diff --git a/books/bookvol10.4.pamphlet b/books/bookvol10.4.pamphlet index 850d053..e90957e 100644 --- a/books/bookvol10.4.pamphlet +++ b/books/bookvol10.4.pamphlet @@ -12638,6 +12638,728 @@ DistinctDegreeFactorize(F,FP): C == T @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{package DFSFUN DoubleFloatSpecialFunctions} +The special functions in this section are developed as special cases +but can all be expressed in terms of generalized hypergeomentric +functions ${}_pF_q$ or its generalization, the Meijer G function. +\cite{Luk169,Luk269} +The long term plan is to reimplement these functions using the +generalized version. +<>= +)set break resume +)sys rm -f DoubleFloatSpecialFunctions.output +)spool DoubleFloatSpecialFunctions.output +)set message test on +)set message auto off +)clear all + +--S 1 of 5 +)show DoubleFloatSpecialFunctions +--R DoubleFloatSpecialFunctions is a package constructor +--R Abbreviation for DoubleFloatSpecialFunctions is DFSFUN +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.4.pamphlet to see algebra source code for DFSFUN +--R +--R------------------------------- Operations -------------------------------- +--R Gamma : DoubleFloat -> DoubleFloat fresnelC : Float -> Float +--R fresnelS : Float -> Float +--R Beta : (DoubleFloat,DoubleFloat) -> DoubleFloat +--R Beta : (Complex DoubleFloat,Complex DoubleFloat) -> Complex DoubleFloat +--R E1 : DoubleFloat -> OnePointCompletion DoubleFloat +--R Ei : OnePointCompletion DoubleFloat -> OnePointCompletion DoubleFloat +--R Ei1 : OnePointCompletion DoubleFloat -> OnePointCompletion DoubleFloat +--R Ei2 : OnePointCompletion DoubleFloat -> OnePointCompletion DoubleFloat +--R Ei3 : OnePointCompletion DoubleFloat -> OnePointCompletion DoubleFloat +--R Ei4 : OnePointCompletion DoubleFloat -> OnePointCompletion DoubleFloat +--R Ei5 : OnePointCompletion DoubleFloat -> OnePointCompletion DoubleFloat +--R Ei6 : OnePointCompletion DoubleFloat -> OnePointCompletion DoubleFloat +--R En : (Integer,DoubleFloat) -> OnePointCompletion DoubleFloat +--R Gamma : Complex DoubleFloat -> Complex DoubleFloat +--R airyAi : Complex DoubleFloat -> Complex DoubleFloat +--R airyAi : DoubleFloat -> DoubleFloat +--R airyBi : DoubleFloat -> DoubleFloat +--R airyBi : Complex DoubleFloat -> Complex DoubleFloat +--R besselI : (DoubleFloat,DoubleFloat) -> DoubleFloat +--R besselI : (Complex DoubleFloat,Complex DoubleFloat) -> Complex DoubleFloat +--R besselJ : (DoubleFloat,DoubleFloat) -> DoubleFloat +--R besselJ : (Complex DoubleFloat,Complex DoubleFloat) -> Complex DoubleFloat +--R besselK : (DoubleFloat,DoubleFloat) -> DoubleFloat +--R besselK : (Complex DoubleFloat,Complex DoubleFloat) -> Complex DoubleFloat +--R besselY : (DoubleFloat,DoubleFloat) -> DoubleFloat +--R besselY : (Complex DoubleFloat,Complex DoubleFloat) -> Complex DoubleFloat +--R digamma : DoubleFloat -> DoubleFloat +--R digamma : Complex DoubleFloat -> Complex DoubleFloat +--R hypergeometric0F1 : (DoubleFloat,DoubleFloat) -> DoubleFloat +--R hypergeometric0F1 : (Complex DoubleFloat,Complex DoubleFloat) -> Complex DoubleFloat +--R logGamma : DoubleFloat -> DoubleFloat +--R logGamma : Complex DoubleFloat -> Complex DoubleFloat +--R polygamma : (NonNegativeInteger,DoubleFloat) -> DoubleFloat +--R polygamma : (NonNegativeInteger,Complex DoubleFloat) -> Complex DoubleFloat +--R +--E 1 + +--S 2 of 5 +pearceyC:=_ +[ [0.00, 0.0000000], [0.25, 0.3964561], [0.50, 0.5502472], [0.75, 0.6531193],_ + [1.00, 0.7217059], [1.25, 0.762404], [1.50, 0.779084], [1.75, 0.774978],_ + [2.00, 0.753302], [2.25, 0.717446], [2.50, 0.670986], [2.75, 0.617615],_ + [3.00, 0.561020], [3.25, 0.504745], [3.50, 0.452047], [3.75, 0.405762],_ + [4.00, 0.368193], [4.25, 0.341021], [4.50, 0.325249], [4.75, 0.321186],_ + [5.00, 0.328457], [5.25, 0.346058], [5.50, 0.372439], [5.75, 0.405610],_ + [6.00, 0.443274], [6.25, 0.482966], [6.50, 0.522202], [6.75, 0.558620],_ + [7.00, 0.590116], [7.25, 0.614951], [7.50, 0.631845], [7.75, 0.640034],_ + [8.00, 0.639301], [8.25, 0.629969], [8.50, 0.612868], [8.75, 0.589271],_ + [9.00, 0.560804], [9.25, 0.529344], [9.50, 0.496895], [9.75, 0.465469],_ + [10.00, 0.436964], [10.25, 0.413053], [10.50, 0.395087], [10.75, 0.384027],_ + [11.00, 0.380390], [11.25, 0.384231], [11.50, 0.395149], [11.75, 0.412319],_ + [12.00, 0.434555], [12.25, 0.460384], [12.50, 0.488146], [12.75, 0.516096],_ + [13.00, 0.542511], [13.25, 0.565798], [13.50, 0.584583], [13.75, 0.597795],_ + [14.00, 0.604721], [14.25, 0.605048], [14.50, 0.598871], [14.75, 0.586682],_ + [15.00, 0.569335], [15.25, 0.547984], [15.50, 0.524009], [15.75, 0.498930],_ + [16.00, 0.474310], [16.25, 0.451659], [16.50, 0.432343], [16.75, 0.417502],_ + [17.00, 0.407985], [17.25, 0.404300], [17.50, 0.406589], [17.75, 0.414627],_ + [18.00, 0.427837], [18.25, 0.445331], [18.50, 0.465972], [18.75, 0.488443],_ + [19.00, 0.511332], [19.25, 0.533222], [19.50, 0.552774], [19.75, 0.568812],_ + [20.00, 0.580389], [20.25, 0.586847], [20.50, 0.587849], [20.75, 0.583401],_ + [21.00, 0.573842], [21.25, 0.559824], [21.50, 0.542266], [21.75, 0.522293],_ + [22.00, 0.501167], [22.25, 0.480207], [22.50, 0.460707], [22.75, 0.443854],_ + [23.00, 0.430662], [23.25, 0.421906], [23.50, 0.418080], [23.75, 0.419367],_ + [24.00, 0.425635], [24.25, 0.436444], [24.50, 0.451078], [24.75, 0.468594],_ + [25.00, 0.487880], [25.25, 0.507725], [25.50, 0.526896], [25.75, 0.544215],_ + [26.00, 0.558626], [26.25, 0.569272], [26.50, 0.575524], [26.75, 0.577038],_ + [27.00, 0.573766], [27.25, 0.565954], [27.50, 0.554127], [27.75, 0.539054],_ + [28.00, 0.521695], [28.25, 0.503146], [28.50, 0.484566], [28.75, 0.467104],_ + [29.00, 0.451832], [29.25, 0.439675], [29.50, 0.431359], [29.75, 0.427366],_ + [30.00, 0.427908], [30.25, 0.432913], [30.50, 0.442034], [30.75, 0.454673],_ + [31.00, 0.470019], [31.25, 0.487100], [31.50, 0.504844], [31.75, 0.522148],_ + [32.00, 0.537944], [32.25, 0.551266], [32.50, 0.561307], [32.75, 0.567471],_ + [33.00, 0.569407], [33.25, 0.567026], [33.50, 0.560508], [33.75, 0.550288],_ + [34.00, 0.537026], [34.25, 0.521566], [34.50, 0.504881], [34.75, 0.488015],_ + [35.00, 0.472012], [35.25, 0.457857], [35.50, 0.446415], [35.75, 0.438375],_ + [36.00, 0.434212], [36.25, 0.434156], [36.50, 0.438182], [36.75, 0.446014],_ + [37.00, 0.457140], [37.25, 0.470848], [37.50, 0.486272], [37.75, 0.502444],_ + [38.00, 0.518359], [38.25, 0.533031], [38.50, 0.545560], [38.75, 0.555182],_ + [39.00, 0.561321], [39.25, 0.563619], [39.50, 0.561957], [39.75, 0.556463],_ + [40.00, 0.547503], [40.25, 0.535653], [40.50, 0.521665], [40.75, 0.506420],_ + [41.00, 0.490870], [41.25, 0.475980], [41.50, 0.462670], [41.75, 0.451755],_ + [42.00, 0.443897], [42.25, 0.439565], [42.50, 0.439006], [42.75, 0.442234],_ + [43.00, 0.449025], [43.25, 0.458938], [43.50, 0.471341], [43.75, 0.485450],_ + [44.00, 0.500382], [44.25, 0.515205], [44.50, 0.529002], [44.75, 0.540923],_ + [45.00, 0.550239], [45.25, 0.556387], [45.50, 0.559004], [45.75, 0.557947],_ + [46.00, 0.553301], [46.25, 0.545374], [46.50, 0.534676], [46.75, 0.521883],_ + [47.00, 0.507802], [47.25, 0.493312], [47.50, 0.479313], [47.75, 0.466670],_ + [48.00, 0.456160], [48.25, 0.448425], [48.50, 0.443930], [48.75, 0.442936],_ + [49.00, 0.445486], [49.25, 0.451406], [49.50, 0.460311], [49.75, 0.471633],_ + [50.00, 0.484658]] +--R +--R +--R (1) +--R [[0.0,0.0], [0.25,0.3964561], [0.5,0.5502472], [0.75,0.6531193], +--R [1.0,0.7217059], [1.25,0.762404], [1.5,0.779084], [1.75,0.774978], +--R [2.0,0.753302], [2.25,0.717446], [2.5,0.670986], [2.75,0.617615], +--R [3.0,0.56102], [3.25,0.504745], [3.5,0.452047], [3.75,0.405762], +--R [4.0,0.368193], [4.25,0.341021], [4.5,0.325249], [4.75,0.321186], +--R [5.0,0.328457], [5.25,0.346058], [5.5,0.372439], [5.75,0.40561], +--R [6.0,0.443274], [6.25,0.482966], [6.5,0.522202], [6.75,0.55862], +--R [7.0,0.590116], [7.25,0.614951], [7.5,0.631845], [7.75,0.640034], +--R [8.0,0.639301], [8.25,0.629969], [8.5,0.612868], [8.75,0.589271], +--R [9.0,0.560804], [9.25,0.529344], [9.5,0.496895], [9.75,0.465469], +--R [10.0,0.436964], [10.25,0.413053], [10.5,0.395087], [10.75,0.384027], +--R [11.0,0.38039], [11.25,0.384231], [11.5,0.395149], [11.75,0.412319], +--R [12.0,0.434555], [12.25,0.460384], [12.5,0.488146], [12.75,0.516096], +--R [13.0,0.542511], [13.25,0.565798], [13.5,0.584583], [13.75,0.597795], +--R [14.0,0.604721], [14.25,0.605048], [14.5,0.598871], [14.75,0.586682], +--R [15.0,0.569335], [15.25,0.547984], [15.5,0.524009], [15.75,0.49893], +--R [16.0,0.47431], [16.25,0.451659], [16.5,0.432343], [16.75,0.417502], +--R [17.0,0.407985], [17.25,0.4043], [17.5,0.406589], [17.75,0.414627], +--R [18.0,0.427837], [18.25,0.445331], [18.5,0.465972], [18.75,0.488443], +--R [19.0,0.511332], [19.25,0.533222], [19.5,0.552774], [19.75,0.568812], +--R [20.0,0.580389], [20.25,0.586847], [20.5,0.587849], [20.75,0.583401], +--R [21.0,0.573842], [21.25,0.559824], [21.5,0.542266], [21.75,0.522293], +--R [22.0,0.501167], [22.25,0.480207], [22.5,0.460707], [22.75,0.443854], +--R [23.0,0.430662], [23.25,0.421906], [23.5,0.41808], [23.75,0.419367], +--R [24.0,0.425635], [24.25,0.436444], [24.5,0.451078], [24.75,0.468594], +--R [25.0,0.48788], [25.25,0.507725], [25.5,0.526896], [25.75,0.544215], +--R [26.0,0.558626], [26.25,0.569272], [26.5,0.575524], [26.75,0.577038], +--R [27.0,0.573766], [27.25,0.565954], [27.5,0.554127], [27.75,0.539054], +--R [28.0,0.521695], [28.25,0.503146], [28.5,0.484566], [28.75,0.467104], +--R [29.0,0.451832], [29.25,0.439675], [29.5,0.431359], [29.75,0.427366], +--R [30.0,0.427908], [30.25,0.432913], [30.5,0.442034], [30.75,0.454673], +--R [31.0,0.470019], [31.25,0.4871], [31.5,0.504844], [31.75,0.522148], +--R [32.0,0.537944], [32.25,0.551266], [32.5,0.561307], [32.75,0.567471], +--R [33.0,0.569407], [33.25,0.567026], [33.5,0.560508], [33.75,0.550288], +--R [34.0,0.537026], [34.25,0.521566], [34.5,0.504881], [34.75,0.488015], +--R [35.0,0.472012], [35.25,0.457857], [35.5,0.446415], [35.75,0.438375], +--R [36.0,0.434212], [36.25,0.434156], [36.5,0.438182], [36.75,0.446014], +--R [37.0,0.45714], [37.25,0.470848], [37.5,0.486272], [37.75,0.502444], +--R [38.0,0.518359], [38.25,0.533031], [38.5,0.54556], [38.75,0.555182], +--R [39.0,0.561321], [39.25,0.563619], [39.5,0.561957], [39.75,0.556463], +--R [40.0,0.547503], [40.25,0.535653], [40.5,0.521665], [40.75,0.50642], +--R [41.0,0.49087], [41.25,0.47598], [41.5,0.46267], [41.75,0.451755], +--R [42.0,0.443897], [42.25,0.439565], [42.5,0.439006], [42.75,0.442234], +--R [43.0,0.449025], [43.25,0.458938], [43.5,0.471341], [43.75,0.48545], +--R [44.0,0.500382], [44.25,0.515205], [44.5,0.529002], [44.75,0.540923], +--R [45.0,0.550239], [45.25,0.556387], [45.5,0.559004], [45.75,0.557947], +--R [46.0,0.553301], [46.25,0.545374], [46.5,0.534676], [46.75,0.521883], +--R [47.0,0.507802], [47.25,0.493312], [47.5,0.479313], [47.75,0.46667], +--R [48.0,0.45616], [48.25,0.448425], [48.5,0.44393], [48.75,0.442936], +--R [49.0,0.445486], [49.25,0.451406], [49.5,0.460311], [49.75,0.471633], +--R [50.0,0.484658]] +--R Type: List List Float +--E 2 + +--S 3 of 5 +[[x.1,x.2,fresnelC(x.1),fresnelC(x.1)-x.2] for x in pearceyC] +--R +--R +--R (2) +--R [[0.0,0.0,0.0,0.0], +--R [0.25,0.3964561,0.3964560954 2000459941,- 0.4579995400 59 E -8], +--R [0.5,0.5502472,0.5502471546 4500637208,- 0.4535499362 79 E -7], +--R [0.75,0.6531193,0.6531193584 292607957,0.5842926079 57 E -7], +--R [1.0,0.7217059,0.7217059242 9260508777,0.2429260508 78 E -7], +--R [1.25,0.762404,0.7624042531 0695862575,0.2531069586 258 E -6], +--R [1.5,0.779084,0.7790837385 0396370968,- 0.2614960362 903 E -6], +--R [1.75,0.774978,0.7749781554 8647675573,0.1554864767 557 E -6], +--R [2.0,0.753302,0.7533023754 6789116559,0.3754678911 656 E -6], +--R [2.25,0.717446,0.7174457114 2671496912,- 0.2885732850 309 E -6], +--R [2.5,0.670986,0.6709858725 0950347483,- 0.1274904965 252 E -6], +--R [2.75,0.617615,0.6176149424 522644261,- 0.5754773557 39 E -7], +--R [3.0,0.56102,0.5610203289 781386693,0.3289781386 693 E -6], +--R [3.25,0.504745,0.5047454684 5758505252,0.4684575850 525 E -6], +--R [3.5,0.452047,0.4520471473 7344948252,0.1473734494 825 E -6], +--R [3.75,0.405762,0.4057621282 0111750857,0.1282011175 086 E -6], +--R [4.0,0.368193,0.3681929762 8097479631,- 0.2371902520 37 E -7], +--R [4.25,0.341021,0.3410206544 3025861592,- 0.3455697413 8408 E -6], +--R [4.5,0.325249,0.3252492294 0997382931,0.2294099738 293 E -6], +--R [4.75,0.321186,0.3211858108 1411496285,- 0.1891858850 372 E -6], +--R [5.0,0.328457,0.3284566248 6755260618,- 0.3751324473 9382 E -6], +--R [5.25,0.346058,0.3460579739 835289212,- 0.2601647107 88 E -7], +--R [5.5,0.372439,0.3724388324 2286847464,- 0.1675771315 254 E -6], +--R [5.75,0.40561,0.4056100692 8402241876,0.6928402241 876 E -7], +--R [6.0,0.443274,0.4432738563 3762333739,- 0.1436623766 626 E -6], +--R [6.25,0.482966,0.4829657793 6269790038,- 0.2206373020 996 E -6], +--R [6.5,0.522202,0.5222015767 8062637928,- 0.4232193736 207 E -6], +--R [6.75,0.55862,0.5586203035 9563381818,0.3035956338 182 E -6], +--R [7.0,0.590116,0.5901160610 939772876,0.6109397728 76 E -7], +--R [7.25,0.614951,0.6149512165 651359762,0.2165651359 762 E -6], +--R [7.5,0.631845,0.6318452111 5510492853,0.2111551049 285 E -6], +--R [7.75,0.640034,0.6400345450 8057441808,0.5450805744 1808 E -6], +--R [8.0,0.639301,0.6393012479 3060490759,0.2479306049 076 E -6], +--R [8.25,0.629969,0.6299689859 2595953795,- 0.1407404046 2 E -7], +--R [8.5,0.612868,0.6128678201 6845088171,- 0.1798315491 183 E -6], +--R [8.75,0.589271,0.5892704028 202327594,- 0.5971797672 406 E -6], +--R [9.0,0.560804,0.5608039810 6395486433,- 0.1893604513 57 E -7], +--R [9.25,0.529344,0.5293438831 4394301245,- 0.1168560569 876 E -6], +--R [9.5,0.496895,0.4968951155 6828252077,0.1155682825 208 E -6], +--R [9.75,0.465469,0.4654692556 4195264614,0.2556419526 4614 E -6], +--R [10.0,0.436964,0.4369639527 2938203483,- 0.4727061796 517 E -7], +--R [10.25,0.413053,0.4130520539 2945147154,- 0.9460705485 2846 E -6], +--R [10.5,0.395087,0.3950866689 6445290526,- 0.3310355470 9474 E -6], +--R [10.75,0.384027,0.3840274319 9186745464,0.4319918674 5464 E -6], +--R [11.0,0.38039,0.3803918718 5818433242,0.0000018718 581843324], +--R [11.25,0.384231,0.3842342501 3415159269,0.0000032501 3415159269], +--R [11.5,0.395149,0.3951525621 4136633426,0.0000035621 4136633426], +--R [11.75,0.412319,0.4123227194 2948890487,0.0000037194 2948890487], +--R [12.0,0.434555,0.4345573415 1310106383,0.0000023415 1310106383], +--R [12.25,0.460384,0.4603851724 4692111457,0.0000011724 469211146], +--R [12.5,0.488146,0.4881459845 7100939501,- 0.1542899060 5 E -7], +--R [12.75,0.516096,0.5160950016 9800402129,- 0.9983019959 7871 E -6], +--R [13.0,0.542511,0.5425104114 0076790311,- 0.5885992320 9689 E -6], +--R [13.25,0.565798,0.5657974628 3445804807,- 0.5371655419 5193 E -6], +--R [13.5,0.584583,0.5845829612 9626646639,- 0.3870373353 36 E -7], +--R [13.75,0.597795,0.5977946491 2254037734,- 0.3508774596 227 E -6], +--R [14.0,0.604721,0.6047209589 3428343112,- 0.4106571656 89 E -7], +--R [14.25,0.605048,0.6050478757 7470272898,- 0.1242252972 71 E -6], +--R [14.5,0.598871,0.5988710711 7868251227,0.7117868251 227 E -7], +--R [14.75,0.586682,0.5866829870 7071159647,0.9870707115 9647 E -6], +--R [15.0,0.569335,0.5693360588 8342021462,0.0000010588 834202146], +--R [15.25,0.547984,0.5479846850 9637199303,0.6850963719 9303 E -6], +--R [15.5,0.524009,0.5240097909 4969920392,0.7909496992 0392 E -6], +--R [15.75,0.49893,0.4989308254 9359679937,0.8254935967 9937 E -6], +--R [16.0,0.47431,0.4743107173 2032792592,0.7173203279 2592 E -6], +--R [16.25,0.451659,0.4516596582 0625475374,0.6582062547 5374 E -6], +--R [16.5,0.432343,0.4323435693 667817725,0.5693667817 725 E -6], +--R [16.75,0.417502,0.4175027376 9772555286,0.7376977255 5286 E -6], +--R [17.0,0.407985,0.4079854159 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3686238137,- 0.0000011002 631376186], +--R [21.0,0.545885,0.5458838021 1302594691,- 0.0000011978 869740531], +--R [21.25,0.562375,0.5623742046 1345743628,- 0.7953865425 6372 E -6], +--R [21.5,0.574811,0.5748105273 3873582708,- 0.4726612641 729 E -6], +--R [21.75,0.582472,0.5824715801 7267885681,- 0.4198273211 432 E -6], +--R [22.0,0.584939,0.5849389064 8781662671,- 0.9351218337 329 E -7], +--R [22.25,0.582119,0.5821190706 5525943405,0.7065525943 405 E -7], +--R [22.5,0.574246,0.5742457362 6788616758,- 0.2637321138 324 E -6], +--R [22.75,0.561862,0.5618616308 0580392102,- 0.3691941960 79 E -6], +--R [23.0,0.545782,0.5457817221 8835794711,- 0.2778116420 529 E -6], +--R [23.25,0.52704,0.5270400684 8894301738,0.6848894301 738 E -7], +--R [23.5,0.506824,0.5068237756 6322281921,- 0.2243367771 808 E -6], +--R [23.75,0.486399,0.4863982517 4424787908,- 0.7482557521 2092 E -6], +--R [24.0,0.467029,0.4670284356 6225373159,- 0.5643377462 6841 E -6], +--R [24.25,0.449901,0.4499008775 2768569685,- 0.1224723143 032 E -6], +--R [24.5,0.436051,0.4360514454 5938610932,0.4454593861 0932 E -6], +--R [24.75,0.426303,0.4263030410 2444838947,0.4102444838 947 E -7], +--R [25.0,0.421217,0.4212170480 2256113561,0.4802256113 561 E -7], +--R [25.25,0.421062,0.4210613602 4178097468,- 0.6397582190 2532 E -6], +--R [25.5,0.425797,0.4257967889 3146047209,- 0.2110685395 279 E -6], +--R [25.75,0.435083,0.4350825056 1623586834,- 0.4943837641 3166 E -6], +--R [26.0,0.4483,0.4483000011 9239677468,0.1192396774 7 E -8], +--R [26.25,0.464594,0.4645939109 1118988398,- 0.8908881011 602 E -7], +--R [26.5,0.482927,0.4829270344 0426767399,0.3440426767 4 E -7], +--R [26.75,0.502146,0.5021460338 2247443232,0.3382247443 23 E -7], +--R [27.0,0.521054,0.5210536692 2170336971,- 0.3307782966 303 E -6], +--R [27.25,0.538483,0.5384830674 2690379321,0.6742690379 321 E -7], +--R [27.5,0.553369,0.5533694357 5487427564,0.4357548742 756 E -6], +--R [27.75,0.564814,0.5648148314 9166491027,0.8314916649 1027 E -6], +--R [28.0,0.572142,0.5721420631 6790330366,0.6316790330 366 E -7], +--R [28.25,0.574935,0.5749345026 8645175971,- 0.4973135482 4029 E -6], +--R [28.5,0.57306,0.5730594822 652482619,- 0.5177347517 3809 E -6], +--R [28.75,0.566674,0.5666739801 1092060771,- 0.1988907939 23 E -7], +--R [29.0,0.556212,0.5562123974 5318039664,0.3974531803 966 E -6], +--R [29.25,0.542357,0.5423573357 3057143592,0.3357305714 359 E -6], +--R [29.5,0.525995,0.5259953183 8591850851,0.3183859185 085 E -6], +--R [29.75,0.50816,0.5081603114 6691447077,0.3114669144 708 E -6], +--R [30.0,0.489969,0.4899686293 6294561315,- 0.3706370543 8685 E -6], +--R [30.25,0.472549,0.4725493003 70056332,0.3003700563 32 E -6], +--R [30.5,0.456974,0.4569742329 8007908723,0.2329800790 872 E -6], +--R [30.75,0.444193,0.4441924732 1202200062,- 0.5267879779 9938 E -6], +--R [31.0,0.434973,0.4349725872 2359219467,- 0.4127764078 0533 E -6], +--R [31.25,0.429857,0.4298566440 1538454345,- 0.3559846154 5655 E -6], +--R [31.5,0.429129,0.4291285439 2130018393,- 0.4560786998 1607 E -6], +--R [31.75,0.432799,0.4327985404 7868362044,- 0.4595213163 7956 E -6], +--R [32.0,0.440605,0.4406047705 8303021964,- 0.2294169697 804 E -6], +--R [32.25,0.452031,0.4520315818 7779157328,0.5818777915 7328 E -6], +--R [32.5,0.466343,0.4663433562 2850339586,0.3562285033 9586 E -6], +--R [32.75,0.482632,0.4826316757 5316315801,- 0.3242468368 4199 E -6], +--R [33.0,0.499873,0.4998728498 5761360978,- 0.1501423863 902 E -6], +--R [33.25,0.516992,0.5169919568 4608062411,- 0.4315391937 59 E -7], +--R [33.5,0.53293,0.5329297341 0588518507,- 0.2658941148 149 E -6], +--R [33.75,0.546708,0.5467080330 374204341,0.3303742043 41 E -7], +--R [34.0,0.55749,0.5574894980 70161353,- 0.5019298386 47 E -6], +--R [34.25,0.564629,0.5646284739 1529463715,- 0.5260847053 6285 E -6], +--R [34.5,0.567709,0.5677093003 3772777525,0.3003377277 753 E -6], +--R [34.75,0.56657,0.5665706285 1241575866,0.6285124157 5866 E -6], +--R [35.0,0.561313,0.5613133551 8174616414,0.3551817461 641 E -6], +--R [35.25,0.552293,0.5522925303 0361142379,- 0.4696963885 762 E -6], +--R [35.5,0.540094,0.5400936428 6312004459,- 0.3571368799 554 E -6], +--R [35.75,0.525495,0.5254947452 6478477127,- 0.2547352152 287 E -6], +--R [36.0,0.509417,0.5094168513 7281277168,- 0.1486271872 283 E -6], +--R [36.25,0.492866,0.4928661334 9354600731,0.1334935460 073 E -6], +--R [36.5,0.476871,0.4768716484 1410150512,0.6484141015 0512 E -6], +--R [36.75,0.46242,0.4624199779 4599876237,- 0.2205400123 76 E -7], +--R [37.0,0.450396,0.4503960869 296054767,0.8692960547 67 E -7], +--R [37.25,0.441528,0.4415290525 3642844985,0.0000010525 364284499], +--R [37.5,0.436345,0.4363459415 0326090466,0.9415032609 0466 E -6], +--R [37.75,0.435144,0.4351444622 142641974,0.4622142641 974 E -6], +--R [38.0,0.437971,0.4379709919 0155134798,- 0.8098448652 02 E -8], +--R [38.25,0.444626,0.4446262686 7031444849,0.2686703144 4849 E -6], +--R [38.5,0.45467,0.4546708406 3699119293,0.8406369911 9293 E -6], +--R [38.75,0.467461,0.4674625380 8725087486,0.0000015380 872508749], +--R [39.0,0.482187,0.4821902969 1952370995,0.0000032969 1952370995], +--R [39.25,0.497924,0.4979286483 1116955857,0.0000046483 1116955857], +--R [39.5,0.51369,0.5136882217 6583353419,- 0.0000017782 341664658], +--R [39.75,0.528507,0.5285123707 6397702394,0.0000053707 6397702393], +--R [40.0,0.541464,0.5414672429 1598081817,0.0000032429 159808182], +--R [40.25,0.551768,0.5517745240 9094539537,0.0000065240 9094539537], +--R [40.5,0.558799,0.5588050506 5912486677,0.0000060506 5912486677], +--R [40.75,0.56214,0.5621369791 6412056122,- 0.0000030208 358794388], +--R [41.0,0.561608,0.5616247711 0814605692,0.0000167711 081460569], +--R [41.25,0.557258,0.5572675955 430330709,0.0000095955 430330709], +--R [41.5,0.549384,0.5493773200 0894645567,- 0.0000066799 9105354433], +--R [41.75,0.538494,0.5385616342 9485623221,0.0000676342 9485623221], +--R [42.0,0.525282,0.5252752111 2342898796,- 0.0000067888 7657101203], +--R [42.25,0.51058,0.5104939314 7702373041,- 0.0000860685 2297626959], +--R [42.5,0.495309,0.4953417684 038968965,0.0000327684 038968965], +--R [42.75,0.480418,0.4806241406 5752533599,0.0002061406 57525336], +--R [43.0,0.466829,0.4667676898 4497320465,- 0.0000613101 5502679535], +--R [43.25,0.455375,0.4553638694 6250373139,- 0.0000111305 374962686], +--R [43.5,0.446755,0.4466829634 3044419959,- 0.0000720365 6955580042], +--R [43.75,0.441487,0.4413450709 5947820527,- 0.0001419290 405217947], +--R [44.0,0.439878,0.4398575523 27691464,- 0.0000204476 72308536], +--R [44.25,0.442007,0.4420222062 7982284284,0.0000152062 798228428], +--R [44.5,0.44772,0.4478984397 6769945092,0.0001784397 676994509], +--R [44.75,0.456645,0.4559273692 7109515615,- 0.0007176307 2890484385], +--R [45.0,0.468209,0.4676487917 6051833044,- 0.0005602082 3948166956], +--R [45.25,0.481681,0.4819138773 9121176935,0.0002328773 912117694], +--R [45.5,0.496215,0.4954833490 8753370091,- 0.0007316509 1246629909], +--R [45.75,0.510904,0.5106859986 5658467469,- 0.0002180013 434153253], +--R [46.0,0.524837,0.5242725127 6971944478,- 0.0005644872 3028055522], +--R [46.25,0.537153,0.5375126919 1563903102,0.0003596919 15639031], +--R [46.5,0.547099,0.5460962647 9105883281,- 0.0010027352 089411672], +--R [46.75,0.55407,0.5524834800 5502643628,- 0.0015865199 449735637], +--R [47.0,0.55765,0.5580597398 2343233094,0.0004097398 234323309], +--R [47.25,0.557635,0.5590882082 4170920981,0.0014532082 417092098], +--R [47.5,0.554044,0.5475186455 874275914,- 0.0065253544 125724086], +--R [47.75,0.54712,0.5455984940 601030917,- 0.0015215059 398969083], +--R [48.0,0.537309,0.5196318344 7879605961,- 0.0176771655 212039404], +--R [48.25,0.525234,0.5131703830 5664910816,- 0.0120636169 433508918], +--R [48.5,0.511657,0.4873016398 0175736065,- 0.0243553601 982426393], +--R [48.75,0.497426,0.5181479895 3691666632,0.0207219895 369166663], +--R [49.0,0.483428,0.4598623177 3318010769,- 0.0235656822 668198923], +--R [49.25,0.470529,0.4397312463 9609993647,- 0.0307977536 039000635], +--R [49.5,0.459523,0.4156435749 1719610661,- 0.0438794250 8280389339], +--R [49.75,0.451084,0.4230292350 336881258,- 0.0280547649 663118742], +--R [50.0,0.445722,0.3044252284 9276788618,- 0.1412967715 072321138]] +--R Type: List List Float +--E 5 + +)spool +)lisp (bye) +@ +<>= +==================================================================== +DoubleFloatSpecialFunctions examples +==================================================================== + +The formula used will agree with the Table of the Fresnel Integral +by Pearcey (1959) to 6 decimal places up to an argument of about 35.0. +After that the summation gets slowly worse, agreeing to only 2 digits +at about 45.0. + +fresnelC(1.5) + 0.7790837385 0396370968 + +fresnelS(1.5) + 0.4154833182 6565542581 + + + +See Also: +o )show DoubleFloatSpecialFunctions + +@ \pagehead{DoubleFloatSpecialFunctions}{DFSFUN} \pagepic{ps/v104doublefloatspecialfunctions.ps}{DFSFUN}{1.00} @@ -12660,10 +13382,12 @@ DistinctDegreeFactorize(F,FP): C == T \cross{DFSFUN}{Ei5} \\ \cross{DFSFUN}{Ei6} & \cross{DFSFUN}{En} & -\cross{DFSFUN}{Gamma} & +\cross{DFSFUN}{fresnelC} & +\cross{DFSFUN}{fresnelS} & +\cross{DFSFUN}{Gamma} \\ \cross{DFSFUN}{hypergeometric0F1} & -\cross{DFSFUN}{logGamma} \\ -\cross{DFSFUN}{polygamma} &&&& +\cross{DFSFUN}{logGamma} & +\cross{DFSFUN}{polygamma} && \end{tabular} <>= @@ -12683,11 +13407,13 @@ DistinctDegreeFactorize(F,FP): C == T ++ real and complex floating point. DoubleFloatSpecialFunctions(): Exports == Impl where - NNI ==> NonNegativeInteger - PI ==> Integer - R ==> DoubleFloat - C ==> Complex DoubleFloat - OPR ==> OnePointCompletion R + NNI ==> NonNegativeInteger + PI ==> Integer + R ==> DoubleFloat + C ==> Complex DoubleFloat + OPR ==> OnePointCompletion R + F ==> Float + LF ==> List Float Exports ==> with Gamma: R -> R @@ -12846,18 +13572,27 @@ DoubleFloatSpecialFunctions(): Exports == Impl where ++ hypergeometric0F1(c,z) is the hypergeometric function ++ \spad{0F1(; c; z)}. + fresnelS : F -> F + ++ fresnelS(f) denotes the Fresnel integral S + ++ + ++X fresnelS(1.5) + + fresnelC : F -> F + ++ fresnelC(f) denotes the Fresnel integral C + ++ + ++X fresnelC(1.5) + Impl ==> add a, v, w, z: C n, x, y: R - -- These are hooks to Bruce's boot code. Gamma z == CGAMMA(z)$Lisp Gamma x == RGAMMA(x)$Lisp @ \subsection{The Exponential Integral} -\subsection{The E1 function} +\subsubsection{The E1 function} (Quoted from Segletes\cite{2}): A number of useful integrals exist for which no exact solutions have @@ -13035,7 +13770,7 @@ the Chebyshev polynomial for computing $E_1$. This agrees with the handbook values to almost the last published digit. See the {\tt e1.input} pamphlet for regression testing against the handbook tables. -\subsection{E1:R$\rightarrow$OPR} +\subsubsection{E1:R$\rightarrow$OPR} The special function E1 below was originally derived from a function written by T.Haavie as the {\tt expint.c} function in the Numlibc library by Lars Erik Lund. Haavie approximates the E1 function by two @@ -13194,7 +13929,7 @@ The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229\cite{4}. @ \subsection{The Ei Function} This function is based on Kin L. Lee's work\cite{8}. See also \cite{21}. -\subsection{Abstract} +\subsubsection{Abstract} The exponential integral Ei(x) is evaluated via Chebyshev series expansion of its associated functions to achieve high relative accuracy throughout the entire real line. The Chebyshev coefficients @@ -13202,7 +13937,7 @@ for these functions are given to 30 significant digits. Clenshaw's\cite{20} method is modified to furnish an efficient procedure for the accurate solution of linear systems having near-triangular coefficient matrices. -\subsection{Introduction} +\subsubsection{Introduction} The evaulation of the exponential integral \begin{equation} Ei(x)=\int_{-\infty}^{X}{\frac{e^u}{u}}\ du=-E_1(-x), x \ne 0 @@ -13251,7 +13986,7 @@ are useful as a master function for finding approximations for (or involving) $Ei(x)$ (e.g. \cite{12,13}) where prescribed accuracy is less than 30 figures. -\subsection{Discussion} +\subsubsection{Discussion} It is proposed here to provide for the evaluation of $Ei(x)$ by obtaining Chebyshev coefficients for the associated functions given by @@ -13320,7 +14055,7 @@ partitioned into 2 and 3 intervals, respectively, to provide approximations to $xe^{-x}Ei(x)$ by polynomials of about the same degree. -\subsection{Expansions in Chebyshev Series} +\subsubsection{Expansions in Chebyshev Series} Let $\phi(t)$ be a differentiable function defined on [-1,1]. To facilitate discussion, denote its Chebyshev series and that of its @@ -13351,7 +14086,7 @@ result in a loss of accuracy if the trial solutions selected are not sufficiently independent. How the difficulty is overcome will be pointed out subsequently. -\subsection{The function $xe^{-x}Ei(x)$ on the Finite Interval} +\subsubsection{The function $xe^{-x}Ei(x)$ on the Finite Interval} We consider first the Chebyshev series expansion of \begin{equation} @@ -13680,7 +14415,7 @@ generalized to solve linear systems having coefficient matrices of order N, the deletion of whose first $r$ ($r < N$) rows and last $r$ columns yields upper triangular matrices of order $N-r$. -\subsection{The Function $(1/x)[Ei(x)-log\vert x\vert-\gamma]$} +\subsubsection{The Function $(1/x)[Ei(x)-log\vert x\vert-\gamma]$} Let \begin{equation} @@ -13756,7 +14491,7 @@ and then determining $\alpha_k$ and $\beta_k$ ($k=M-1, M-2, \ldots, 0$) by backward recurrence by means of equation 33. The arbitrary constant $c$ is determined by substituting 34 into 32. -\subsection{The Function $xe^{-x}Ei(x)$ on the Infinite Interval} +\subsubsection{The Function $xe^{-x}Ei(x)$ on the Infinite Interval} Let \begin{equation} f(x)=xe^{-x}Ei(x),\quad -\infty < x \le b < 0,\quad or 0 < b \le x < \infty @@ -13840,7 +14575,7 @@ and computing $\alpha_k$ (k=0,1,$\ldots$,M-1) by means of equation 43 by backward recurrence. The substitution of equation 46 into 42 then enables one to determine $c$ from the resulting equation. -\subsection{Remarks on Convergence and Accuracy} +\subsubsection{Remarks on Convergence and Accuracy} The Chebyshev coefficients of table 3 were computed on the IBM 7094 with 50-digit normalized floating-point arithmetic. In order to assure @@ -14697,6 +15432,101 @@ $\infty$ & -1.000 & 0.100000000 0000000000 00000000001 E 01\\ 32 & 1.000 & 0.103341356 4216241049 43493552567 E 01\\ \end{tabular} +\subsection{The Fresnel Integral\cite{PEA56,LOS60}} +The Fresnel function is +\[C(x) - iS(x) = \int_0^x{i^{-t^2}}~dt = \int_0^x{\exp(-i\pi{}t^2/2)}~dt\] + +We compare Axiom's results to Pearcey's tables which show the fresnel +results to 6 decimal places. Computation of these values requires floats +as the range quickly exceeds DoubleFloat. In each decade of the range we +increase the number of terms by a factor of 10. So we compute with 10 +terms in the range 0.0-10.0, 100 terms in 10.0-20.0, etc. +\subsubsection{fresnelC} +The fresnelC is the real portion of the Fresnel integral, C(u), is defined as: +\[C(\sqrt{2x/\pi}) = +\frac{1}{2}\int_0^x{J_{-\frac{1}{2}}(t)}~dt = +\frac{1}{\sqrt{(2\pi)}}\int_0^x{\frac{\cos(t)}{\sqrt{t}}}~dt\] +where ${J_{-\frac{1}{2}}(t)}$ is the Bessel function of the first kind of +order $-\frac{1}{2}$. + +This is related to the better known definition of C(u), namely: +\[C(u)=\int_0^u{\cos{\frac{\pi{}t^2}{2}}}~dt\] +where $x=\pi{}u^2/2$, or $u=(2x/\pi{})^{1/2}$ + +fresnelC is an analytic function of z with z=0 as a two-sheeted branch point. +Along the positive real axis the real definition gives: +\[C(0) = 0\] +\[\lim_{x\rightarrow{}+\infty{}} C(x)=\frac{1}{2}\] + +The asymptotic behavior of the function in the corner +$|\rm{arc\ }z| \le \pi-\epsilon$, ($\epsilon > 0$), for $|z| \gg 1$ is given by +\[C(z) \approx \frac{1}{2} + \frac{\sin z}{\sqrt{2\pi{}z}} +\left(1-\frac{1\cdot{}3}{(2z)^2}+\frac{1\cdot{}3\cdot{}5\cdot{}7}{(2z)^4}- +\cdots\right)-\frac{\cos z}{\sqrt{2\pi{}z}}\left(\frac{1}{(2z)}- +\frac{1\cdot{}3\cdot{}5}{(2z)^3}+\cdots\right)\] +(Note: Pearcey has a sign error for the second term (\cite{PEA56},p7) + +The first approximation is +\[C(z) \approx \frac{1}{2} + \frac{\sin z}{\sqrt{2\pi{}z}}\] + +Axiom uses the power series at the zero point: +\[C(z)=\sqrt{\frac{2z}{\pi}}\sum_{k=0}^n{(-1)^k\frac{z^{2k}}{(4k+1)(2k)!}}\] + +<>= + fresnelC(z:F):F == + z < 0 => error "fresnelC not defined for negative argument" + z = 0 => 0 + n:PI:= 100 + sqrt((2.0/pi()$F)*z)*_ + reduce(_+,[(-1)**k*z**(2*k)/(factorial(2*k)*(4*k+1))_ + for k in 0..n])$LF + +@ + +\subsubsection{fresnelS} +The fresnelS is the complex portion of the Fresnel integral, +S(u), is defined as: +\[S(\sqrt{2x/\pi}) = +\frac{1}{2}\int_0^x{J_{\frac{1}{2}}(t)}~dt = +\frac{1}{\sqrt{(2\pi)}}\int_0^x{\frac{\sin(t)}{\sqrt{t}}}~dt\] +where ${J_{\frac{1}{2}}(t)}$ is the Bessel function of the first kind of +order $\frac{1}{2}$. + +This is related to the better known definition of S(u), namely: +\[S(u)=\int_0^u{\sin{\frac{\pi{}t^2}{2}}}~dt\] +where $x=\pi{}u^2/2$, or $u=(2x/\pi{})^{1/2}$ + +fresnelS is an analytic function of z with z=0 as a two-sheeted branch point. +Along the positive real axis the real definition gives: +\[S(0) = 0\] +\[\lim_{x\rightarrow{}+\infty{}} S(x)=\frac{1}{2}\] + +The asymptotic behavior of the function in the corner +$|\rm{arc\ }z| \le \pi-\epsilon$, ($\epsilon > 0$), for $|z| \gg 1$ is given by +\[S(z) \approx \frac{1}{2} - \frac{\cos z}{\sqrt{2\pi{}z}} +\left(1-\frac{1\cdot{}3}{(2z)^2}+\frac{1\cdot{}3\cdot{}5\cdot{}7}{(2z)^4}- +\cdots\right)-\frac{\sin z}{\sqrt{2\pi{}z}}\left(\frac{1}{(2z)}- +\frac{1\cdot{}3\cdot{}5}{(2z)^3}+\cdots\right)\] + +The first approximation is +\[S(z) \approx \frac{1}{2} - \frac{\cos z}{\sqrt{2\pi{}z}}\] + +Axiom uses the power series at the zero point: +\[S(z)= +\sqrt{\frac{2z}{\pi}}\sum_{k=0}^n{(-1)^k\frac{z^{2k+1}}{(4k+3)(2k+1)!}}\] + +<>= + fresnelS(z:F):F == + z < 0 => error "fresnelS not defined for negative argument" + z = 0 => 0 + n:PI:= 100 + sqrt((2.0/pi()$F)*z)*_ + reduce(_+, + [(-1)**k*(z**(2*k+1))/(factorial(2*k+1)*(4*k+3)) _ + for k in 0..n])$LF + +@ + <>= polygamma(k,z) == CPSI(k, z)$Lisp diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet index 9e49165..b48d01c 100644 --- a/books/bookvolbib.pamphlet +++ b/books/bookvolbib.pamphlet @@ -757,6 +757,15 @@ alg\'{e}brique. {\sl Journal de l'Ecole Polytechnique}, 14:124-148, 1833 Joseph Liouville. Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales dont la valeur est alg\'{e}brique. {\sl Journal de l'Ecole Polytechnique}, 14:149-193, 1833 +\bibitem[Los60]{Los60} +L\"osch, Friedrich ``Tables of Higher Functions'' +McGraw-Hill Book Company 1960 +\bibitem[Luk169]{Luk169} +Luke, Yudell L. ``The Special Functions and their Approximations'' Volume I +Academic Press (1969) Mathematics in Science and Engineering Volume 53-I +\bibitem[Luk269]{Luk269} +Luke, Yudell L. ``The Special Functions and their Approximations'' Volume II +Academic Press (1969) Mathematics in Science and Engineering Volume 53-II \bibitem[Mul97]{Mul97} Thom Mulders. ``A note on subresultants and a correction to the lazard/rioboo/trager formula in rational function integration'' @@ -766,6 +775,13 @@ M.W. Ostrogradsky. De l'int\'{e}gration des fractions rationelles. {\sl Bulletin de la Classe Physico-Math\'{e}matiques de l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,} IV:145-167,286-300, 1845 +\bibitem[Pea56]{Pea56} +Pearcey, T. ``Table of the Fresnel Integral'' +Cambridge University Press 1956 +\bibitem[PTVF95]{PTVF95} +Press, William H., Teukolsky, Saul A., Vetterling, William T., +Flannery, Brian P. ``Numerical Recipes in C'' +Cambridge University Press (1995) ISBN 0-521-43108-5 \bibitem[Pu09]{Pu09} Puffinware LLC ``Singular Value Decomposition (SVD) Tutorial'' \verb|www.puffinwarellc.com/p3a.htm| diff --git a/changelog b/changelog index f625740..7e71500 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20100718 tpd src/algebra/Makefile add help and test for DFSFUN +20100718 tpd books/bookvolbib add Los60, Luk169, Luk269, Pea56, PTVF95 +20100718 tpd books/bookvol10.4 add fresnelC, fresnelS to DFSFUN 20100717 tpd src/axiom-website/patches.html 20100717.01.tpd.patch 20100717 tpd src/algebra/Makefile handle case-insensitive MAC filesystem 20100713 wxh src/axiom-website/patches.html 20100713.01.wxh.patch diff --git a/src/algebra/Makefile.pamphlet b/src/algebra/Makefile.pamphlet index 7b27206..601de6e 100644 --- a/src/algebra/Makefile.pamphlet +++ b/src/algebra/Makefile.pamphlet @@ -17418,6 +17418,7 @@ SPADHELP=\ ${HELP}/DivisorCategory.help \ ${HELP}/DoubleFloat.help \ ${HELP}/DoubleFloatMatrix.help \ + ${HELP}/DoubleFloatSpecialFunctions.help \ ${HELP}/DoubleFloatVector.help \ ${HELP}/DoublyLinkedAggregate.help \ ${HELP}/DrawOption.help \ @@ -18110,6 +18111,7 @@ REGRESS= \ DivisorCategory.regress \ DoubleFloat.regress \ DoubleFloatMatrix.regress \ + DoubleFloatSpecialFunctions.regress \ DoubleFloatVector.regress \ DoublyLinkedAggregate.regress \ DrawOption.regress \ @@ -20319,6 +20321,18 @@ ${HELP}/DoubleFloatMatrix.help: ${BOOKS}/bookvol10.3.pamphlet >${INPUT}/DoubleFloatMatrix.input @echo "DoubleFloatMatrix (DFMAT)" >>${HELPFILE} +${HELP}/DoubleFloatSpecialFunctions.help: ${BOOKS}/bookvol10.4.pamphlet + @echo 7210 create DoubleFloatSpecialFunctions.help from \ + ${BOOKS}/bookvol10.4.pamphlet + @${TANGLE} -R"DoubleFloatSpecialFunctions.help" \ + ${BOOKS}/bookvol10.4.pamphlet \ + >${HELP}/DoubleFloatSpecialFunctions.help + @cp -f ${HELP}/DoubleFloatSpecialFunctions.help ${HELP}/DFSFUN.help + @${TANGLE} -R"DoubleFloatSpecialFunctions.input" \ + ${BOOKS}/bookvol10.4.pamphlet \ + >${INPUT}/DoubleFloatSpecialFunctions.input + @echo "DoubleFloatSpecialFunctions (DFSFUN)" >>${HELPFILE} + ${HELP}/DoubleFloatVector.help: ${BOOKS}/bookvol10.3.pamphlet @echo 7210 create DoubleFloatVector.help from \ ${BOOKS}/bookvol10.3.pamphlet diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 9815880..15953d2 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -2996,5 +2996,7 @@ src/input/derivefail.input failing integrals from derive 6.10
books/bookvol10.* add fresnelS, fresnelC to LF, COMMONOP, LIMITPS, EXPR
20100717.01.tpd.patch src/algebra/Makefile handle case-insensitive MAC filesystem
+20100718.01.tpd.patch +books/bookvol10.4 add fresnelC, fresnelS to DFSFUN