Mathematica Moravica Vol. 19-2 (215), 65 73 Convolutions Involving the Eponential Function and the Eponential Integral Brian Fisher and Fatma Al-Sirehy Abstract. The eponential integral ei(λ) and its associated functions ei (λ) and ei (λ) are defined as locally summable functions on the real line and their derivatives are found as distributions. The convolutions r ei () s e and r ei () s e are evaluated. 1. Introduction and Results The eponential integral ei() is defined for > by (1) ei() u 1 e u d u, see Sneddon [8, the integral diverging for. It was pointed out in [1 that equation (1) can be rewritten in the form (2) ei() u 1 [e u H(1 u) d u H(1 ) ln, where H denotes Heaviside s function. The integral in this equation is convergent for all and so we use equation (2) to define ei() on the real line. More generally, see [1, if λ, we define ei(λ) in the obvious way by (3) ei(λ) λ u 1 [e u H(1 u) d u H(1 λ) ln λ. Further, we define the functions ei (λ) and ei (λ) by so that ei (λ) H() ei(λ), ei (λ) H() ei(λ) (4) ei(λ) ei (λ) ei (λ). (5) In particular, if λ >, we have ei(λ) u 1 [e λu H(1 λu) d u H(1 λ) ln λ, 21 Mathematics Subject Classification. Primary: 33B1, 46F1. Key words and phrases. eponential integral, convolution. 65 c 215 Mathematica Moravica
66 Convolutions Involving the Eponential Function... (6) (7) where and ei (λ) ei (λ) γ(λ) u 1 e λu d u, >, γ u 1 (e λu 1) d u ln, <, γ(λ) γ ln λ u 1 [e λu H(1 λu) d u is Euler s constant. The derivatives of these functions are given by (8) for all λ. In particular, we have (9) where ei() ei () [ei(λ) e λ 1, [ei (λ) e λ 1 γ(λ)δ(), [ei (λ) e λ 1 ei () γ γ γ(λ)δ()), u 1 [e u H(1 u) d u H(1 ) ln, u 1 e u d u, >, u 1 (e u 1) d u ln, <, u 1 [e u H(1 u) d u is Euler s constant. The derivatives of these functions are given by (1) [ei() e 1, [ei () e 1 γδ(), [ei () e 1. The classical definition of the convolution of two functions f and g is as follows: Definition 1. Let f and g be functions. defined by (f g)() for all points for which the integral eist. Then the convolution f g is f(t)g( t) d t
Brian Fisher and Fatma Al-Sirehy 67 It follows easily from the definition that if f g eists then g f eists and (11) f g g f and if (f g) and f g (or f g) eists, then (12) (f g) f g ( or f g). Definition 1 can be etended to define the convolution f g of two distributions f and g in D with the following definition, see Gel fand and Shilov [7. Definition 2. Let f and g be distributions in D. Then the convolution f g is defined by the equation (f g)(), ϕ f(y), g(), ϕ( y) for arbitrary ϕ in D, provided f and g satisfy either of the conditions (a) either f or g has bounded support, (b) the supports of f and g are bounded on the same side. It follows that if the convolution f g eists by this definition then equations (1) and (11) are satisfied. The locally summable functions e and e are defined by e H()e e H()e. In the following we need the following lemma, which is easily proved by induction. Lemma 1. for, 1, 2,.... t e t d t t e 2t d t i i! ui e u!,! 2 i1 ui e 2u! 2 1, We now prove the following theorem.
68 Convolutions Involving the Eponential Function... Theorem 1. The convolution r ei () s e eists and (13) r ei () s e r )(1) (r )! s[ i1 (i 1)! 2 ij j! j e (i 1)! 2 i e j ( ) s (1) (r )! s [e ei (2) e ei () ln 2 e )(1) (r )! s[ r i (1 e ) ei (), for r, s, 1, 2,... and r, s not both zero. In particular, (14) for r 1, 2,... and r ei () e r! r [ i1 j (i 1)! 2 ij j! j e (i 1)! 2 i e r![e ei (2) e ei () ln 2 e [ r i r! (1 e ) ei (), (15) ei () e ei () e ei (2) ln 2 e. Proof. The convolution r ei () s e if < and so when >, we have r ei () s e t r ( t) s e t u 1 e u d u d t (16) where (17) t u 1 e u t r ( t) s e t d t d u I 1 I 2, t r ( t) s e t d t u 1 e u t r ( t) s e t d t d u ( s u )(1) s t r e t d t )(1) (r )! s[ r u i eu (e u 1)
Brian Fisher and Fatma Al-Sirehy 69 and in particular when r s, (18) e t d t e u 1, on using the lemma. Hence ( ) s I 1 (1) (r )! s e (19) (2) [r u i1 e 2u u 1 (e 2u e u ) d u r ( s (r )! )(1) s e u i1 e 2u d u ( s )(1) (r )! s e u 1 (e 2u e u ) d u. Further, we have u i1 on using the lemma, and (21) (22) Similarly i1 e 2u d u j u 1 [e u H(1 u) d u (i 1)! 2 ij j! j e 2 u 1 [e u H(1 u) d u u 1 e u d u γ ei () (i 1)! 2 i, u 1 H(1 u) d u u 1 H(1 u) d u. u 1 [e 2u H(1 2u) d u γ ei (2) It follows from equations (21) and (22) that (23) u 1 (e u e 2u ) d u ei (2) ei () ei (2) ei () ln 2. u 1 H[1 2u d u. u 1 [H(1 u) H(1 2u) d u
7 Convolutions Involving the Eponential Function... In particular, when r s, we have (24) I 1 [ei (2) ei () ln 2e. Net, as in equation (17), we have t r ( t) s e t d t (25) )(1) (r )! s[ r i e (e 1) and so (26) I 2 )(1) (r )! s[ r i (1 e ) ei (). In particular, when r s, we have (27) I 2 (e 1) u 1 e u d u (e 1) ei (). Equation (13) now follows from equations (2), (21), (22), (25) and (26). Equation (14) follows on putting s in equation (13) and equation (15) follows on putting r in equation (14). In the corollary, the distribution 2 as in Gel fand and Shilov. is defined by 2 (1 ) and not Corollary 1.1. The convolutions (e 1 ) e and (e 2 ) e eist and (28) (29) (e 1 ) e e ei (2) γ(2)e (e 2 ) e 2e ei (2) 2γ(2)e e 1. Proof. The convolution (e 1 ) e eists by Definition 2, since e 1 and e are both bounded on the left. From equation (12), we have [ei () e [e 1 γδ() e (e 1 ) e γe ei () [e δ() e ei (2) ln 2 e and equation (28) follows. From equations (12) and (28), we now have [(e 1 ) e (e 1 e 2 ) e e ei (2) γ(2)e (e 2 ) e (e 1 ) [e δ() e ei (2) γ(2)e e 1
Brian Fisher and Fatma Al-Sirehy 71 and equation (29) follows. Theorem 2. The convolution r ei () s e eists and r ( ) s (1) r ei () s e (r )! 2 i s e (3) ( ) s (1) ln 2(r )! s e, for r, s, 1, 2,... and r, s not both zero. In particular r (31) r ei () e r! 2 i e ln 2r!e, for r 1, 2,... and (32) ei () e ln 2 e (33) ei () e ln 2 e ln 2 e 1 2 e. Proof. We have r ei () s e t r ( t) s e t u 1 e u d u d t t u 1 e u t r ( t) s e t d t d u r )(1) (r )! s e u i1 e 2u d u ( s )(1) (r )! s e (u 1 e 2u u 1 e u ) d u r ( ) s (1) (r )! 2 i s e i ( ) s (1) ln 2(r )! s e on maing use of equation (17), the lemma and noting that u 1 (e 2u e u ) d u ln u(2e 2u e u ) d u Γ (1) ln 2 Γ (1) ln 2, proving equation (3). Equation (31) follows on putting s in equation (3) and equation (32) follows on putting r in equation (31).
72 Convolutions Involving the Eponential Function... Equation (31) follows on putting r and s 1 in equation (3). Corollary 2.1. The convolution (e n ) e eists and (34) e n e (1)n 2 n1 γ(2)e (n 1)! for n 1, 2,.... In particular, (35) e 1 e γ(2)e 1 2 e. Proof. Differentiating equation (32), we get [e 1 γδ() e (e 1 ) e γe ln 2e and we see that equation (34) is true when n 1. Now assume that equation (34) is true for some n. Then differentiating equation (34), we get It follows that (e n ne n1 ) e (e n ) e. ne n1 e 2(e n ) e (1)n1 2 n γ(2)e (n 1)! and so equation (33) is true for n1. Equation (34) now follows by induction. Differentiating equation (33), we get and equation (35) follows. [e 1 γδ() e ln 2 e 1 2 e For further results involving the eponential integral, see [2, 3, 4, 5 and [6. References [1 B. Fisher and J.D. Nicholas, The eponential integral and the convolution, SUT J. Math., 33(2)(1997), 139-148. [2 B. Fisher and J.D. Nicholas, The neutri convolution product of ei(λ) and r, Math. Montisnigri, 9(1998), 51-64. [3 B. Fisher and J.D Nicholas, On the eponential integral, Hacet. Bull. Nat. Sci. Eng. Ser. B, 28(1999), 55-64. [4 B. Fisher and J.D Nicholas, The eponential integral and the neutri convolution product, J. Nat. Sci. Math., 4(1,2)(2), 23-36. [5 B. Fisher and J.D. Nicholas, On the eponential integral and the non-commutative convolution product, Integral Transforms Spec. Funct., 11(4)(21), 353-362.
Brian Fisher and Fatma Al-Sirehy 73 [6 B. Fisher, E. Özçaḡ and Ü. Gülen, The eponential integral and the commutative neutri convolution product, J. Analysis, 7(1999), 7-2. [7 I.M. Gel fand and G.E. Shilov, Generalized functions, Vol. I, Academic Press (1964). [8 I.N. Sneddon, Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd, (1961). Brian Fisher Department of Mathematics University of Leicester Leicester, LE1 7RH England E-mail address: fbr@le.ac.u Fatma Al-Sirehy Department of Mathematics King Abdulaziz University Jeddah Saudi Arabia E-mail address: falserehi@au.edu.sa