Asymptotic Formulas for Diffraction By Parabolic Surfaces

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Asymptotic Formulas for Diffraction By Parabolic Surfaces
Harry Hochstadt
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2iky^^) wP(-2iky^^) |P(-2ikv, ^) w^(-2iky|^) m P(-2ikr, ) inP(-2ikC) w^ (-2ik£ ) 1 K) ) mP (-2ik^^) w^ (-2ik4) s I a I — s w — B =0 'o ^ai ^0=^ ^ ^°' 0 ^o • The integrals under the summation can be evaluated by closing the contour on the left and suimning over the residues of P(s + -5^). Heference to the asymptotic fonmilaB '- -» for the parabolic functions with respect to the parameter s shows that this is permissible provided that This leads to 9G^ %-\ sir t y'''""' t ^^^"^ ■!.^" la p«-co n«o... ^ n+ n+ — - n+— ij*- It now follows from the definition of the function in^(z) that 7- B-, _(-2ik£ ) n+ 1+p^ -P/2 -"^ W^ii^^a) -^oV i^^Vna^ ^Fi(-n;l. P|-2ik4^) - 9 - The KuDuner functions appearing in the fraction are polynoraiais of degree n.

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