Related

expintegral_ei

? expintegral_ei;

Calculate

expintegral_ei-numer

expintegral_ei(10),nu...

Calculate

expintegral_ei-taylor

taylor(expintegral_ei...

Calculate

expintegral_ei-float-taylor

float(taylor(expinteg...

Calculate

expintegral_ei-numer-taylor

taylor(expintegral_ei...

Calculate

expintegral_ei-numer

expintegral_ei(20),nu...

Calculate

expintegral_ei-numer-taylor

taylor(expintegral_ei...

Calculate

expintegral_ei-numer

expintegral_ei(10),nu...

Calculate

expintegral_ei-numer

for x in [10, 20, 30,...

Calculate

expintegral_ei-numer-taylor

taylor(expintegral_ei...

Calculate

expintegral_ei

Run Example
```(%i1)rat(taylor(expintegral_ei(x), x, -22, 5));
2               3
x + 22    23 (x + 22)    265 (x + 22)
(%o1)/T/ expintegral_ei(- 22) - ------- - ------------ - -------------
22           22              22
22 %e       968 %e        31944 %e
4                 5
6119 (x + 22)    35401 (x + 22)
- -------------- - --------------- + . . .
22                22
2811072 %e       77304480 %e
(%i2) ```
Run Example
```a1(x) := float(0.57721566490153286060651209008240243104215933593992 + log(abs(x)) + sum(x ^ k / (k * k!), k, 1, 128));
k
x
(%o1) a1(x) := float(0.57721566490153 + log(abs(x)) + sum(----, k, 1, 128))
k k!
(%i2) a2(x) := float(%e ^ x / x * (1 + sum((k - 1)! / x ^ k, k, 1, 128)));
x
%e           (k - 1)!
(%o2)         a2(x) := float(--- (1 + sum(--------, k, 1, 128)))
x               k
x
(%i3) a3(x) := float(%e ^ x / x);
x
%e
(%o3)                         a3(x) := float(---)
x
(%i4) for x in delete(0, makelist(x, x, -100, 100)) do(    ref : float(expintegral_ei(x)),    err1 : abs(abs(a1(x) - ref) / ref),    err2 : abs(abs(a2(x) - ref) / ref),    err3 : abs(abs(a3(x) - ref) / ref),    besterr : min(err1, err2, err3),    print('x=x, if besterr=err1 then "a1" elseif besterr=err2 then "a2" elseif besterr=err3 then "a3", besterr));
x = - 100 a2 9.7123329191636934E-5
x = - 99 a2 9.90668969011002E-5
x = - 98 a2 1.0106940526561632E-4
x = - 97 a2 1.0313326231402627E-4
x = - 96 a2 1.0526100034502512E-4
x = - 95 a2 1.0745528369848329E-4
x = - 94 a2 1.0971891711213437E-4
x = - 93 a2 1.120548546911335E-4
x = - 92 a2 1.1446620956448921E-4
x = - 91 a2 1.1695626426210979E-4
x = - 90 a2 1.1952848191064378E-4
x = - 89 a2 1.2218651828385486E-4
x = - 88 a2 1.2493423481450228E-4
x = - 87 a2 1.2777571263614977E-4
x = - 86 a2 1.30715267761129E-4
x = - 85 a2 1.3375746749394874E-4
x = - 84 a2 1.3690714819866308E-4
x = - 83 a2 1.4016943455207192E-4
x = - 82 a2 1.4354976042198152E-4
x = - 81 a2 1.4705389152831934E-4
x = - 80 a2 1.5068795007774837E-4
x = - 79 a2 1.544584415405104E-4
x = - 78 a2 1.5837228382742841E-4
x = - 77 a2 1.6243683906546727E-4
x = - 76 a2 1.666599482945217E-4
x = - 75 a2 1.7104996934601776E-4
x = - 74 a2 1.7561581828052153E-4
x = - 73 a2 1.803670147413239E-4
x = - 72 a2 1.853137316836488E-4
x = - 71 a2 1.9046684993683784E-4
x = - 70 a2 1.9583801816556179E-4
x = - 69 a2 2.0143971884706725E-4
x = - 68 a2 2.0728534095313864E-4
x = - 67 a2 2.1338926014172302E-4
x = - 66 a2 2.1976692734959984E-4
x = - 65 a2 2.2643496680435556E-4
x = - 64 a2 2.3341128463329311E-4
x = - 63 a2 2.4071518938350604E-4
x = - 62 a2 2.4836752597820114E-4
x = - 61 a2 2.5639082484716919E-4
x = - 60 a2 2.6480946822970264E-4
x = - 59 a2 2.7364987592847162E-4
x = - 58 a2 2.8294071298309549E-4
x = - 57 a2 2.9271312061924002E-4
x = - 56 a2 3.0300095690009003E-4
x = - 55 a2 3.1384087266218158E-4
x = - 54 a2 3.2527028752566545E-4
x = - 53 a2 3.3730124854433163E-4
x = - 52 a2 3.4960860106068614E-4
x = - 51 a2 3.5767609231079813E-4
x = - 50 a2 3.0283169972404117E-4
x = - 49 a2 6.0931311110812606E-4
x = - 48 a2 0.013712226616403
x = - 47 a3 0.020850406033685
x = - 46 a3 0.021294761046627
x = - 45 a3 0.021758483661381
x = - 44 a3 0.022242869330456
x = - 43 a3 0.022749331765
x = - 42 a3 0.023279416754854
x = - 41 a3 0.023834817974559
x = - 40 a3 0.024417395116647
x = - 39 a3 0.025029194762399
x = - 38 a3 0.025672474485012
x = - 37 a3 0.026349730785259
x = - 36 a3 0.027063731590523
x = - 35 a3 0.027817554212086
x = - 34 a3 0.028614629862002
x = - 33 a3 0.029458796092659
x = - 32 a3 0.030354358856075
x = - 31 a3 0.03130616630901
x = - 30 a3 0.032319697045024
x = - 29 a1 0.0058984693852964
x = - 28 a1 0.013937794270229
x = - 27 a1 0.0022156181864484
x = - 26 a1 0.0015125735275583
x = - 25 a1 3.8618572789692924E-4
x = - 24 a1 4.668658386231302E-4
x = - 23 a1 1.8134285580727845E-4
x = - 22 a1 3.7675677509784753E-5
x = - 21 a1 2.3109784330628769E-6
x = - 20 a1 1.535919454053257E-6
x = - 19 a1 3.315431878593789E-6
x = - 18 a1 3.7253355342658653E-7
x = - 17 a1 1.6848390457743497E-7
x = - 16 a1 6.36037951841017E-8
x = - 15 a1 6.9600320643282772E-9
x = - 14 a1 7.0411854948012859E-11
x = - 13 a1 1.8634074986585538E-10
x = - 12 a1 1.1589586988414338E-9
x = - 11 a1 3.8303522624562454E-10
x = - 10 a1 3.8542019136807368E-11
x = - 9 a1 3.2535700916810707E-12
x = - 8 a1 3.4066960115158543E-12
x = - 7 a1 4.9630046712846585E-12
x = - 6 a1 2.4915885024202082E-13
x = - 5 a1 4.8719887566683262E-14
x = - 4 a1 1.813050752359538E-14
x = - 3 a1 1.5554622847392893E-14
x = - 2 a1 2.8379637772426651E-16
x = - 1 a1 5.0606395937068529E-16
x = 1 a1 2.3433329897345802E-16
x = 2 a1 1.7927662598845908E-16
x = 3 a1 3.5763776503602066E-16
x = 4 a1 0.0
x = 5 a1 1.7681668956325575E-16
x = 6 a1 1.6526217029955487E-16
x = 7 a1 1.4841256114795748E-16
x = 8 a1 5.1631256486365746E-16
x = 9 a1 4.3815094212277121E-16
x = 10 a1 3.6493223955063252E-16
x = 11 a1 1.4979967502306353E-16
x = 12 a1 3.6478199769536566E-16
x = 13 a1 1.9560241266079398E-16
x = 14 a1 3.1229794456500611E-16
x = 15 a1 1.2386935736311241E-16
x = 16 a1 0.0
x = 17 a1 4.6055286743483411E-16
x = 18 a1 1.2008065377827354E-16
x = 19 a1 3.7436689986935731E-16
x = 20 a1 1.4543023155873645E-16
x = 21 a1 3.3801138416223459E-16
x = 22 a1 8.7067631616153058E-16
x = 23 a1 6.7127432195120522E-16
x = 24 a1 1.0329061581942695E-15
x = 25 a1 0.0
x = 26 a1 4.8638607246511363E-16
x = 27 a1 3.7222919051127893E-16
x = 28 a1 1.4221973464595905E-16
x = 29 a1 4.3410743511561288E-16
x = 30 a1 1.6541893745632124E-16
x = 31 a1 5.0366542016353165E-16
x = 32 a1 9.57397858691116E-16
x = 33 a1 2.9087560114267316E-16
x = 34 a1 8.8286626276563862E-16
x = 35 a1 1.0039504750911471E-15
x = 36 a1 1.267386433019123E-16
x = 37 a1 0.0
x = 38 a1 4.352171597856851E-16
x = 39 a1 0.0
x = 40 a1 1.6557063696578522E-16
x = 41 a1 1.4993769017626193E-15
x = 42 a1 7.5386424472255566E-16
x = 43 a1 8.5231185449546261E-16
x = 44 a1 4.2802998230490625E-16
x = 45 a1 1.6112959832996161E-16
x = 46 a1 9.6999899732587253E-16
x = 47 a1 1.0943425750361731E-15
x = 48 a1 1.2340403828324033E-15
x = 49 a1 1.4424556869857484E-15
x = 50 a1 1.5477576039321208E-16
x = 51 a1 4.648148877240191E-16
x = 52 a1 5.232559461470988E-16
x = 53 a1 3.9254516350416207E-16
x = 54 a1 3.9250283328503266E-16
x = 55 a1 1.1769614030858001E-15
x = 56 a1 2.2050191717728532E-16
x = 57 a1 1.4866923906688642E-15
x = 58 a1 3.7113106469001029E-16
x = 59 a1 0.0
x = 60 a1 2.6341909492338221E-15
x = 61 a1 9.7512572219094192E-15
x = 62 a1 3.538472606351384E-14
x = 63 a1 1.0433787473268398E-13
x = 64 a1 3.0230724874076371E-13
x = 65 a1 8.5116771702945877E-13
x = 66 a1 2.3136818268189764E-12
x = 67 a1 6.1046171651780541E-12
x = 68 a1 1.5649429618094339E-11
x = 69 a1 3.9018962093173204E-11
x = 70 a1 9.4679598756244864E-11
x = 71 a1 2.2378217458813728E-10
x = 72 a1 5.1557745560023495E-10
x = 73 a1 1.1587095378430077E-9
x = 74 a1 2.5419623562560444E-9
x = 75 a1 5.4470876286683079E-9
x = 76 a1 1.1408739598309355E-8
x = 77 a1 2.3369809826402493E-8
x = 78 a1 4.6846036910394982E-8
x = 79 a1 9.1946955539634584E-8
x = 80 a1 1.7680164607730312E-7
x = 81 a1 3.3323362058667337E-7
x = 82 a1 6.1595059475256182E-7
x = 83 a1 1.1170968948009239E-6
x = 84 a1 1.9887937639766822E-6
x = 85 a1 3.4772964833575857E-6
x = 86 a1 5.973643626383514E-6
x = 87 a1 1.0087157905253089E-5
x = 88 a1 1.6749838310841035E-5
x = 89 a1 2.736147081847143E-5
x = 90 a1 4.3987031197447875E-5
x = 91 a1 6.9619441477385992E-5
x = 92 a1 1.085216859486735E-4
x = 93 a2 1.1953475615223983E-4
x = 94 a2 1.1696220583726812E-4
x = 95 a2 1.1447183942281202E-4
x = 96 a2 1.1206019226221599E-4
x = 97 a2 1.0972398042593128E-4
x = 98 a2 1.074600894958742E-4
x = 99 a2 1.0526556418912717E-4
x = 100 a2 1.0313759868520487E-4
(%o4)                                done
(%i5) ```
Run Example
```a1(x) := float(0.57721566490153286060651209008240243104215933593992 + log(abs(x)) + sum(x ^ k / (k * k!), k, 1, 64));
k
x
(%o1) a1(x) := float(0.57721566490153 + log(abs(x)) + sum(----, k, 1, 64))
k k!
(%i2) a2(x) := float(%e ^ x / x * (1 + sum((k - 1)! / x ^ k, k, 1, 64)));
x
%e           (k - 1)!
(%o2)          a2(x) := float(--- (1 + sum(--------, k, 1, 64)))
x               k
x
(%i3) a3(x) := -1.9000649792328929E-88*(x+22.0)^64-4.2464944297754076E-87*(x+22.0)^63                                         -9.492969805902987E-86*(x+22.0)^62                                         -2.122690297573232E-84*(x+22.0)^61                                         -4.747750632182922E-83*(x+22.0)^60                                         -1.0622086159797149E-81*(x+22.0)^59                                         -2.3771496265913751E-80*(x+22.0)^58                                         -5.3214788120040256E-79*(x+22.0)^57                                         -1.1916311475588821E-77*(x+22.0)^56                                         -2.6692537665372524E-76*(x+22.0)^55                                         -5.9811056396782957E-75*(x+22.0)^54                                         -1.3406704593745883E-73*(x+22.0)^53                                         -3.0061956173979742E-72*(x+22.0)^52                                         -6.7433090322890202E-71*(x+22.0)^51                                         -1.5131983634130564E-69*(x+22.0)^50                                         -3.3969749819854333E-68*(x+22.0)^49                                         -7.6290349656894803E-67*(x+22.0)^48                                         -1.71409577402792E-65*(x+22.0)^47                                         -3.8529781760903084E-64*(x+22.0)^46                                         -8.6648679893868591E-63*(x+22.0)^45                                         -1.9495714486559344E-61*(x+22.0)^44                                         -4.3886953272556278E-60*(x+22.0)^43                                         -9.8845410518905115E-59*(x+22.0)^42                                         -2.2274346520072279E-57*(x+22.0)^41                                         -5.0220104350911686E-56*(x+22.0)^40                                         -1.1328209370728269E-54*(x+22.0)^39                                         -2.5563869157183307E-53*(x+22.0)^38                                         -5.7705750935285106E-52*(x+22.0)^37                                         -1.3027081726313322E-50*(x+22.0)^36                                         -2.9401295401332217E-49*(x+22.0)^35                                         -6.6307393471521297E-48*(x+22.0)^34                                         -1.4932329033057209E-46*(x+22.0)^33                                         -3.3546438095276633E-45*(x+22.0)^32                                         -7.5088574900127436E-44*(x+22.0)^31                                         -1.6719594020592552E-42*(x+22.0)^30                                         -3.6963600853649327E-41*(x+22.0)^29                                         -8.0956652525475791E-40*(x+22.0)^28                                         -1.7521309989150719E-38*(x+22.0)^27                                         -3.7369164106189495E-37*(x+22.0)^26                                         -7.8307222629120482E-36*(x+22.0)^25                                         -1.6072117465568993E-34*(x+22.0)^24                                         -3.2204627369322888E-33*(x+22.0)^23                                         -6.2790044895737295E-32*(x+22.0)^22                                         -1.1871701081687363E-30*(x+22.0)^21                                         -2.1690829567804746E-29*(x+22.0)^20                                         -3.8162342302509827E-28*(x+22.0)^19                                         -6.4416284083168799E-27*(x+22.0)^18                                         -1.0391987171454787E-25*(x+22.0)^17                                         -1.5958645313722511E-24*(x+22.0)^16                                         -2.3228608182298272E-23*(x+22.0)^15                                         -3.1897948682357527E-22*(x+22.0)^14                                         -4.1114993505279957E-21*(x+22.0)^13                                         -4.9461528270720219E-20*(x+22.0)^12                                         -5.5178525122197877E-19*(x+22.0)^11                                         -5.666176809540679E-18*(x+22.0)^10                                         -5.3095141232869844E-17*(x+22.0)^9                                         -4.493142901350178E-16*(x+22.0)^8                                         -3.3903894170769208E-15*(x+22.0)^7                                         -2.2448974370045494E-14*(x+22.0)^6                                         -1.277415745577136E-13*(x+22.0)^5                                         -6.0719737026532761E-13*(x+22.0)^4                                         -2.3140779007333616E-12*(x+22.0)^3                                         -6.6278684024778167E-12*(x+22.0)^2                                         -1.2679400422131475E-11*(x+22.0)                                         -1.2149378956204365E-11;
64
(%o3) a3(x) := (- 1.9000649792328929E-88) (x + 22.0)
63                                       62
- 4.2464944297754076E-87 (x + 22.0)   + (- 9.492969805902987E-86) (x + 22.0)
61
+ (- 2.122690297573232E-84) (x + 22.0)
60
+ (- 4.747750632182922E-83) (x + 22.0)
59
+ (- 1.0622086159797149E-81) (x + 22.0)
58
+ (- 2.3771496265913751E-80) (x + 22.0)
57
+ (- 5.3214788120040256E-79) (x + 22.0)
56
+ (- 1.1916311475588821E-77) (x + 22.0)
55
+ (- 2.6692537665372524E-76) (x + 22.0)
54
+ (- 5.9811056396782957E-75) (x + 22.0)
53
+ (- 1.3406704593745883E-73) (x + 22.0)
52
+ (- 3.0061956173979742E-72) (x + 22.0)
51
+ (- 6.7433090322890202E-71) (x + 22.0)
50
+ (- 1.5131983634130564E-69) (x + 22.0)
49
+ (- 3.3969749819854333E-68) (x + 22.0)
48
+ (- 7.6290349656894803E-67) (x + 22.0)
47
+ (- 1.71409577402792E-65) (x + 22.0)
46
+ (- 3.8529781760903084E-64) (x + 22.0)
45
+ (- 8.6648679893868591E-63) (x + 22.0)
44
+ (- 1.9495714486559344E-61) (x + 22.0)
43
+ (- 4.3886953272556278E-60) (x + 22.0)
42
+ (- 9.8845410518905115E-59) (x + 22.0)
41
+ (- 2.2274346520072279E-57) (x + 22.0)
40
+ (- 5.0220104350911686E-56) (x + 22.0)
39
+ (- 1.1328209370728269E-54) (x + 22.0)
38
+ (- 2.5563869157183307E-53) (x + 22.0)
37
+ (- 5.7705750935285106E-52) (x + 22.0)
36
+ (- 1.3027081726313322E-50) (x + 22.0)
35
+ (- 2.9401295401332217E-49) (x + 22.0)
34
+ (- 6.6307393471521297E-48) (x + 22.0)
33
+ (- 1.4932329033057209E-46) (x + 22.0)
32
+ (- 3.3546438095276633E-45) (x + 22.0)
31
+ (- 7.5088574900127436E-44) (x + 22.0)
30
+ (- 1.6719594020592552E-42) (x + 22.0)
29
+ (- 3.6963600853649327E-41) (x + 22.0)
28
+ (- 8.0956652525475791E-40) (x + 22.0)
27
+ (- 1.7521309989150719E-38) (x + 22.0)
26
+ (- 3.7369164106189495E-37) (x + 22.0)
25
+ (- 7.8307222629120482E-36) (x + 22.0)
24
+ (- 1.6072117465568993E-34) (x + 22.0)
23
+ (- 3.2204627369322888E-33) (x + 22.0)
22
+ (- 6.2790044895737295E-32) (x + 22.0)
21
+ (- 1.1871701081687363E-30) (x + 22.0)
20
+ (- 2.1690829567804746E-29) (x + 22.0)
19
+ (- 3.8162342302509827E-28) (x + 22.0)
18
+ (- 6.4416284083168799E-27) (x + 22.0)
17
+ (- 1.0391987171454787E-25) (x + 22.0)
16
+ (- 1.5958645313722511E-24) (x + 22.0)
15
+ (- 2.3228608182298272E-23) (x + 22.0)
14
+ (- 3.1897948682357527E-22) (x + 22.0)
13
+ (- 4.1114993505279957E-21) (x + 22.0)
12
+ (- 4.9461528270720219E-20) (x + 22.0)
11
+ (- 5.5178525122197877E-19) (x + 22.0)
10
+ (- 5.666176809540679E-18) (x + 22.0)
9
+ (- 5.3095141232869844E-17) (x + 22.0)
8
+ (- 4.493142901350178E-16) (x + 22.0)
7
+ (- 3.3903894170769208E-15) (x + 22.0)
6
+ (- 2.2448974370045494E-14) (x + 22.0)
5
+ (- 1.277415745577136E-13) (x + 22.0)
4
+ (- 6.0719737026532761E-13) (x + 22.0)
3
+ (- 2.3140779007333616E-12) (x + 22.0)
2
+ (- 6.6278684024778167E-12) (x + 22.0)
+ (- 1.2679400422131475E-11) (x + 22.0) - 1.2149378956204365E-11
(%i4) e1(x) := block([ref : float(expintegral_ei(x))], abs(abs(a1(x) - ref) / ref));
!abs(a1(x) - ref)!
(%o4) e1(x) := block([ref : float(expintegral_ei(x))], !----------------!)
!      ref       !
(%i5) e2(x) := block([ref : float(expintegral_ei(x))], abs(abs(a2(x) - ref) / ref));
!abs(a2(x) - ref)!
(%o5) e2(x) := block([ref : float(expintegral_ei(x))], !----------------!)
!      ref       !
(%i6) e3(x) := block([ref : float(expintegral_ei(x))], abs(abs(a3(x) - ref) / ref));
!abs(a3(x) - ref)!
(%o6) e3(x) := block([ref : float(expintegral_ei(x))], !----------------!)
!      ref       !
(%i7) for x in float(delete(0, makelist(-20.7 + x/100, x, -10, 10))) do(    ref : float(expintegral_ei(x)),    err1 : e1(x),    err2 : e2(x),    err3 : e3(x),    besterr : min(err1, err2, err2),    print('x=x, if besterr=err1 then "a1" elseif besterr=err2 then "a2" elseif besterr=err3 then "a3", besterr));
x = - 20.8 a2 687.4707338779738
x = - 20.79 a2 709.0563167757665
x = - 20.78 a2 731.3305065790395
x = - 20.77 a2 754.3156172799389
x = - 20.76 a2 778.0346971877046
x = - 20.75 a2 802.5115534641305
x = - 20.74 a2 827.7707774913587
x = - 20.73 a2 853.8377711003491
x = - 20.72 a2 880.7387736899708
x = - 20.71 a2 908.5008902671124
x = - 20.7 a2 937.152120439873
x = - 20.69 a2 966.7213883963711
x = - 20.68 a1 966.3219205813016
x = - 20.67 a1 912.7656552536257
x = - 20.66 a1 896.0506380839912
x = - 20.65 a1 844.8141438901995
x = - 20.64 a1 804.5568802649279
x = - 20.63 a1 825.7827316753246
x = - 20.62 a1 769.0031844454268
x = - 20.61 a1 768.6171407113646
x = - 20.6 a1 708.7103058797691
(%o7)                                done
(%i8) ```

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