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

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1 ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition E t t t ; Re 0 Specific values Specialied values For fixed E 0 ; Re E 0 ; Re E Subfactorial For fixed E E Π erf

2 E Π erfc E n E n Π n n n erfc Π erfc 3 k n k ; n E n n n erfc n n kn ; n nk E Ei log log log E E Shi log log E n n n Ei log log n log ; n n k k k E n n n n, ; n E E n n n kn k ; n n E n n Ei n log n log log n kn n kn k k n k ; n E n n erfc n n k n n kn kn n k k ; n Values at infinities E 0

3 3 General characteristics Domain and analyticity E is an analytical function of and which is defined in. For fixed, it is an entire function of E Symmetries and periodicities Mirror symmetry E E ;, 0 Periodicity No periodicity Poles and essential singularities With respect to For fixed, the function E has an essential singularity at. At the same time, the point is a branch point for generic ing E. With respect to For fixed, the function E has only one singular point at. It is an essential singular point ing E, Branch points With respect to For fixed, not being a nonpositive integer, the function E has two branch points: 0,. At the same time, the point is an essential singularity E 0, E, 0 log ; E p, 0 q ; p q gcdp, q q

4 E, log ; E p, q ; p q gcdp, q q With respect to For fixed, the function E does not have branch points E Branch cuts With respect to For fixed, not being a nonpositive integer, the function E is a single-valued function on the -plane cut along the interval, 0, where it is continuous from above E, 0, lim E x Ε E x ; x 0 Ε lim E x Ε E x Π x ; x 0 Ε0 With respect to For fixed, the function E does not have branch cuts E Series representations Generalied power series Expansions at generic point 0 For the function itself E E F 0, 0 ; 0, 0 ; 0 H 0 log H 0 log Ψ 0 0 3F 3 0, 0, 0 ; 0, 0, 0 ; 0 ; 0

5 E E F 0, 0 ; 0, 0 ; 0 H 0 log H 0 log Ψ 0 0 3F 3 0, 0, 0 ; 0, 0, 0 ; 0 O 0 3 E k s s 0 k k s log ks s 0 0 sj s j 0 s j s j 0 js log j sj F sja, a,, a sj ; a, a,, a sj ; 0 k ; a a a k 0 k E E 0 O 0 Expansions at generic point For the function itself E E arg arg, arg arg 3 0 3, 3 arg arg ; E E arg arg, arg arg 3 0 3, 3 arg arg O E k k arg arg k k F, ; k, ; k

6 E E arg arg O Expansions on branch cuts For the function itself E E x x argx x argx, x x x3 argx 3 3, x x ; x x x E E x x argx x argx, x x x3 argx 3 3, x x O x 3 ; x x E k x k argx k x k F, ; k, ; x x k ; x x E E x x argx O x ; x x 0 Expansions at 0 For the function itself General case E E k k E k k E ; O3 F ; ;

7 E O E F, ; n k k F n, k k n n, n n F, n ; n, n 3; n Summed form of the truncated series expansion. Special cases E log O E log O E n n n n Ψn log k k k n k n n n O n ; n n k k E log k k k E n n n Ψn log k n k k ; n k n k E n n F, ;, n ; n n n n Ψn log k k ; n k n k E log O ; E O log ; E 3 O ; E n n O Ψn log ; 0 n n n

8 n k E n n n ; n k E n n n O ; 0 n Asymptotic series expansions E F 0, ; ; ; E O ; E ; Residue representations E res s j 0 s s j s E res s s s s res s j 0 s s j s Integral representations On the real axis Of the direct function E t t t ; arg Π Contour integral representations E Π s s s s s E Γ s s Π s s ; 0 Γ Re arg Π Γ s

9 E Π s ss s s E Γ s s s Π s ; max Re, 0 Γ arg Π Γ s Continued fraction representations E 3 ;, 0 E k k k k k k, k ;, E ;, E k k k, k ;, E ;, 0

10 E k k k k, k ;, E E k k, k E E k k k k k k, k Differential equations Ordinary linear differential equations and wronskians For the direct function itself w w w 0 ; w c E c W, E

11 w g g g g g w g w 0 ; w c E g c g g W E g, g g g g w g g g g g h h h g g h g g h 0 ; w c h E g c h g h w W h E g, h g g g h g h g h g g h h h w w a r r r s r w s a r r s r w 0 ; w c s a r c s E a r W s E a r, s a r a r r s a r w a r logr logs w a r logr logs logr logs w 0 ; w c s E a r c s a r W s E a r, s a r a r a r s logr Transformations Transformations and argument simplifications Argument involving basic arithmetic operations E E E E E n n E n ; n n n k n E n n n E n k k ; n k

12 Identities Recurrence identities Consecutive neighbors E E E E Distant neighbors E n n n E n k k ; n E n n n E n n k k ; n Functional identities Relations of special kind k E n n n n E ; n n k k Differentiation Low-order differentiation With respect to E log Ψ F, ;, ; E Π cot log cotπ Π csc log Ψ cotπ log Ψ Ψ 3 3 F 3,, ;,, ; With respect to

13 E E E E Symbolic differentiation With respect to n E n n n E n n k logk nk nk k n ; n nk k n k n k n k lognk k kj k j 0 k j k j kj log j kj F kja, a,, a kj ; a, a,, a kj ; ; a a a n n With respect to n E n n n n n E kn k ; n n E n E n ; n n n E Π cscπ n n n F, ; n, ; ; n n Fractional integro-differentiation With respect to Α E Α Α k k F k k ; k Α ; log Α k k Α I Α k ; Α E Α Α t t logt Α QΑ, 0, logtt ; Re 0 With respect to

14 Α E Α Α exp, Α Α F, ; Α, ; Integration Indefinite integration Involving only one direct function E a E a a E E Involving one direct function and elementary functions Involving power function Α Α E a Α E a E Α a Α Α E Α E E Α Involving only one direct function with respect to E k k F k ; k ; log k I k k k ; Integral transforms Fourier cos transforms c t E tx Π csc Π x Π x F, ; ; ; x Re 0 x Fourier sin transforms s t E tx Π x sgnx sec Π Π x F, ; 3 ; x ; x Re Laplace transforms

15 t E t F, ; ; Π cscπ ; Re 0 Re 0 Representations through more general functions Through hypergeometric functions Involving F E F ; ; ; Involving F E Involving hypergeometric U E U,, F ; ; Through Meijer G Classical cases for the direct function itself E G,, 0, E G,0,, 0 Classical cases involving exp E G,, 0 0, Classical cases for products of exponential integrals E E E Π G 4,,4 4 0, 0,,, E E Π 4, G,4 4 0, 0,,, Representations through equivalent functions

16 6 With inverse function E Q, Q, With related functions E, E Q,

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