ExpIntegralEi. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

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1 ExpItegralEi Notatios Traditioal ame Expoetial itegral Ei Traditioal otatio Ei Mathematica StadardForm otatio ExpItegralEi Primary defiitio Ei k k 2 log log k Specific values Values at fixed poits Ei0 Values at ifiities Ei Ei Ei Π Ei Π Ei

2 2 Geeral characteristics Domai ad aalyticity Ei is a aalytical fuctio of which is defied over the whole complex -plae Ei Symmetries ad periodicities Mirror symmetry Ei Ei Periodicity No periodicity Poles ad essetial sigularities The fuctio Ei has a essetial sigularity at. At the same time, the poit is a brach poit ig Ei. Brach poits The fuctio Ei has two brach poits: 0,. At the same time, the poit is a essetial sigularity Ei 0, Ei, 0 log Ei, log Brach cuts The fuctio Ei is a sigle-valued fuctio o the -plae cut alog the iterval, 0 where it has discotiuities from both sides Ei, 0, lim Eix Ε Eix Π ; x 0 Ε lim Eix Ε Eix Π ; x 0 Ε0

3 3 Series represetatios Geeralied power series Expasios at geeric poit 0 For the fuctio itself arg Π Ei Ei 0 Π 2 log 0 arg 0 log 0 log 0 log log ; arg Π Ei Ei 0 Π 2 log 0 arg 0 log 0 log 0 log log O arg Π Ei Ei 0 Π arg 0 log 0 log 0 2 log 0 log 0 log 0 k 0 k k k, 0 0 k arg Π Ei Ei 0 Π arg 0 log 0 log 0 2 log k kj jk log 0 log k 0 j 0 k j arg Π Ei Ei 0 Π arg 0 log 0 log 0 2 log 0 log 0 log 0 0 k k j 0 j j 0 j k j 0 k arg Π Ei Ei 0 Π arg 0 log 0 log 0 2 log 0 log 0 log 0 k k k 0 0 2F 2, ; 2, 2 k; 0 0 k k k

4 arg Π Ei Ei 0 Π 2 log 0 log 0 log 0 arg 0 log 0 log 0 O 0 Expasios o brach cuts For the fuctio itself arg Π Ei Eix Π 2 Π arg x Π x x x x x 2 x 2 x 2 ; x x x arg Π Ei Eix Π 2 Π arg x Π x x x x x 2 x 2 x 2 O x 3 ; x x Ei Eix Π Π arg Π 2 Π arg x k x k k, x x k ; x x 0 k Ei Eix Π Π arg Π 2 Π arg x k kj x jk x x k ; x x 0 k j j Ei Eix Π Π arg Π 2 Π arg x k x k j 0 j j x j k j x k ; x x Ei Eix Π Π arg Π 2 Π arg x k x k k xk k 2F 2, ; 2, 2 k; x x k ; x x arg Π Ei Eix Π 2 Π arg x Π O x ; x x 0 Expasios at 0 For the fuctio itself Ei 2 log log Ei 2 log log ; O Ei 2 log log k k k

5 Ei log k k k Π arg Π 0 True Ei 2 F 2, ; 2, 2; 2 log log Ei 2 log log O ; Ei F ; F 2 log log k 2 Ei k 0 k 2 k 2 2 2F 2, 2; 3, 3; Summed form of the trucated series expasio k k k Ei j k 2 log log j Asymptotic series expasios Ei 2 log log log log 2 6 ; Ei 2F 0, ; ; 2 log log log log ; Ei 2 log log log log k 0 k ; k Ei k Π sgim ; k k Ei 2 log log log log O ; Ei Π Π arg 0 arg Π arg 0 ; True

6 Ei Π arg Π 2 Π Π arg Π 2 ; True Residue represetatios Ei 2 log log res s j 0 s 2 s s j s Ei log 2 log log res s s s s 0 res s j s s j s Other series represetatios L Eix x k x ; x 0 k 0 k Itegral represetatios O the real axis Of the direct fuctio t Ei t 0 t 2 log log t Eix t ; x x t x t Eix t ; x t Cotour itegral represetatios Ei 2 log Ei 2 log log log 2Π s s2 s 2 2Π Γ Γ s s 2 s s s 2 s s ; Γ 0 arg Π 2

7 Ei log 2 log Ei log 2 log log 2Π s2 s s s log Γ s 2 2Π Γ s s s ; 0 Γ arg Π 2 Cotiued fractio represetatios Ei Π sgim ; arg Π Ei Π sgim k k 2, 2 k ; arg Π Ei Π sgim ; arg Π Ei Π sgim k k 2, 2 k ; arg Π Differetial equatios Ordiary liear differetial equatios ad wroskias For the direct fuctio itself w 3 2 w w 0 ; w c Ei c 2 Ei c 3

8 W, Ei, Ei w 3 2 w w 0 ; w c Ei c 2 Shi c W, Ei, Shi w 3 2 w w 0 ; w c Ei c 2 Chi c W, Ei, Chi w 3 2 g g 3 g g w g 2 3 g 2 2 g g 2 g g3 g w 0 ; w c Eig c 2 Eig c W Eig, Eig, 2 g w 3 2 g 3 h g h g 2 4 h g g h g 2 3 g g w 6 h 2 3 g 2 6 h g h 2 g 2 h g 2 g g 3 h h g3 g w 6 h 3 4 g h 2 6 g h 2 6 h h 3 g 2 h 2 h g 2 g h h 3 g h 2 h 2 g h 2 h g 2 g h 3 g h h g 3 h g g 2 h h 3 h w 0 ; w c Eig h c 2 Eig h c 3 h W h Eig, h Eig, h 2 h3 g 3 g w 3 r 3 s 3 2 w a 2 r 2 2 r 3 s 2 r 2 r s 3 s w s a 2 r 2 2 r s 2 r s w 0 ; w c s Eia r c 2 s Eia r c 3 s W s Eia r, s Eia r, s 2 a r 3 3r3 s w 3 logr 3 logs w a 2 log 2 r r 2 3 log 2 s 2 logr logs w logs a 2 log 2 r r 2 log 2 s logr logs w 0 ; w c s Eia r c 2 s Eia r c 3 s

9 W s Eia r, s Eia r, s 2 a r s 3 log 3 r Trasformatios Trasformatios ad argumet simplificatios Argumet ivolvig basic arithmetic operatios Ei Ci Si log 2 log log Ei Ci Si log 2 log log Ei 2 Ei 2 log log log log 2 Shi Related trasformatios Eilog li Complex characteristics Real part ReEix y Chix logx 2 logx2 y 2 j y 2 j2 j 0 2 j 2 j 2 F 2 j ; x2, j 2; 2 4 x j y 2 j j 0 2 j 2 j F 2 j 2 ; 3 2, j 3 2 ; x ReEix y 2 logx2 y 2 k k x k y 2 x 2 k2 cos k ta y x ReEix y j y 2 j x k2 j 2 logx2 y 2 k 2 j k 2 j k 2 j 0 ReEix y 2 Ei x x y2 Ei x x y2 x 2 x 2

10 0 Imagiary part ImEix y 20 k y 2 k x 2 k 2 k 0 F 2 k ; 3 x2 20, k 2; 2 4 k y 2 k k 0 2 k 2 k F 2 k 2 ; 2, k 3 2 ; x2 4 2 ta x, y ta x, y y2 ImEix y k k x k x 2 k2 si k ta y x 2 ta x, y ta x, y j y 2 j x ImEix y k 2 j k 2 j 2 ta x, y ta x, y k j 0 k2 j ImEix y x y2 Ei x x y2 Ei x x y2 2 y x 2 x 2 x 2 Absolute value Eix y Ei x x y2 x 2 Ei x x y2 x 2 Argumet argeix y ta 2 Ei x x y2 x 2 Ei x x y2 x 2, x y2 Ei x x y2 Ei x x y2 2 y x 2 x 2 x 2 Cojugate value Eix y 2 Ei x x y2 Ei x x y2 x 2 x 2 x y2 Ei x x y2 Ei x x y2 2 y x 2 x 2 x 2 Sigum value

11 sgeix y y2 x Ei x x y2 Ei y2 x x Ei y2 x x Ei x x y2 y x 2 x 2 x 2 x 2 x 2 2 Ei x x y2 x 2 Ei y2 x 2 x x Differetiatio Low-order differetiatio Ei Ei 2 2 Symbolic differetiatio Ei k k Ei ; k k Ei 2 log log log, ; Ei Ei Boole 0, k k k k k 0 ; Ei Ei Boole 0,, ; Ei 2 F 2, ; 2, 2 ; ; Fractioal itegro-differetiatio Α Ei Α Α 2 F 2, ; 2, 2 Α; Α Α log Α Α 2 log log Itegratio

12 2 Idefiite itegratio Ivolvig oly oe direct fuctio Eib a b a Eib a ba a Eia Eia a a Eia Eia a a Ei Ei Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig power fuctio Ivolvig power Liear argumet Α Eia Α Α, a a Α Eia Α Α Eia Α Α Α, a a Α Eia Α Ei Α Α Α, Α Ei Eia a2 Eia 2 a a 2 a Eia a 3 F 3,, ; 2, 2, 2; a log 2 Eia 2 0, a log 2 loga Eia a Eia a 2

13 3 Eib a a b Eia b a Eib a b Power argumets Α Eia r Α Α Α r, a r a r Α r Eia r Ivolvig expoetial fuctio Ivolvig exp Liear argumet b Eia b Eia Eia b b a Eia loga a Eia a Power argumets a r Eia r 2 r a r r r, a r log r a 0, 2 loga r loga r 2 G 2,2 2,3 a r r 0, 0, r Ivolvig expoetial fuctio ad a power fuctio Ivolvig exp ad power Liear argumets b Eia b b b k a b k k, a b Eia b Eia, b ; k a Eia a a k Eia, a log k k ;

14 b Eia b ab a b b b Eia a b Eia b b 2 a b b Eia b 3 b b b 2 Eia 2 Eia b a a b 2 b2 b ab b b 3 a b b Eia b 4 b b b b 6 Eia 6 a a b 3 b3 Eia b b ab b b 6 a 2 b 2 b b 5 a b 2 b b Α a Eia 2 Α Α, a 2 loga loga log a a Α Α G 2,2 2,3 a 0, Α 0, 0, Α a Eia a a k Eia, a log k k ; a Eia a a a Eia log a c Eib a a c 2 a c b c a ac ba a a c a c a b c Ei a c b a a a a c c b a c Eib a a Eia Eia 2 2 Power argumets Α a r Eia r Α 2 r a r Α r Α r, a r log r a 0, Α 2 loga r loga r 2 G 2,2 2,3 a r r 0, 0, Α r Ivolvig trigoometric fuctios Ivolvig si cosb Eia Eia b Eia b sib Eia 2 b

15 5 Ivolvig cos Eia b Eia b 2 Eia sib cosb Eia 2 b Ivolvig trigoometric fuctios ad a power fuctio Ivolvig si ad power b sib Eia 2 b b k b a k k, b a Eia b Eia, b k b k b a k k, a b Eia b Eia, b k ; sib Eia Eia b Eia b 2 Eia sib b cosb 2 b a a cosb b sib 2 b 2 a 2 b sib Eia a a 2 a 2 b 2 a 3 b 2 cosb b 2 a b 2 a 2 b 2 2 a a 2 2 b 2 sib b Eia b b 3 a 2 b 2 2 a 2 b 2 2 Eia b Eia b cosb 2 b sib sib Eia Eia b 6 b 4 b2 2 cosb 3 b sib 3 Eia b Eia b a 2 b 2 3 a2 b 2 3 b a a a 2 b b 2 a 2 b 2 a 2 5 b 2 b 2 2 a 3 a 4 8 b 2 a 2 9 b 4 cosb b 2 a 3 a 4 6 b 2 a 2 a 2 b 2 3 a 2 7 b 2 a b 4 b 2 a 2 b sib Ivolvig cos ad power

16 cosb Eia 2 b b k b a k k, b a b Eia b Eia, b k b k b a k k, a b Eia b Eia, b k ; cosb Eia Eia b Eia b 2 b a a sib b cosb 2 Eia cosb b sib 2 b 2 a 2 b cosb Eia a a 2 a 2 b 2 a 3 b 2 sib b 2 a b 2 a 2 b 2 2 a a 2 2 b 2 cosb b b 3 a 2 b 2 2 a 2 b 2 2 Eia b Eia b Eia 2 b cosb b sib cosb Eia b 4 a 2 b 2 3 b2 a 3 a 4 6 b 2 a 2 a 2 b 2 3 a 2 7 b 2 a b 4 b 2 a 2 b cosb 3 Eia b Eia b b a 2 b 2 3 a a a 2 b b 2 a 2 b 2 a 2 5 b 2 b 2 2 a 3 a 4 8 b 2 a 2 9 b 4 sib Eia 3 b cosb b b sib Ivolvig hyperbolic fuctios Ivolvig sih coshb Eia Eia b Eia b sihb Eia 2 b siha Eia a a Eia Ei2 a loga 2 a Ivolvig cosh Eia b Eia b 2 Eia sihb coshb Eia 2 b

17 Chi2 a loga 2 Eia siha Shi2 a cosha Eia 2 a Ivolvig hyperbolic fuctios ad a power fuctio Ivolvig sih ad power sihb Eia 2 b b k b a k k, b a Eia b Eia, b k b k a b k k, a b Eia b Eia, b k ; sihb Eia 2 a Eia, a, a 2 k k, 2 a a k Ei2 a log k k k ; sihb Eia Eia b Eia b 2 Eia b coshb sihb 2 b a b sihb a coshb 2 b 2 a 2 b sihb Eia a a 3 a 2 b 2 a 3 b 2 coshb b 2 a b 4 4 a b 2 a 2 b 2 2 a 3 sihb b Eia b b 3 a 2 b 2 2 a 2 b 2 2 Eia b Eia 2 b sihb b coshb sihb Eia 3 Eia b Eia b b 4 a b 3 a b 3 b2 a 2 3 b a a a 2 b b 2 a 4 6 b 2 a 2 5 b 4 b 2 2 a 3 a 4 8 b 2 a 2 9 b 4 coshb b 2 a 3 a 4 6 b 2 a 2 3 a 4 0 b 2 a 2 7 b 4 a b 4 b 2 a 2 b sihb Eia b b coshb 3 b sihb Ivolvig cosh ad power

18 coshb Eia 2 b b k b a k k, b a Eia b Eia, b k b k a b k k, a b Eia b Eia, b k ; coshb Eia 2 a Eia, a, a 2 k k, 2 a a k Ei2 a log k k k ; coshb Eia Eia b Eia b 2 Eia b sihb coshb 2 b a b coshb a sihb 2 b 2 b 2 a coshb Eia b 3 a 3 b 2 a a 2 a b 2 sihb b 2 a b 4 4 a b 2 a 2 b 2 2 a 3 coshb b Eia b a 2 b 2 2 a 2 b 2 2 Eia b Eia b sihb 2 b coshb coshb Eia Eia b b 4 b2 2 6 sihb 3 b coshb 3 Eia b Eia b a b 3 a b 3 a2 b 2 3 b 2 a 3 a 4 6 b 2 a 2 3 a 4 0 b 2 a 2 7 b 4 a b 4 b 2 a 2 b coshb b a a a 2 b b 2 a 4 6 b 2 a 2 5 b 4 b 2 2 a 3 a 4 8 b 2 a 2 9 b 4 sihb Ivolvig logarithm Ivolvig log logb Eia Eia a a logb a logb a Ivolvig logarithm ad a power fuctio

19 9 Ivolvig log ad power Α logb Eia Α 3 Α a Α 2 F 2 Α, Α; Α, Α ; a a Α Α Eia Α logb a Α Α log Α, a Α logb log Eia log Eia log Eia 4 a 2 Eia a2 2 2 a 2 log 2 2 a a 2 2 a log 9 a 3 Eia a3 3 3 a 3 log 3 6 a a 2 2 a 3 a a 2 log 7 6 a 4 Eia a4 4 4 a 4 log 4 24 a a 3 3 a a 4 a a a 6 log 38 Ivolvig fuctios of the direct fuctio Ivolvig elemetary fuctios of the direct fuctio Ivolvig powers of the direct fuctio Eia 2 a Eia 2 2 a Eia 2 Ei2 a a Ivolvig products of the direct fuctio Eia Eia a Eia Eia a Eia a a Eia Eib a b b a Eib a Eia b b Eib a b Eia b Ivolvig fuctios of the direct fuctio ad elemetary fuctios Ivolvig elemetary fuctios of the direct fuctio ad elemetary fuctios Ivolvig powers of the direct fuctio ad a power fuctio

20 Eia 2 2a Ei2 2 k k, 2 a a Eia, a Eia 2 ; k Eia 2 a2 2 Eia 2 2 a a Eia 2 a 2 Ei2 a Eia Eia 2 2 a 2 6 a 3 2 a3 Eia a 2 a 5 4 a a a 2 Eia 8 Ei2 a 4 a 4 a4 Eia a a a 8 2 a a a a 6 Eia 2 Ei2 a Ivolvig products of the direct fuctio ad a power fuctio Eia Eib a k a b k k, a b Eia b a a Eib, a a Eia k a b b b k a b k k, a b Eia b Eia, b k ; Eia Eia a a a Eia Eia Eia, a a k a k Eia, a log log k k k k ; Eia Eib 2 a 2 b 2 Eia b b b 2 2 Eib a 2 a b ab b 2 a a Eib a 2 b 2 Eia b Eia Eia 2 a 2 a a Eia a Eia a 2 Eia 2 a a 2 a log Eia Eib a b ab 2 a 2 b a b a b a 2 b 2 3 a 3 b 3 a b a b Eia b 3 3 Eib b b b 2 2 a 3 b 3 a a a 2 2 Eib 2 a 3 b 3 Eia b

21 Eia Eia 3 a 3 a 4 a a a a 2 Eia a Eia a a a 2 a 3 3 Eia Eia Eib Eia 4 a 4 b 4 b4 Eib 4 b 6 b b b 3 6 a 4 a b a b 2 ab b b 3 6 a 4 b b 2 b 5 9 a 3 b 2 b b 5 2 a 2 3 b 3 b 3 a 6 b 4 b 4 a a a a Eib 6 a 4 b 4 Eia b Eia Eia 4 a 4 a a a a 6 Eia a Eia a a a a 6 a 4 4 Eia 3 a a log Defiite itegratio Ivolvig the direct fuctio log t Eitt ; Re 0 Itegral trasforms Laplace trasforms log t Eit ; Re Operatios Limit operatio Π 0 arga Π Π argb Π Ima 0 Π argb Π 2 2 lim Eia b x x Π 0 arga Π argb Π Π arga 0 argb Π argb Π Ima 0 argb Π True Represetatios through more geeral fuctios Through hypergeometric fuctios

22 22 Ivolvig p F q Ei 2 F 2, ; 2, 2; 2 log log Ivolvig hypergeometric U Ei U,, log 2 log log Through Meijer G Classical cases for the direct fuctio itself Ei 2 log log G,2 2,3,, 0, Ei log 2 log log G 2,0,2 0, 0 Classical cases ivolvig exp Ei 2 log 2 log log G 2,,2 0 0, Ei 2 log log Π G 2, 2,3 0, 2 0, 0, Ei Π G 2, 2,3 0, 2 0, 0, 2 ;, 0 Classical cases for products of Ei Ei Ei Π 2 G 2,4 3, 2 4 0, 0, 0, 0, 2 ; Im 0 Through other fuctios Ei 0, 2 log log log Ei E 2 log log log

23 23 Represetatios through equivalet fuctios With related fuctios Ei Chi Shi 2 log log Ei Ci Si 2 log log log Ei li ; Π Im Π Theorems The real part of the Heiseberg-Euler Lagragia of quatum electrodyamics The real part of the Heiseberg Euler Lagragia of quatum electrodyamics L eff E, B ca be expressed as the followig series. ReL eff E, B c Α Π c 2coth Π k Η 2 Ξ Ci k Ξ cos k Ξ Si k Ξ si k Ξ coth Π k Ξ Η Ei k Ξ exp k Ξ Ei k Ξ exp k Ξ ; Ξ e Π c c 2 c 2 2 Η e Π c c 2 c 2 2 c 2 E2 B 2 c 2 E B, where E is the electric field, B is the magetic iductio ad Α is the fie structure costat. The Nambu-Goldstoe modes for weakly iteractig fermios i a magetic field The Nambu-Goldstoe modes for weakly iteractig fermios i a magetic field are described by m 2 VrΨr 0, where i 3+ dimesioal ladder quatum electrodyamics the potetial Vr takes the r 2 form Vr 2l 2 Ei r2, where l is the magetic legth. 2 2l History A. M. Legedre (8) O. Schlomilch (846) F. Ardt (847) J. W. L. Glaisher (870) itroduced the otatio Ei

24 24 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a key to the otatios used here, see Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for example: To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: This documet is curretly i a prelimiary form. If you have commets or suggestios, please commets@fuctios.wolfram.com , Wolfram Research, Ic.

Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values EulerPhi Notatios Traditioal ame Euler totiet fuctio Traditioal otatio Φ Mathematica StadardForm otatio EulerPhi Primary defiitio 3.06.02.000.0 Φ gcd,k, ; For oegative iteger, the Euler totiet fuctio Φ