JOURNAL OF RESEARCH of the National Bureau af Standards - B. Mathematics and Mathematical Science Vol. 73B, No. 3, July-September 1969 A Table of Integrals of the Exponential Integral* Murray Geller** and Edward W. Ng** (March 11, 1969) This is a compendium of indefinite and definite integrals of products of the Exponential Integral with elementary or transcend..,ntal functions. A substantial portion of the results are new. Key words: Diffusion theory; exponential integral; indefinite integrals; quantum mechanics; radiative equilibrium; special fun ctions; transport problems. 1. Introduction Integrals of the exponential integral occur in a wide variety of applications. Examples of applications can be cited from diffusion theory [12], I transport problems [12] , the study of the radiative equilibrium of stellar atmospheres [9] , and in the evaluation of exchange integrals occurring in quantum mechanics [11] . This paper is an attempt to give an up·to·date exhaustive tabulation of such integrals. All formulas for indefinite integrals in section 4 were derived from integration by parts and checked by differentiation of the resulting expressions. The formulas given in [1 , 4,5,6,7, 10, 14, and 15] have all been checked and included, with the omission of trivial duplications. Additional formulas were obtained either from the various integral representations, from the hypergeometric series for the exponential integrals, from multiple integrals involving elementary functions, from the existing literature [2,3, 12 , and 13], or by specialization of parameters of integrals over con· fluent hypergeometric functions [4] and [6]. Throughout this paper, we have adhered to the notations used in the NBS Handbook [8] and we have also assumed the reader's familiarity with the properties of the exponential integral. In addition, the reader should also attend to the following conventions: (i) The integration constants have been omitted for the indefinite integrals; (ii) the parameters a, h, and c are real and positive except where otherwise stated; (iii) unless otherwise specified, the parameters nand k represent the integers 0, 1, 2 ... , whereas the parameters p, q, and p. and lJ may be nonintegral; (iv) the integration symbol +d enotes a Cauchy principal value; (v) x, y, and t represent real variables. 2. Glossary of Functions and Notations ber (x), bei (x) Thomson functions Bx(p, q) Incomplete beta function Ci(x) Cosine integral -foe cos t dt x t Truncated exponential • An invited paper. This paper present s th e results of one phase of research carried ou l at the let Propulsion Laboratory, Ca lifornia In s titut e of Techn ology, under Contract No. NAS7- 100 s pon sored by the National Aeronautics and Space Administrat ion . "Present address: Je l Propuls ion La boratory, California Institute of Technology, Pasade na, Calif. 9] 103. ] Figures in brac kets indicate the literature reterences at the end of this paper. 191 oo e - t Exponential integral f -dt x t oo e - t Ei(x) Exponential integral - f -dt -x t erf (x) Error function erfc (x) Error function Confluent hypergeometric function xl! J'I (a, b; c; x) Hypergeometric function f (a)//(b)" 1l= 0 (c) // n! pFq Generalized hypergeometric function Hp(x) Struve function /p(x) Bessel function lof imaginary argument }p(x) Bessel function Kp(x) Bessel function of imaginary argument x dt li(x) Logarithmic integral f -l-'x> I o n t x In (1- t) Euler's dilogarithm - dt fo t eX dl! Ln(x) Laguerre polynomial -- (x"e- X ) n!dx" Mp,q(x) Whittaker function Binomial coefficient m! (:) n!(m-n)! P~(x) Associated Legendre function of the first kind r(p+ n) (p )" Pochhammer's symbol rep) 00 sin t d si(x) Sine integral - -- t Jx t sgnx Sign of the real number xl Ix I Wp,q(x) Whittaker function Yp(x) Neumann function y Euler's constant 0.57721 56649 y(a, x) Incomplete gamma function rea, x) Incomplete gamma function rea) Gamma function 00 I ~(p ) Riemann zeta function L111 = 0 (m+ IV 192 00 xm .. <I> (x, p, q) ,£;o (m + q)l' t/J(p) Psi function d dp In f(p) t/J(a ; b; x) Confluent hypergeometric function. 3. Definition, Special Values, and Integral Representations 3.1. Definition and Other Notations '" -t 1.EI(x)= J ~dt X>o x t '" e-t 2. Ei(x) =-f -dt -x t x et = f -dt X>o. _00 t 3. Some authors use [-Ei(-x) ] for EI(x). 4. Some authors useE*(x) or Ei(x) forEi(x) .. 5. Integrals involving li(x) can be transformed into integrals over Ei(x) since li(x) =lx Idt =Ei(lnx) x > 1. Jo nt 3.2. Special Values 1. Ei (0.372507 ... ) =0. 2. Ei(x) =EI(x) at x=0.523823 3. Lim [xPEI (x)] = Lim (xpEi(x» = 0 P > 0. x--+O x-o 4. Lim [lnx + EI(x)] =-y. x-o 6. Lim [e- xEi(x) ] = 0. X~ OO 7. Inflection point of Ei(x) at x= 1. 3.3. Integral Representations x dt 1. EI(x)=-y- In x+ I (l-e- t )- _ 0 t I 1 dt 4. EI(x) =rX ( I ) Io x- n t 193 10. E1(x)=-1 f 00 sintIn (t2)1+2 -dt 7T 0 x t 14. Ei(x) =eXf (x+1In t) dt 00 1 dt 15. Ei(x) =eX J ( I ) 2 1 X- n t t rOO e-xt 16. Ei(x)=exJo (l_t)dt 18 .. Ei(x) = e; [f (l ~xI: t)2 dt+ 1] 19. Ei(x) =-eX fo oo (t~i:X) dt e - xt x '" 20. [E1 (x))2=2e- fI (l+t) In t dt· 21. E1(ax)E1(bx)+E1[(a+b)x] In (ab)=e-( a+b)x 10'''' (a~~X~t)In [(a+t)(b+t)]dt 22. e- XEi(x) +exE1(x) =2 10 00 (t2~X2) sintdt 23. e- xEi(x) +exE1(x) =2e- x In x-.; 10'" (t2~X2) cos tIn tdt 24. e-xEi(x) -exE1(x) =-2 fo'" (t2~X2) cos tdt 25. e- xEi(x) -exE1(x) =2e- x In x-.; 10'" (t2~X2) sin tIn tdt 194 4. Integrals of the Exponential Integral With Other Functions 4.1. Combination of Exponential Integral With Powers 1. J EI(ax)dx=xE,(ax) -~ e-ax 2. JEi(ax)dx=;=xEi(ax) -~ eax 1 3. '" E1(ax)dx=- Jo a r'" 1 5. Jo xE,(ax)dx=2a2 n+1 n' 1 ( ) X () . ( ) - ax 6 . J x",E ax d x= (n+1) E 1 ax - (n+1) an+' ell ax e II n! x"- m ( b)m+l 7. J x IlE 1 (ax+b)dx=E1 (ax+b) ,t;o (-l)m(n_m)! (m+1)! x+~ n n! 1 x n- m m -e-(ax+b) ~ -- ~ (-l)k(m-k)!(ax+b)k n7::o (n-m)! (m+1)! a,n+l 6'0 n! 1 (n+ 1) an+! 10. JaxilE,'" (x-y)dx= e-(a-y)];on (n)m (m~m' 1) yn- me",(a-y) n (n) 11 - 1/1 -E1 (a-y) L y (a-y)m+1 y<a 111 =0 m (m+ 1) n 11. '" xllE, (x-a)dx= L (n) m.' a 11 - 111 Ja m= O m (m+ 1) -e-(y- a) ~II (n) m! (-1) ",yn - mem(y-a) t:o m (m+ 1) +E,(y-a) Ln (n) (-1)'" y'!-1I1( y -a)m+' y>a 111 =0 m (m+ 1) 13. J'" xnE,(ly-xl)dx=Ln (n) (~1)m' [1+ (_1)m)yn-m o m= O m m Xp+l 1 1 14. J xpE1(ax)dx= (p + 1) EI (ax) + (p + 1) ap+1 y(p + 1, ax) p > -I ( 00 1 1 15. Jo xPE 1 (ax)dx= (p+I) aP+1f(p+I) p>-I p > -I 22 E ( ) dx -- 1 [_I_E ( ) +J - ax dx] • J 1 ax x n+2- (n+ 1) x n+1 1 ax e x n+2 n - e- ab L (n - m) ! (- ab) m] m=O dx (a)n+l} 24. J E 1 (ax+b) x n+2= (n+1)1 {Ixn+1+(-I)n b E 1 (ax+b) (_1)n (a)n+l - b + (n+ 1) b en(b)e E 1 (ax) (-I)n (a)n+1 n 1 ( b)m m- l + - e -(ax+b) L ---; -- L (m-k-I) !(-ax)k (n+l) b m=l m . ax k =O 25 E ( ) dx -- 1 [_I_E ( ) +J - ax dx] p > 1 • J 1 ax x P - (p -1) x p- I 1 ax e x P ( 00 dx 1 1 a n+p f(p) 26. Jb El (ax) xn+p+1 = (n + p) . bn+p EI (ab) + (-1) n (n + p) f(n + p + 1) f (1-p, ab) 1 1 e- ab n - (n+p) ' bn+p' f(n+p+I)~o (-ab)mf(n-m+p) O< p < I ( 00 1 1 f(p+ 1) 28. Jo xp(x + a) - p- 2E 1 (bx)dx = -;;. ~. (p + 1) eab/2W- (P +I/2) ,o(ab) p>-I 196 30 [ '" P ( + ) - p {I 2X} E ( ) d - _1_ . f (p + 1) a/2W ( ) p ·Jo xx a +a(p+l) IX x-Va (p+l)e 1/2- Jl . - la > -1 31. 10'" xP( x +a)p{I+ 2:} El(X) dx=ap - I / 2~~:I~) ea/2Wl/2. _Jl _I(a) p > - 1 32. f"xP(x+a) - p- 2El(x+a)dx=a-3/2f(p+1)e- a/2W_(/J +3/2J,O(a ) p > - 1 4.2.