where

$\begin{array}{l}\text{\tau}(\text{\xi})=\pm 2\sqrt{{a}^{2}+2c}\left(\text{\xi}+\frac{ab+d}{{a}^{2}+2c}\right),\\ \text{\beta}(\text{\lambda})=2\text{\lambda}-{b}^{2}+\frac{{(ab+d)}^{2}}{{a}^{2}+2c}\pm \sqrt{{a}^{2}+2c}.\end{array}$

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 *a*^{2} + 2*c* = 0 and *ab* + *d* = 0 for the various coefficients, Equation (10.26) reduces to the normal form

$\left(\frac{{d}^{2}}{d{\text{\xi}}^{2}}+2\text{\lambda}-{b}^{2}\right){v}_{\text{\lambda}}(\text{\xi})=0,$

yielding ${v}_{\text{\lambda}}(\text{\xi})={v}_{0}{e}^{\pm i\sqrt{2\text{\lambda}-{b}^{2}}\text{\xi}}$. Indeed, it can readily be seen that, under the aforementioned relations for the coefficients, Equation (10.9) ultimately yields the ...

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