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4 we have that where vg is as previously defined. Hence Multidimensional Inverse Scattering Theory 45 where Vg is the Herglotz wave function with kernel g. By the Schwarz inequality we now have that Since |*4(x)l2 < ^\\9\\2 for &U x e K2. 4). Remark. Let r be the radius of the smallest circle with center on Im A = —Re A, Im A > 0, and passing through the origin that contains all the eigenvalues of F. 2. 3. 3. 7. The Inverse Scattering Problem The inverse scattering problem we are concerned with is to determine the index of refraction re(x) from a knowledge of Uoo(x;d) for x,d e £l,k fixed.

Now the field was open to Hertz's experiments on radio waves and to Marconi's wireless telegraph. We recommend [5] and [7] for further study of electromagnetic phenomena. References [1] T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, MA, 1957. [2] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1992. [3] L. Hormander, Linear Partial Differential Operators III, Springer-Verlag, Berlin, 1985. [4] R. Kress, Linear Integral Equations, Springer-Verlag, Berlin, 1989.

Consider the function To this end, For fixed z this is an analytic function of t in any annulus It therefore has a Laurent expansion where and C is a positively oriented simple closed curve enclosing the origin. Setting t — 2u/z and deforming C gives and, computing the residue, 30 Inverse Problems for n > 0. 7) we see that this is also true for n < 0. 9) gives Finally, it can be shown that the following asymptotic expansions are valid for from which expansions for Yn(r) and Hn (r) can be deduced.