ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied values For fixed 06.34.03.000.0 E 0 ; Re 06.34.03.000.0 E 0 ; Re 06.34.03.007.0 E Subfactorial For fixed E 0 06.34.03.0003.0 06.34.03.0004.0 E Π erf
http://functions.wolfram.com 06.34.03.0005.0 E Π erfc E n E 06.34.03.0006.0 n Π n n n erfc 06.34.03.0007.0 Π erfc 3 k n k ; n 06.34.03.0008.0 E n n n erfc n n kn ; n nk 06.34.03.0009.0 E Ei log log log 06.34.03.000.0 E E Shi log log 06.34.03.00.0 E n n n Ei log log n log ; n n k k k 06.34.03.008.0 E n n n n, ; n 06.34.03.00.0 E 06.34.03.003.0 E n n n kn k ; n 06.34.03.005.0 n E n n Ei n log 06.34.03.006.0 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 06.34.03.004.0 E 0
http://functions.wolfram.com 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. 06.34.04.000.0 E Symmetries and periodicities Mirror symmetry 06.34.04.000.0 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. 06.34.04.0003.0 ing E. With respect to For fixed, the function E has only one singular point at. It is an essential singular point. 06.34.04.0004.0 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. 06.34.04.0005.0 E 0, 06.34.04.0006.0 E, 0 log ; 06.34.04.0007.0 E p, 0 q ; p q gcdp, q q
http://functions.wolfram.com 4 06.34.04.0008.0 E, log ; 06.34.04.0009.0 E p, q ; p q gcdp, q q With respect to For fixed, the function E does not have branch points. 06.34.04.000.0 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. 06.34.04.00.0 E, 0, 06.34.04.00.0 lim E x Ε E x ; x 0 Ε0 06.34.04.003.0 lim E x Ε E x Π x ; x 0 Ε0 With respect to For fixed, the function E does not have branch cuts. 06.34.04.004.0 E Series representations Generalied power series Expansions at generic point 0 For the function itself 06.34.06.007.0 E E 0 0 0 F 0, 0 ; 0, 0 ; 0 H 0 log 0 0 0 H 0 log Ψ 0 0 3F 3 0, 0, 0 ; 0, 0, 0 ; 0 ; 0
http://functions.wolfram.com 5 06.34.06.008.0 E E 0 0 0 F 0, 0 ; 0, 0 ; 0 H 0 log 0 0 0 H 0 log Ψ 0 0 3F 3 0, 0, 0 ; 0, 0, 0 ; 0 O 0 3 E 06.34.06.009.0 0 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 06.34.06.000.0 E E 0 O 0 Expansions at generic point For the function itself E 06.34.06.00.0 E arg arg, arg arg 3 0 3, 3 arg arg ; 06.34.06.00.0 E E arg arg, arg arg 3 0 3, 3 arg arg O 3 06.34.06.003.0 E k k arg arg k k F, ; k, ; k
http://functions.wolfram.com 6 06.34.06.004.0 E E arg arg O Expansions on branch cuts For the function itself 06.34.06.005.0 E E x x argx x argx, x x x3 argx 3 3, x x ; x x x 0 06.34.06.006.0 E E x x argx x argx, x x x3 argx 3 3, x x O x 3 ; x x 0 06.34.06.007.0 E k x k argx k x k F, ; k, ; x x k ; x x 0 06.34.06.008.0 E E x x argx O x ; x x 0 Expansions at 0 For the function itself General case 06.34.06.000.0 E 06.34.06.009.0 E 06.34.06.000.0 k k E k k 06.34.06.0003.0 E ; 0 3 3 O3 F ; ;
http://functions.wolfram.com 7 06.34.06.0004.0 E O 06.34.06.0030.0 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 06.34.06.003.0 E log O 06.34.06.003.0 E log O3 06.34.06.0033.0 E n n n n Ψn log k k k n k n n n O n ; n n 06.34.06.0034.0 k k E log k k k 06.34.06.0005.0 E n n n Ψn log k n 06.34.06.0006.0 k k ; n k n k E n n F, ;, n ; n n n n Ψn log k k ; n k n k 06.34.06.0007.0 E log O ; 0 06.34.06.0008.0 E O log ; 0 06.34.06.0009.0 E 3 O ; 0 06.34.06.000.0 E n n O Ψn log ; 0 n n n
http://functions.wolfram.com 8 06.34.06.00.0 n k E n n n ; n k 06.34.06.00.0 E n n n O ; 0 n Asymptotic series expansions 06.34.06.003.0 E F 0, ; ; ; 06.34.06.004.0 E O ; 06.34.06.0035.0 E ; Residue representations 06.34.06.005.0 E res s j 0 s s j s 06.34.06.006.0 E res s s s s res s j 0 s s j s Integral representations On the real axis Of the direct function 06.34.07.000.0 E t t t ; arg Π Contour integral representations 06.34.07.000.0 E Π s s s s s 06.34.07.0003.0 E Γ s s Π s s ; 0 Γ Re arg Π Γ s
http://functions.wolfram.com 9 06.34.07.0004.0 E Π s ss s s 06.34.07.0005.0 E Γ s s s Π s ; max Re, 0 Γ arg Π Γ s Continued fraction representations 06.34.0.000.0 E 3 ;, 0 E 06.34.0.000.0 k k k k k k, k ;, 0 06.34.0.0003.0 E 4 3 4 3 6 5 4 8 0 ;, 0 06.34.0.0004.0 E k k k, k ;, 0 06.34.0.0005.0 E 3 4 3 5 6 ;, 0
http://functions.wolfram.com 0 06.34.0.0006.0 E k k k k, k ;, 0 06.34.0.0007.0 E 3 3 4 4 5 5 6 06.34.0.0008.0 E k k, k 06.34.0.0009.0 E 3 4 3 5 6 06.34.0.000.0 E k k k k k k, k Differential equations Ordinary linear differential equations and wronskians For the direct function itself 06.34.3.000.0 w w w 0 ; w c E c 06.34.3.000.0 W, E
http://functions.wolfram.com 06.34.3.0003.0 w g g g g g w g w 0 ; w c E g c g g 06.34.3.0004.0 W E g, g g g g 06.34.3.0005.0 w g g g g g h h h g g h g g h 0 ; w c h E g c h g 06.34.3.0006.0 h w W h E g, h g g g h g 06.34.3.0007.0 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 06.34.3.0008.0 W s E a r, s a r a r r s a r 06.34.3.0009.0 w a r logr logs w a r logr logs logr logs w 0 ; w c s E a r c s a r 06.34.3.000.0 W s E a r, s a r a r a r s logr Transformations Transformations and argument simplifications Argument involving basic arithmetic operations 06.34.6.000.0 E E 06.34.6.000.0 E E 06.34.6.0003.0 E n n E n ; n n n k 06.34.6.0004.0 n E n n n E n k k ; n k
http://functions.wolfram.com Identities Recurrence identities Consecutive neighbors 06.34.7.000.0 E E 06.34.7.000.0 E E Distant neighbors 06.34.7.0003.0 E n n n E n k k ; n E 06.34.7.0004.0 n n n E n n k k ; n Functional identities Relations of special kind 06.34.7.0005.0 k E n n n n E ; n n k k Differentiation Low-order differentiation With respect to 06.34.0.000.0 E log Ψ F, ;, ; 06.34.0.000.0 E Π cot log cotπ Π csc log Ψ cotπ log Ψ Ψ 3 3 F 3,, ;,, ; With respect to
http://functions.wolfram.com 3 06.34.0.0003.0 E E 06.34.0.0004.0 E E Symbolic differentiation With respect to 06.34.0.0005.0 n E n n 06.34.0.0006.0 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 06.34.0.00.0 n n n n n E kn k ; n 06.34.0.0007.0 n E n E n ; n n 06.34.0.0008.0 n E Π cscπ n n n F, ; n, ; ; n n Fractional integro-differentiation With respect to Α E Α 06.34.0.0009.0 Α k k F k k ; k Α ; log Α k k Α I Α k ; 06.34.0.000.0 Α E Α Α t t logt Α QΑ, 0, logtt ; Re 0 With respect to
http://functions.wolfram.com 4 06.34.0.00.0 Α E Α Α exp, Α Α F, ; Α, ; Integration Indefinite integration Involving only one direct function 06.34..000.0 E a E a a 06.34..000.0 E E Involving one direct function and elementary functions Involving power function 06.34..0003.0 Α Α E a Α E a E Α a 06.34..0004.0 Α Α E Α E E Α Involving only one direct function with respect to 06.34..0005.0 E k k F k ; k ; log k I k k k ; Integral transforms Fourier cos transforms 06.34..000.0 c t E tx Π csc Π x Π x F, ; ; ; x Re 0 x Fourier sin transforms 06.34..000.0 s t E tx Π x sgnx sec Π Π x F, ; 3 ; x ; x Re Laplace transforms
http://functions.wolfram.com 5 06.34..0003.0 t E t F, ; ; Π cscπ ; Re 0 Re 0 Representations through more general functions Through hypergeometric functions Involving F 06.34.6.000.0 E F ; ; ; Involving F 06.34.6.000.0 E Involving hypergeometric U 06.34.6.0003.0 E U,, F ; ; Through Meijer G Classical cases for the direct function itself 06.34.6.0004.0 E G,, 0, 06.34.6.0005.0 E G,0,, 0 Classical cases involving exp 06.34.6.0006.0 E G,, 0 0, Classical cases for products of exponential integrals E E E 06.34.6.0007.0 Π G 4,,4 4 0, 0,,, 06.34.6.0008.0 E E Π 4, G,4 4 0, 0,,, Representations through equivalent functions
http://functions.wolfram.com 6 With inverse function 06.34.7.000.0 E Q, Q, With related functions 06.34.7.000.0 E, 06.34.7.0003.0 E Q,
http://functions.wolfram.com 7 Copyright This document was downloaded from functions.wolfram.com, a comprehensive online compendium of formulas involving the special functions of mathematics. For a key to the notations used here, see http://functions.wolfram.com/notations/. Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for example: http://functions.wolfram.com/constants/e/ To refer to a particular formula, cite functions.wolfram.com followed by the citation number. e.g.: http://functions.wolfram.com/0.03.03.000.0 This document is currently in a preliminary form. If you have comments or suggestions, please email comments@functions.wolfram.com. 00-008, Wolfram Research, Inc.