai(1,0.0) |
| 289 | -0.258819403792807 |
| 290 | |
| 291 | Plots:: |
| 292 | |
| 293 | sage: plot(airy_ai(x),(x,-10,5))+plot(airy_ai_prime(x),(x,-10,5),color='red') |
| 294 | |
| 295 | **References** |
| 296 | |
| 297 | - Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10" |
| 298 | |
| 299 | - http://en.wikipedia.org/wiki/Airy_function |
| 300 | """ |
| 301 | |
| 302 | #We catch the case with no alpha |
| 303 | if x==None: |
| 304 | x=alpha |
| 305 | return airy_ai_simple(x, **kwds) |
| 306 | #We raise an error if there are too many arguments |
| 307 | if len(args) > 0: |
| 308 | raise TypeError, "Symbolic function airy_ai takes at most 3 arguments (%s given)"%(len(args)+3) |
| 309 | |
| 310 | #We take care of all other cases. |
| 311 | if hold_derivative: |
| 312 | return airy_ai_general(alpha,x,**kwds) |
| 313 | elif alpha==0: |
| 314 | return airy_ai_simple(x, **kwds) |
| 315 | elif alpha==1: |
| 316 | return airy_ai_prime(x, **kwds) |
| 317 | elif alpha>1: |
| 318 | #we use a different variable here because if x is a |
| 319 | #particular value, we would be differentiating a constant |
| 320 | #which would return 0. What we want is the value of |
| 321 | #the derivative at the value and not the derivative of |
| 322 | #a particular value of the function. |
| 323 | v=var('v') |
| 324 | return derivative(airy_ai_simple(v,**kwds),v,alpha).subs(v=x) |
| 325 | else: |
| 326 | return airy_ai_general(alpha,x,**kwds) |
| 327 | |
| 328 | ######################################################################## |
| 329 | ######################################################################## |
| 330 | |
| 331 | class FunctionAiryBiSimple(BuiltinFunction): |
| 332 | """ |
| 333 | The class for the Airy Bi function. |
| 334 | |
| 335 | Examples:: |
| 336 | |
| 337 | sage: airy_bi_simple= FunctionAiryBiSimple() |
| 338 | sage: f=airy_bi_simple(x); f |
| 339 | airy_bi(x) |
| 340 | """ |
| 341 | def __init__(self): |
| 342 | BuiltinFunction.__init__(self, "airy_bi", latex_name=r'\operatorname{Bi}', |
| 343 | conversions=dict(mathematica='AiryBi')) |
| 344 | |
| 345 | def _derivative_(self, x, diff_param=None): |
| 346 | """ |
| 347 | Examples:: |
| 348 | |
| 349 | sage: derivative(airy_bi(x),x) |
| 350 | airy_bi_prime(x) |
| 351 | """ |
| 352 | return airy_bi_prime(x) |
| 353 | |
| 354 | def _eval_(self, x): |
| 355 | """ |
| 356 | Examples:: |
| 357 | |
| 358 | sage: airy_bi(0) |
| 359 | 1/3*3^(5/6)/gamma(1/3) |
| 360 | sage: airy_bi(0.0) |
| 361 | 0.614926627446001 |
| 362 | sage: airy_bi(I) |
| 363 | airy_bi(I) |
| 364 | sage: airy_bi(1.0*I) |
| 365 | 0.648858208330395 + 0.344958634768048*I |
| 366 | """ |
| 367 | |
| 368 | if not isinstance(x,Expression): |
| 369 | if is_inexact(x): |
| 370 | return self._evalf_(x, parent(x)) |
| 371 | elif x==0: |
| 372 | return 1/(3**(R('1/6'))*gamma(R('2/6'))) |
| 373 | else: |
| 374 | return None |
| 375 | |
| 376 | def _evalf_(self, x, parent=None): |
| 377 | """ |
| 378 | Examples:: |
| 379 | |
| 380 | sage: airy_bi(0.0) |
| 381 | 0.614926627446001 |
| 382 | sage: airy_bi(1.0*I) |
| 383 | 0.648858208330395 + 0.344958634768048*I |
| 384 | """ |
| 385 | return mpmath_utils.call(airy_bi_mpmath, x, parent=RR) |
| 386 | |
| 387 | class FunctionAiryBiPrime(BuiltinFunction): |
| 388 | """ |
| 389 | The derivative of the Airy Bi function. |
| 390 | |
| 391 | Examples:: |
| 392 | |
| 393 | sage: x,n=var('x n') |
| 394 | sage: airy_bi_prime(x) |
| 395 | airy_bi_prime(x) |
| 396 | sage: airy_bi_prime(0) |
| 397 | 3^(1/6)/gamma(1/3) |
| 398 | """ |
| 399 | def __init__(self): |
| 400 | BuiltinFunction.__init__(self, "airy_bi_prime", |
| 401 | latex_name=r"\operatorname{Bi}'", |
| 402 | conversions=dict(mathematica='AiryBiPrime')) |
| 403 | |
| 404 | def _derivative_(self, x, diff_param=None): |
| 405 | """ |
| 406 | Examples:: |
| 407 | |
| 408 | sage: derivative(airy_bi_prime(x),x) |
| 409 | x*airy_bi(x) |
| 410 | """ |
| 411 | return x*airy_bi_simple(x) |
| 412 | |
| 413 | def _eval_(self, x): |
| 414 | """ |
| 415 | Examples:: |
| 416 | |
| 417 | sage: airy_bi(1,0) |
| 418 | 3^(1/6)/gamma(1/3) |
| 419 | sage: airy_bi(1,0.0) |
| 420 | 0.448288357353826 |
| 421 | """ |
| 422 | if not isinstance(x,Expression): |
| 423 | if is_inexact(x): |
| 424 | return self._evalf_(x, parent(x)) |
| 425 | elif x==0: |
| 426 | return 3**(R('1/6'))/gamma(R('1/3')) |
| 427 | else: |
| 428 | return None |
| 429 | |
| 430 | def _evalf_(self, x, parent=None): |
| 431 | """ |
| 432 | Examples:: |
| 433 | |
| 434 | sage: airy_bi(1,0.0) |
| 435 | 0.448288357353826 |
| 436 | """ |
| 437 | return mpmath_utils.call(airy_bi_mpmath, x, derivative=1, parent=RR) |
| 438 | |
| 439 | class FunctionAiryBiGeneral(BuiltinFunction): |
| 440 | """ |
| 441 | The generalized derivative of the Airy Bi function |
| 442 | |
| 443 | INPUT: |
| 444 | |
| 445 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to `z`. For `\alpha = n = 1,2,3,\ldots` this gives the derivative `\operatorname{Bi}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots` this gives the `n`-fold iterated integral |
| 446 | |
| 447 | .. math :: |
| 448 | |
| 449 | f_0(z) = \operatorname{Bi}(z) |
| 450 | |
| 451 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 452 | |
| 453 | - ``x``.- The argument of the function. |
| 454 | |
| 455 | - ``hold``.- Whether or not to stop from returning higher derivatives in terms of `\operatorname{Bi}(x)` and `\operatorname{Bi}'(x)`. |
| 456 | |
| 457 | Examples:: |
| 458 | |
| 459 | sage: x,n=var('x n') |
| 460 | sage: airy_bi(-2,x) |
| 461 | airy_bi(-2, x) |
| 462 | sage: derivative(airy_bi(-2,x),x) |
| 463 | airy_bi(-1, x) |
| 464 | sage: airy_bi(n,x) |
| 465 | airy_bi(n, x) |
| 466 | sage: derivative(airy_bi(n,x),x) |
| 467 | airy_bi(n + 1, x) |
| 468 | sage: airy_bi(2,x,True) |
| 469 | airy_bi(2, x) |
| 470 | sage: derivative(airy_bi(2,x,hold_derivative=True),x) |
| 471 | airy_bi(3, x) |
| 472 | """ |
| 473 | def __init__(self): |
| 474 | BuiltinFunction.__init__(self, "airy_bi", nargs=2, |
| 475 | latex_name=r"\operatorname{Bi}") |
| 476 | |
| 477 | def _derivative_(self, alpha, *args, **kwds): |
| 478 | """ |
| 479 | Examples:: |
| 480 | |
| 481 | sage: x,n=var('x n') |
| 482 | sage: derivative(airy_bi(n,x),x) |
| 483 | airy_bi(n + 1, x) |
| 484 | """ |
| 485 | |
| 486 | x=args[0] |
| 487 | return airy_bi_general(alpha+1,x) |
| 488 | |
| 489 | def _eval_(self, alpha, *args): |
| 490 | """ |
| 491 | Examples:: |
| 492 | |
| 493 | sage: x,n=var('x n') |
| 494 | sage: airy_bi(-2,1.0) |
| 495 | 0.388621540699059 |
| 496 | sage: airy_bi(n,1.0) |
| 497 | airy_bi(n, 1.00000000000000) |
| 498 | """ |
| 499 | |
| 500 | |
| 501 | x=args[0] |
| 502 | if not isinstance(x,Expression) and not isinstance(alpha,Expression): |
| 503 | if is_inexact(x): |
| 504 | return self._evalf_(alpha,x, parent(x)) |
| 505 | else: |
| 506 | return None |
| 507 | |
| 508 | def _evalf_(self, alpha, x, parent=None): |
| 509 | """ |
| 510 | Examples:: |
| 511 | sage: airy_bi(-2,1.0) |
| 512 | 0.388621540699059 |
| 513 | """ |
| 514 | return mpmath_utils.call(airy_bi_mpmath, x, derivative=alpha, parent=RR) |
| 515 | |
| 516 | airy_bi_simple= FunctionAiryBiSimple() |
| 517 | airy_bi_prime= FunctionAiryBiPrime() |
| 518 | airy_bi_general=FunctionAiryBiGeneral() |
| 519 | |
| 520 | def airy_bi(alpha,x=None, hold_derivative=False, *args, **kwds): |
| 521 | r""" |
| 522 | The Airy Bi function `\operatorname{Bi}(x)` is one of the two |
| 523 | linearly independent solutions the Airy differental equation `f''(z) +f(z)x=0`, |
| 524 | defined by the initial conditions: |
| 525 | |
| 526 | .. math :: |
| 527 | \operatorname{Bi}(0)=\frac{1}{3^{1/6} \Gamma(\frac{2}{3})}, |
| 528 | |
| 529 | \operatorname{Bi}'(0)=\frac{3^{1/6}}{ \Gamma(\frac{1}{3})}. |
| 530 | |
| 531 | Another way to define the Airy Bi function is: |
| 532 | |
| 533 | .. math:: |
| 534 | \operatorname{Bi}(x)=\frac{1}{\pi}\int_0^\infty |
| 535 | \left[ \exp\left( xt -\frac{t^3}{3} \right) |
| 536 | +\sin\left(xt + \frac{1}{3}t^3\right) \right ] dt. |
| 537 | |
| 538 | INPUT: |
| 539 | |
| 540 | - ``alpha``.- Return the `\alpha`-th order fractional derivative with respect to `z`. For `\alpha = n = 1,2,3,\ldots` this gives the derivative `\operatorname{Bi}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots` this gives the `n`-fold iterated integral |
| 541 | |
| 542 | .. math :: |
| 543 | |
| 544 | f_0(z) = \operatorname{Bi}(z) |
| 545 | |
| 546 | f_n(z) = \int_0^z f_{n-1}(t) dt. |
| 547 | |
| 548 | - ``x``.- The argument of the function. |
| 549 | |
| 550 | - ``hold_derivative``.- Whether or not to stop from returning higher derivatives in terms of `\operatorname{Bi}(x)` and `\operatorname{Bi}'(x)`. |
| 551 | |
| 552 | |
| 553 | EXAMPLES:: |
| 554 | |
| 555 | sage: n,x=var('n x') |
| 556 | sage: airy_bi(x) |
| 557 | airy_bi(x) |
| 558 | |
| 559 | It can return derivatives or integrals:: |
| 560 | |
| 561 | sage: airy_bi(1,x) |
| 562 | airy_bi_prime(x) |
| 563 | sage: airy_bi(2,x) |
| 564 | x*airy_bi(x) |
| 565 | sage: airy_bi(2,x,True) |
| 566 | airy_bi(2, x) |
| 567 | sage: airy_bi(-2,x) |
| 568 | airy_bi(-2, x) |
| 569 | sage: airy_bi(n, x) |
| 570 | airy_bi(n, x) |
| 571 | |
| 572 | It can be evaluated symbolically or numerically for real or complex values:: |
| 573 | |
| 574 | sage: airy_bi(0) |
| 575 | 1/3*3^(5/6)/gamma(1/3) |
| 576 | sage: airy_bi(0.0) |
| 577 | 0.614926627446001 |
| 578 | sage: airy_bi(I) |
| 579 | airy_bi(I) |
| 580 | sage: airy_bi(1.0*I) |
| 581 | 0.648858208330395 + 0.344958634768048*I |
| 582 | |
| 583 | And the derivatives can be evaluated:: |
| 584 | |
| 585 | sage: airy_bi(1,0) |
| 586 | 3^(1/6)/gamma(1/3) |
| 587 | sage: airy_bi(1,0.0) |
| 588 | 0.448288357353826 |
| 589 | |
| 590 | Plots:: |
| 591 | |
| 592 | sage: plot(airy_bi(x),(x,-10,5))+plot(airy_bi_prime(x),(x,-10,5),color='red') |
| 593 | |
| 594 | **References** |
| 595 | |
| 596 | - Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10" |
| 597 | |
| 598 | - http://en.wikipedia.org/wiki/Airy_function |
| 599 | """ |
| 600 | |
| 601 | #We catch the case with no alpha |
| 602 | if x==None: |
| 603 | x=alpha |
| 604 | return airy_bi_simple(x, **kwds) |
| 605 | #We raise an error if there are too many arguments |
| 606 | if len(args) > 0: |
| 607 | raise TypeError, "Symbolic function airy_ai takes at most 3 arguments (%s given)"%(len(args)+3) |
| 608 | |
| 609 | #We take care of all other cases. |
| 610 | if hold_derivative: |
| 611 | return airy_bi_general(alpha,x,**kwds) |
| 612 | elif alpha==0: |
| 613 | return airy_bi_simple(x, **kwds) |
| 614 | elif alpha==1: |
| 615 | return airy_bi_prime(x, **kwds) |
| 616 | elif alpha>1: |
| 617 | #we use a different variable here because if x is a |
| 618 | #particular value, we would be differentiating a constant |
| 619 | #which would return 0. What we want is the value of |
| 620 | #the derivative at the value and not the derivative of |
| 621 | #a particular value of the function. |
| 622 | v=var('v') |
| 623 | return derivative(airy_bi_simple(v,**kwds),v,alpha).subs(v=x) |
| 624 | else: |
| 625 | return airy_bi_general(alpha,x,**kwds) |
| 626 | |