The process of setting up problems of wave propagation through turbulence and reducing the expressions to integrals is typically lengthy. Furthermore, to yield useful results the integrals must be evaluated numerically, except for the simplest problems. Here procedures are given for quickly writing an integral expression and easily evaluating it analytically, yielding a series solution that requires only a few terms to yield accurate results. The solution can also be expressed as a finite sum of generalized hypergeometric functions. The approach uses the Rytov approximation and filter functions in the spatial domain to express quantities of interest such as Zernike modes and effects of anisoplanatism for single or counterpropagating or copropagating plane or spherical waves in integral form. The integrals are readily evaluated with Mellin transforms. We illustrate the technique by deriving the tilt jitter of a single wave and the jitter between two waves with outer-scale effects present. It is shown that outer scale has a significant effect on tilt even for large outer-scale sizes. The effect of outer scale on tilt anisoplanatism is less pronounced.
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