Fractional Integro-Differentiation for the HP-41
Overview
1°) A few Elementary Functions
2°) A few Special Functions
-If a function f is defined by a power series: f(x) = SUMk=0,1,2,..... ck xk , its fractional integro-differentiation may be computed by
dµ f / dxµ = Dµ f(x) = SUMk=0,1,2,..... ck [ Gam(k+1) / Gam(k+1-µ) ] xk-µ where µ is a real number ( integer or fractional )
-If the function may be expressed in terms of hypergeometric functions pFq , the following relation is very useful too:
Dµ pFq ( a1 , .......... , ap ; b1 , .......... bq ; x ) = x -µ Gam(b1).........Gam(bq) p+1F~q+1 ( 1 , a1 , .......... , ap ; 1-µ , b1 , .......... bq ; x )
where Gam = Euler's Gamma function and pF~q is the regularized generalized hypergeometric function ( cf "Hypergeometric Functions for the HP-41" )
-We have D0 f(x) = f(x)
-If µ = 1 , 2 , 3 , ... we get the 1st , 2nd , 3rd
, ... derivatives
and if µ = -1 , -2 , -3 , .... the results
are the repeated integrals of the function f - usually those that
vanish for x = 0.
1°) A few Elementary Functions
Formulae:
• Hyperbolic Sine Dµ Sinh x = 2µ-1 sqrt(PI) x1-µ1F~2 ( 1 ; (2-µ)/2 , (3-µ)/2 ; x2/4 )
• Hyperbolic Cosine Dµ Cosh x = (2/x)µ sqrt(PI) 1F~2 ( 1 ; (1-µ)/2 , (2-µ)/2 ; x2/4 )
• Sine Dµ Sin x = 2µ-1 sqrt(PI) x1-µ1F~2 ( 1 ; (2-µ)/2 , (3-µ)/2 ; -x2/4 )
• Cosine Dµ Cos x = (2/x)µ sqrt(PI) 1F~2 ( 1 ; (1-µ)/2 , (2-µ)/2 ; -x2/4 )
• Logarithm Dµ Ln x = x -µ FC(µ)log (x)
where
FC(µ)log (x) = (-1)µ-1 (µ-1)
!
if µ is a positive integer
and
FC(µ)log (x) = [ Ln x - Psi(1-µ) - gamma
] / Gam(1-µ) otherwise
Psi = Digamma Function , gamma = Euler's constant = 0.5772156649... and Gam = Gamma Function.
• Exponential
Dµ Exp x = x -µ 1F~1
( 1 ; 1-µ ; x )
Data Registers: R00 to R04: temp
Flags: F09 F10
Subroutines: "1/G+" ( or "GAM+" ...
) "PSI" ( cf "Gamma Function for the HP-41" )
-The M-Code function HGF+ may be replaced by XEQ "HGF+"
( cf "Hypergeometric Functions" )
but in this case, register R00 must be replaced by another -
unused - data register.
| 01 LBL "DSH"
02 CF 09 03 CF 10 04 GTO 00 05 LBL "DCH" 06 SF 09 07 CF 10 08 GTO 00 09 LBL "DSIN" 10 CF 09 11 SF 10 12 GTO 00 13 LBL "DCOS" 14 SF 09 15 SF 10 16 LBL 00 17 STO 00 18 CLX 19 SIGN 20 STO 01 21 FC? 09 22 ST+ X |
23 X<>Y
24 STO 04 25 - 26 STO 02 27 STO 03 28 1 29 ST+ 03 30 FC?C 09 31 ST- 04 32 2 33 ST/ 00 34 ST/ 02 35 ST/ 03 36 CHS 37 RCL 00 38 X^2 39 FS?C 10 40 CHS 41 HGF+ 42 RCL 00 43 RCL 04 44 Y^X |
45 /
46 PI 47 SQRT 48 * 49 RTN 50 LBL "DLN" 51 STO 00 52 X<>Y 53 STO 01 54 FRC 55 X#0? 56 GTO 00 57 LASTX 58 X<=0? 59 GTO 00 60 1 61 CHS 62 ST+ Y 63 X<>Y 64 Y^X 65 LASTX 66 FACT |
67 *
68 GTO 01 69 LBL 00 70 1 71 LASTX 72 - 73 STO 02 74 XEQ "PSI" 75 RCL 00 76 LN 77 X<>Y 78 - 79 .5772156649 80 - 81 X<> 02 82 XEQ "1/G+" 83 RCL 02 84 * 85 LBL 01 86 RCL 00 87 RCL 01 88 Y^X |
89 /
90 RTN 91 LBL "DEXP" 92 STO 00 93 CLX 94 SIGN 95 STO 01 96 X<>Y 97 STO 03 98 - 99 STO 02 100 1 101 ENTER^ 102 CHS 103 RCL 00 104 HGF+ 105 RCL 00 106 RCL 03 107 Y^X 108 / 109 END |
( 197 bytes / SIZE 005 )
| STACK | INPUTS | OUTPUTS |
| Y | µ | / |
| X | x | (Dµ f) (x) |
Examples:
• Hyperbolic Sine
3.14 ENTER^
1.28 XEQ "DSH" >>>>
D3.14 Sinh ( 1.28 ) = 1.999005451
• Hyperbolic Cosine
3.14 ENTER^
1.28 XEQ "DCH" >>>>
D3.14 Cosh ( 1.28 ) = 1.502958219
• Sine
3.14 ENTER^
1.28 XEQ "DSIN" >>>>
D3.14 Sin ( 1.28 ) = -0.019142092
• Cosine
3.14 ENTER^
1.28 XEQ "DCOS" >>>>
D3.14 Cos ( 1.28 ) = 0.888787267
• Logarithm
3.14 ENTER^
1.28 XEQ "DLN" >>>>
D3.14 Ln ( 1.28 ) = 1.138569850
• Exponential
3.14 ENTER^
1.28 XEQ "DEXP" >>>>
D3.14 Exp ( 1.28 ) = 3.501963669
2°) A few Special Functions
Formulae:
• Sine Integral
Dµ Si x = 2µ-2 PI x1-µ2F~3
( 1/2 , 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; -x2/4 )
• Hyperbolic Sine Integral
Dµ Shi x = 2µ-2 PI x1-µ2F~3
( 1/2 , 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; x2/4 )
• Cosine Integral
Dµ Ci x = [ FC(µ)log (x) +
gamma / Gam(1-µ) ] x -µ - 2µ-3
sqrt(PI) x2-µ 2F~3
( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; -x2/4 )
• Hyperbolic Cosine Integral
Dµ Chi x = [ FC(µ)log (x)
+ gamma / Gam(1-µ) ] x -µ + 2µ-3
sqrt(PI) x2-µ 2F~3
( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; x2/4 )
• Exponential Integral Dµ
Ei x = [ FC(µ)log (x) + gamma / Gam(1-µ)
] x -µ + x1-µ 2F~2
( 1 , 1 ; 2 , 2-µ ; x )
• Fresnel Cosine Integral
Dµ C(x) = 22µ-3/2 PI3/2 x1-µ3F~4
[ 1/4 , 3/4 , 1 ; (2-µ)/4 , (3-µ)/4 , (4-µ)/4 , (5-µ)/4
; -(PI)2 x4/16 ]
• Fresnel Sine Integral Dµ
S(x) = 22µ-11/2 PI5/2 x3-µ3F~4
[ 3/4 , 1 , 5/4 ; (4-µ)/4 , (5-µ)/4 , (6-µ)/4 , (6-µ)/4
; -(PI)2 x4/16 ]
• Spherical Bessel Function - 1st kind
Dµ jn (x) = 2µ-2n-1 PI
xn-µ Gam(n+1) 2F~3
[ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+3/2 ; -x2/4
]
• Modified Bessel Function - 1st kind
Dµ In (x) = 2µ-2n sqrt(PI)
xn-µ Gam(n+1) 2F~3
[ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; x2/4
]
• Bessel Function - 1st kind
Dµ Jn (x) = 2µ-2n sqrt(PI)
xn-µ Gam(n+1) 2F~3
[ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; -x2/4
]
• Modified Bessel Function - 2nd kind
Dµ Kn (x) = 2µ-2n-1 (PI)3/2 x-µ-n csc(n.PI) { 16n Gam(1-n) 2F~3 [ (1-n)/2 , (2-n)/2 ; (1-µ-n)/2 , (2-µ-n)/2 , 1-n ; x2/4 ]
- x2n Gam(n+1) 2F~3 [ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; x2/4 ] } where n is not an integer.
• Bessel Function - 2nd kind
Dµ Yn (x) = 2µ-2n (PI)1/2 x-µ-n csc(n.PI) { -16n Gam(1-n) 2F~3 [ (1-n)/2 , (2-n)/2 ; (1-µ-n)/2 , (2-µ-n)/2 , 1-n ; -x2/4 ]
+ x2n Cos(n.PI) Gam(n+1) 2F~3 [ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; -x2/4 ] } where n is not an integer.
• Generalized Laguerre's Functions Dµ
Lan (x) = [ Gam(n+a+1)/Gam(n+1) ] x -µ2F~2
( 1 , -n ; a+1 , 1-µ ; x )
• Airy Functions
Dµ Ai(x) = 3µ-4/3 x -µ { 32/3 Gam(1/3) 2F~3 [ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x3/9 ] - x Gam(2/3) 2F~3 [ 2/3 , 1 ; (4-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x3/9 ] }
Dµ Bi(x) = 3µ-5/6 x -µ { 32/3 Gam(1/3) 2F~3 [ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x3/9 ] + x Gam(2/3) 2F~3 [ 2/3 , 1 ; (4-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x3/9 ] }
• Error Function Dµ
Erf (x) = 2µ x1-µ 2F~2
[ 1/2 , 1 ; (2-µ)/2 , (3-µ)/2 ; -x2 ]
• Hermite Function
Dµ Hn (x) = [ 2n+µ (PI) x-µ / Gam((1-n)/2) ] 2F~2 [ 1 , -n/2 ; (1-µ)/2 , (2-µ)/2 ; x2 ] - [ 2n+µ (PI) x1-µ / Gam((-n)/2) ] 2F~2 [ 1 , (1-n)/2 ; 1-µ/2 , (3-µ)/2 ; x2 ]
• Kummer's Function
Dµ F(a;b;x) = x -µ Gam(b) 2F~2
( 1 , a ; 1-µ , b ; x )
Data Registers: R00 to R09: temp
Flags: F01
Subroutines: "1/G+" ( or "GAM+" ...
) "PSI" = digamma function ( cf "Gamma Function for the HP-41" )
-The M-Code function HGF+ may be replaced by XEQ "HGF+"
( cf "Hypergeometric Functions" )
but in this case, register R00 must be replaced by another -
unused - data register.
-LBL 08 ( lines 644 to 675 ) is a subroutine that is called by
"DCI" "DCHI" & "DEI"
| 01 LBL "DSI"
02 CF 01 03 GTO 00 04 LBL "DSHI" 05 SF 01 06 LBL 00 07 STO 00 08 1 09 STO 02 10 STO 04 11 .5 12 STO 01 13 + 14 STO 03 15 STO 05 16 R^ 17 STO 06 18 LASTX 19 * 20 ST- 04 21 ST- 05 22 LASTX 23 1/X 24 3 25 CHS 26 RCL 00 27 LASTX 28 * 29 X^2 30 FC? 01 31 CHS 32 HGF+ 33 RCL 00 34 1 35 RCL 06 36 - 37 Y^X 38 * 39 2 40 RCL 06 41 2 42 - 43 Y^X 44 * 45 PI 46 * 47 RTN 48 LBL "DCI" 49 CF 01 50 GTO 00 51 LBL "DCHI" 52 SF 01 53 LBL 00 54 STO 06 55 CLX 56 2 57 STO 03 58 STO 05 59 SIGN 60 STO 01 61 STO 02 62 STO 04 63 X<>Y 64 STO 07 65 LASTX 66 1/X 67 ST+ 04 68 * 69 ST- 04 70 ST- 05 71 3 72 CHS 73 RCL 06 74 2 75 STO T 76 / 77 X^2 78 FC? 01 79 CHS 80 HGF+ 81 RCL 06 82 2 83 ST/ Y 84 ST/ Z 85 RCL 07 86 - 87 Y^X 88 * 89 PI 90 SQRT 91 * 92 STO 08 93 XEQ 08 94 RCL 06 95 RCL 07 96 Y^X 97 / 98 RCL 08 99 FC? 01 100 CHS 101 + 102 RTN 103 LBL "DEI" 104 STO 06 105 CLX 106 2 107 STO 03 108 X<>Y 109 STO 07 110 - 111 STO 04 112 1 113 STO 01 114 STO 02 115 2 116 ENTER^ 117 CHS |
118 RCL 06
119 HGF+ 120 RCL 06 121 * 122 STO 05 123 XEQ 08 124 RCL 05 125 + 126 RCL 06 127 RCL 07 128 Y^X 129 / 130 RTN 131 LBL "DCX" 132 CF 01 133 GTO 00 134 LBL "DSX" 135 SF 01 136 LBL 00 137 STO 00 138 X<>Y 139 STO 08 140 CHS 141 4 142 / 143 STO 04 144 STO 05 145 STO 06 146 STO 07 147 3 148 LASTX 149 1/X 150 STO 01 151 STO 09 152 * 153 STO 02 154 ST+ 05 155 1 156 STO 03 157 ST+ 06 158 FC? 01 159 GTO 00 160 ST+ 01 161 ST+ 04 162 ST+ 05 163 LBL 00 164 LASTX 165 + 166 ST+ 07 167 .5 168 ST+ 04 169 3 170 4 171 CHS 172 RCL 00 173 X^2 174 PI 175 * 176 4 177 / 178 ST* 09 179 X^2 180 CHS 181 HGF+ 182 RCL 00 183 1 184 RCL 08 185 - 186 Y^X 187 * 188 RCL 09 189 X<>Y 190 FS? 01 191 * 192 PI 193 1.5 194 STO 09 195 Y^X 196 * 197 2 198 RCL 08 199 ST+ X 200 RCL 09 201 - 202 Y^X 203 * 204 RTN 205 LBL "DSB1" 206 STO 00 207 RDN 208 STO 06 209 1 210 + 211 STO 05 212 .5 213 STO 02 214 STO 04 215 ST+ 05 216 * 217 STO 01 218 STO 03 219 ST+ 02 220 ST+ 04 221 X<>Y 222 STO 07 223 LASTX 224 * 225 ST- 03 226 ST- 04 227 2 228 3 229 CHS 230 RCL 00 231 LASTX 232 * 233 X^2 234 CHS |
235 HGF+
236 STO 01 237 RCL 06 238 1 239 + 240 XEQ "1/G+" 241 ST/ 01 242 RCL 00 243 RCL 06 244 RCL 07 245 - 246 Y^X 247 RCL 01 248 * 249 PI 250 * 251 2 252 RCL 07 253 RCL 06 254 ST+ X 255 - 256 1 257 - 258 Y^X 259 * 260 RTN 261 LBL "DINX" 262 SF 01 263 GTO 00 264 LBL "DJNX" 265 CF 01 266 LBL 00 267 STO 06 268 RDN 269 STO 07 270 1 271 + 272 STO 05 273 2 274 / 275 STO 01 276 STO 03 277 LASTX 278 1/X 279 + 280 STO 02 281 X<>Y 282 CHS 283 STO 08 284 LASTX 285 * 286 ST+ 03 287 + 288 STO 04 289 3 290 CHS 291 RCL 06 292 2 293 STO T 294 / 295 X^2 296 FC? 01 297 CHS 298 HGF+ 299 STO 01 300 RCL 05 301 XEQ "1/G+" 302 ST/ 01 303 RCL 06 304 RCL 07 305 RCL 08 306 + 307 Y^X 308 RCL 01 309 * 310 2 311 RCL 07 312 ST+ X 313 RCL 08 314 + 315 Y^X 316 / 317 PI 318 SQRT 319 * 320 RTN 321 LBL "DKNX" 322 SF 01 323 GTO 00 324 LBL "DYNX" 325 CF 01 326 LBL 00 327 STO 06 328 RDN 329 STO 07 330 1 331 X<>Y 332 - 333 STO 03 334 2 335 / 336 STO 01 337 STO 04 338 LASTX 339 1/X 340 + 341 STO 02 342 X<>Y 343 CHS 344 STO 08 345 LASTX 346 * 347 ST+ 04 348 + 349 STO 05 350 3 351 CHS |
352 RCL 06
353 2 354 STO T 355 / 356 X^2 357 FC? 01 358 CHS 359 STO 00 360 HGF+ 361 STO 09 362 RCL 03 363 XEQ "1/G+" 364 ST/ 09 365 16 366 RCL 07 367 ST+ 01 368 ST+ 02 369 ST+ 04 370 ST+ 05 371 Y^X 372 ST* 09 373 RCL 07 374 1 375 + 376 STO 03 377 2 378 3 379 CHS 380 RCL 00 381 HGF+ 382 STO 01 383 RCL 03 384 XEQ "1/G+" 385 ST/ 01 386 RCL 06 387 RCL 07 388 ST+ X 389 Y^X 390 RCL 01 391 * 392 1 393 CHS 394 ACOS 395 RCL 07 396 * 397 STO 05 398 FS? 01 399 CLX 400 COS 401 * 402 RCL 09 403 - 404 2 405 RCL 08 406 RCL 07 407 ST+ X 408 + 409 Y^X 410 / 411 PI 412 SQRT 413 * 414 RCL 06 415 RCL 08 416 RCL 07 417 - 418 Y^X 419 * 420 RCL 05 421 SIN 422 / 423 PI 424 2 425 / 426 CHS 427 X<>Y 428 FS? 01 429 * 430 RTN 431 LBL "DLANX" 432 STO 00 433 RDN 434 CHS 435 STO 02 436 CLX 437 SIGN 438 STO 01 439 + 440 STO 04 441 X<>Y 442 CHS 443 STO 05 444 LASTX 445 + 446 STO 03 447 2 448 ENTER^ 449 CHS 450 RCL 00 451 HGF+ 452 RCL 00 453 RCL 05 454 Y^X 455 * 456 X<> 01 457 RCL 02 458 - 459 XEQ "1/G+" 460 ST* 01 461 RCL 04 462 RCL 02 463 - 464 XEQ "1/G+" 465 ST/ 01 466 X<> 01 467 RTN 468 LBL "DAIRY" |
469 STO 00
470 1 471 STO 02 472 STO 04 473 ST+ X 474 3 475 / 476 STO 01 477 STO 03 478 ST+ X 479 STO 05 480 R^ 481 STO 06 482 LASTX 483 / 484 ST- 03 485 ST- 04 486 ST- 05 487 2 488 LASTX 489 CHS 490 RCL 00 491 3 492 Y^X 493 9 494 / 495 STO 07 496 HGF+ 497 RCL 00 498 * 499 STO 08 500 RCL 01 501 XEQ "1/G+" 502 ST/ 08 503 1 504 ST- 05 505 3 506 1/X 507 STO 01 508 2 509 LASTX 510 CHS 511 RCL 07 512 HGF+ 513 9 514 RCL 01 515 Y^X 516 * 517 STO 07 518 RCL 01 519 XEQ "1/G+" 520 RCL 07 521 X<>Y 522 / 523 STO Y 524 RCL 08 525 ST- Z 526 + 527 3 528 RCL 00 529 / 530 RCL 06 531 Y^X 532 ST* Z 533 * 534 243 535 6 536 1/X 537 Y^X 538 / 539 X<>Y 540 81 541 RCL 01 542 Y^X 543 / 544 RTN 545 LBL "DERF" 546 STO 05 547 X<>Y 548 CHS 549 STO 06 550 3 551 + 552 2 553 / 554 STO 04 555 1 556 STO 02 557 RCL 06 558 LASTX 559 1/X 560 STO 01 561 * 562 + 563 STO 03 564 2 565 ENTER^ 566 CHS 567 RCL 05 568 X^2 569 CHS 570 HGF+ 571 RCL 05 572 ST* Y 573 RCL 01 574 * 575 RCL 06 576 Y^X 577 * 578 RTN 579 LBL "DHMT" 580 STO 00 581 CLX 582 SIGN 583 STO 01 584 STO 03 585 .5 |
586 STO 02
587 + 588 STO 04 589 RDN 590 STO 05 591 LASTX 592 * 593 ST- 02 594 X<>Y 595 STO 06 596 LASTX 597 * 598 ST- 03 599 ST- 04 600 2 601 ENTER^ 602 CHS 603 RCL 00 604 X^2 605 HGF+ 606 RCL 00 607 * 608 STO 08 609 RCL 02 610 XEQ "1/G+" 611 STO 07 612 RCL 05 613 CHS 614 2 615 / 616 STO 02 617 XEQ "1/G+" 618 ST* 08 619 1 620 ST- 04 621 2 622 ENTER^ 623 CHS 624 RCL 00 625 X^2 626 HGF+ 627 RCL 07 628 * 629 RCL 08 630 - 631 2 632 RCL 05 633 RCL 06 634 + 635 Y^X 636 * 637 RCL 00 638 RCL 06 639 Y^X 640 / 641 PI 642 * 643 RTN 644 LBL 08 645 RCL 07 646 FRC 647 X#0? 648 GTO 00 649 LASTX 650 X<=0? 651 GTO 00 652 1 653 CHS 654 ST+ Y 655 X<>Y 656 Y^X 657 LASTX 658 FACT 659 * 660 RTN 661 LBL 00 662 1 663 LASTX 664 - 665 STO 04 666 XEQ "PSI" 667 RCL 06 668 LN 669 X<>Y 670 - 671 X<> 04 672 XEQ "1/G+" 673 RCL 04 674 * 675 RTN 676 LBL "DKUM" 677 STO 00 678 X<>Y 679 STO 05 680 CHS 681 RCL 02 682 STO 04 683 CLX 684 SIGN 685 STO 02 686 + 687 STO 03 688 2 689 ENTER^ 690 CHS 691 RCL 00 692 HGF+ 693 X<> 04 694 STO 02 695 XEQ "1/G+" 696 ST/ 04 697 RCL 04 698 RCL 00 699 RCL 05 700 Y^X 701 / 702 END |
( 1008 bytes / SIZE 010 )
| STACK | INPUTS1 | INPUTS2 | INPUTS3 | OUTPUTS |
| T | / | / | µ | / |
| Z | / | µ | a | / |
| Y | µ | n | n | / |
| X | x | x | x | (Dµ f) (x) |
µ is always to be entered first, then the order/index - if any - and finally, x in register X
Examples:
• Sine Integral
3.14 ENTER^
1.28 XEQ "DSI" >>>>
D3.14 Si ( 1.28 ) = -0.045395644
• Hyperbolic Sine Integral
3.14 ENTER^
1.28 XEQ "DSHI" >>>>
D3.14 Shi ( 1.28 ) = 0.576495211
• Cosine Integral
3.14 ENTER^
1.28 XEQ "DCI" >>>>
D3.14 Ci ( 1.28 ) = 1.367323895
• Hyperbolic Cosine Integral
3.14 ENTER^
1.28 XEQ "DCHI" >>>>
D3.14 Chi ( 1.28 ) = 1.405640394
• Exponential Integral
3.14 ENTER^
1.28 XEQ "DEI" >>>>
D3.14 Ei ( 1.28 ) = 1.982135606
• Fresnel Cosine Integral
3.14 ENTER^
1.28 XEQ "DCX" >>>>
D3.14 C ( 1.28 ) = 16.95612253
• Fresnel Sine Integral
3.14 ENTER^
1.28 XEQ "DSX" >>>>
D3.14 S ( 1.28 ) = -11.20302776
• Spherical Bessel Function - 1st kind
3.14 ENTER^
2.41 ENTER^
1.28 XEQ "DSB1" >>>>
D3.14 j2.41 ( 1.28 ) = -0.064451622
• Modified Bessel Function - 1st kind , n # -1 , -2 , -3 , .................
3.14 ENTER^
2.41 ENTER^
1.28 XEQ "DINX" >>>>
D3.14 I2.41 ( 1.28 ) = 0.352247279
• Bessel Function - 1st kind , n # -1 , -2 , -3 , .................
3.14 ENTER^
2.41 ENTER^
1.28 XEQ "DJNX" >>>>
D3.14 J2.41 ( 1.28 ) = -0.150524582
• Modified Bessel Function - 2nd kind - non-integer order
3.14 ENTER^
2.41 ENTER^
1.28 XEQ "DKNX" >>>>
D3.14 K2.41 ( 1.28 ) = -38.98469314
• Bessel Function - 2nd kind - non-integer order
3.14 ENTER^
2.41 ENTER^
1.28 XEQ "DYNX" >>>>
D3.14 Y2.41 ( 1.28 ) = 25.49308580
• Generalized Laguerre's Functions
3.14 ENTER^
1.76 ENTER^
2.41 ENTER^
1.28 XEQ "DLANX"
>>>> D3.14 L1.762.41
( 1.28 ) = -1.767203465
• Airy Functions
3.14 ENTER^
1.28 XEQ "DAIRY"
>>>> D3.14 Ai ( 1.28 ) = -0.162004857
X<>Y D3.14 Bi ( 1.28 ) = 3.432592624
• Error Function
3.14 ENTER^
1.28 XEQ "DERF" >>>>
D3.14 Erf ( 1.28 ) = 1.250557023
• Hermite Function
3.14 ENTER^
2.41 ENTER^
1.28 XEQ "DHMT" >>>>
D3.14 H2.41 ( 1.28 ) = 3.537707646
• Kummer's Functions With a = sqrt(2) & b = sqrt(3)
2 SQRT STO 01 3 SQRT STO 02
3.14 ENTER^
1.28 XEQ "DKUM" >>>>
D3.14 F ( 21/2 ; 31/2 ; 1.28
) = 2.075891500
Notes:
-For Kummer's Functions, R01 & R02 must be initialized first.
-R02 is modified during the calculations, but its original content
is restored at the end.
-As usual when a function is evaluated by a power series, the results
are not very accurate for large arguments,
-They may even be meaningless ... unless all the terms have the same
sign !
Reference: