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04) where ex is a constant such that ex > -1. (x) being reserved for L~O)(x). (x). ,. 2 Explicit expressions for the foregoing polynomials are supplied by Rodrigues' formulas: (-)" d' 2"«! 07) e^x'' d" n! 09) That each of these expressions represents a polynomial of degree η is perceivable from Leibniz's theorem. 07), for example, let φ„{χ) denote the right-hand side and w{x) be any polynomial. Then by repeated partial integrations we arrive at 1 (i-xy^%\-{-xy^f'w^"\x)dx. 2»«! J -1 The last integral vanishes when the degree of w(x) is less than n.

1. This result has been established for positive z . However, if b is an arbitrary small positive number and if begins in the sector - i π + < 5 < p h ^ ; < - i π - < 5 and ends at infinity in the conjugate sector, then at the extremities of Se the factor exp(i;^/3) dominates 54 2 Introduction to Special Functions Fig. 1 V plane. 02) converges absolutely and uniformly in any compact ζ domain. 02) supplies the analytic con tinuation of Ai(z) to the whole ζ plane; moreover, Ai(z) is entire. 2 To obtain the Maclaurin expansion of Ai(z) we use the following general theorem concerning the integration of an infinite series over an infinite interval, or over an interval in which terms become infinite.

Ex. 1 Show that Si(z) = ^ s= 0 i-Y z^^^' . Ci(z) = l n 2 + y + -, 25+1(25+1)! Ex. 2 Show that jl'^txp(-ze'') dt z'^ ^ - 25 (25)! =-si(z)-i{Ci(zHEi{z)}. Ex. 3 If a is real and b is positive, prove that Ex. 4 Verify the following Laplace transforms when Rep > 0: ttan-*/? Too e-'^smdt 0 = -- 2p Ex. 5 The generalized exponential integral is defined by E„(z) = Cooe-" dt (Λ = 1 , 2 , . . ) , when Rez > 0, and by analytic continuation elsewhere. Show that the only singularity of £'„(z) is a branch point at ζ = 0.