The occurrence of the double sign in the preceding expressions makes evident that the transformation from (10.9) to (10.26) is not unique (Nikiforov and Uvarov 1988); the specific choice may eventually be suggested by the problem at hand. Plane Waves, Parabolic Waves, and Gaussian Beams  If τ′(ξ) = 0, which for the specific form of τ(ξ) amounts to τ(ξ) = 0 as well, and hence to the relations a2 + 2c = 0 and ab + d = 0 for the various coefficients, Equation (10.26) reduces to the normal form


yielding vλ(ξ)=v0e±i2λb2ξ. Indeed, it can readily be seen that, under the aforementioned relations for the coefficients, Equation (10.9) ultimately yields the ...

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