478 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



10. FORMULAS FOR THE SADDLE-POINT METHOD 



Much of our work involves the behavior of the parabolic cylinder 

 functions as functions of n when n is a large complex number. Although 

 this subject has been studied by several writers," •" • " their results 

 are not in the form we require. As the work of Sections 6 and 7 shows, 

 the paths of steepest descent for the integrals in our electromagnetic 

 problem are intimately connected with the function /(/o) — f(h)- In 

 turn, this function is closely related to the saddle point method of e\'alu- 

 ating Un(z), etc., for large values of n. For the sake of completeness, 

 we shall outline this method. We shall pay special attention to the rela- 

 tive importance of the two saddle points as n moves about in its complex 

 plane. 



When we write the integrand of the integrals (9.1) as exp [f(t)] we 

 obtain expressions of the form 



Uniz) = 1-f exp [fit)] dl, 

 Zirl Ju 



(10.1) 



/(/) = -f- ^ 2zt - m log /, m = n + I. 



The saddle points of the integrand are at the points U and t\ in the com- 

 plex ^plane where /'(/) is zero: 



2/o - 2zto + m = 0, tl - zio = -m/2, 



to + h = z, 



h = ^^^ — , 2/o/i = m. 



(10.2) 



Let the path of integration U of (10.1), for example, be deformed so 

 as to pass through a saddle point, say ^o, along a path of steepest descent. 

 Let 



00 



fit) = fito) - E hit - t,Y/k!. (10.3) 



2 



Then, if 62 is not too small, the contribution of the region around to 



20 Nathan Schwid, The Asymptotic Forms of the Hermite and Weber Functions, 

 Amer. Math. Soc. Trans. 37, pp. 339-362, 1935. References to earlier work will be 

 found in this paper. Schwid's work is based on R. Langer's studj' of the asymptotic 

 solutions of second order differential equations. 



-1 O. E. H. Rvdbeck, The Propagation of Radio Waves, Trans, of Chalmers Univ. 

 of Tech. 34, 1944. 



2^ G. N. Watson, Harmonic Functions Associated with Parabolic Cylinder 

 Functions, Proc. London Math. Soc. (2) 17, pp. 116-148, 1918. 



