Note on the WKB method

In the phase integral WKB method, a solution u1(z) of a second-order linear differential equation is represented in terms of its logarithmic derivative iy(z) which satisfies a simple nonlinear first-order equation. This representation does not in general lead directly to an independent second solution of the original equation. However, if y(z) is expressed in the form q(z) + iq′(z)/2q(z), where q(z) satisfies a nonlinear second-order equation, then q(z) can be used to determine a second solution to the original equation. These two solutions remain linearly independent throughout their domain of definition. It is shown that q(z) is given by the sum of alternate terms in the well-known asymptotic expansion of y(z). Any two linearly independent solutions of the original equation, normalized so that the Wronskian is -2i, give q(z) in the form (u 1u2)-1.