where

τ(ξ)=±2a2+2cξ+ab+da2+2c,β(λ)=2λb2+(ab+d)2a2+2c±a2+2c.

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.

10.3.2.1.2.2.1 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

d2dξ2+2λb2vλ(ξ)=0,

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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