 By F. W. J. Olver

ISBN-10: 012525850X

ISBN-13: 9780125258500

A vintage reference, meant for graduate scholars mathematicians, physicists, and engineers, this booklet can be utilized either because the foundation for educational classes and as a reference instrument

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Additional resources for Asymptotics and Special Functions

Example text

Ex. ι(^Ό: (i) along the join of - 1 and 1; (ii) around the other three sides of the square having vertices at - 1 , 1, 1 + 2 / , - 1 + 2/; (iii) around the path conjugate to (ii). Ex. ,z„, arranged in order. Show that ir^(/) = sup''xi/(zs^i)-/(zs)|, for all η and all possible modes of subdivision, provided that ζ"(τ) is continuous on each arc of ^ . 4 Historical details in this subsection were obtained from Bromwich (1926, §104). Further information is contained in this reference. §§4-6 For further results concerning integration and differentiation of asymptotic and order relations, and the asymptotic solution of transcendental equations see de Bruijn (1961), Berg (1968), Dieudonne (1968), and Riekstiiis (1968).

04) e'^dt. It is easily verified that F{z) = —e-^' erf(/z). 2 Corresponding integrals of oscillatory type (when the variables are real) are the Fresnel integrals cos(iπ/^)Λ, C(z) = S(z) = un{\nt^)dt. 06) They, too, are entire. In terms of the error function, C(z) + iS{z) = i ( l + 0 erf {^π^/^(1 - 0 ^ } . 07) Ex. 04). Ex. 2 Let a and Z> be positive and e x p ( - ö / ^ ) ( / 2 + 6^)-^i/r. / = By considering d{t\p{-ab^)I)lda prove that / = ^nb'^ exp(a^^) erfc(Z>a'/*). Ex. 3 Show that C(oo) = 5 ( o o ) = i .

Ex. 6 Verify that for n ;;. _ , (x) = 0, and (n+ I)L~O~ I (x) + (x-2n-a-I)L~")(x) + (n+ a)L~'21 (x) = O. Ex. 7 Show that Ex. (x). Ex. (x) = n- I f"{x + (X 2 _1)1I2 cosO}·dO. 1 For real values of χ the Airy integral is defined by π J cos(^t^-l·xt)dt. 01) Although the integrand does not die away as ί -»c», its increasingly rapid oscilla­ tions induce convergence of the integral. This can be confirmed by partial integration, as follows. We have cos(it^+xt)dt t^+x + 2 sin(it^+xt) tdt {t^+x) 2 * As r 00 the first term on the right-hand side vanishes, and the last integral con­ verges absolutely.