## The asymptotic solutions of an ordinary differential equation in which the coefficient of the parameter is singular |

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analytic function bounded arc length arg JUL asymptotic forms asymptotic representations asymptotic solutions becomes infinite behavior boundary bounded from zero bounded in length boundedness coefficients complex plane considered by Langer converges uniformly curves corresponding curves of integration decreasing with respect defined denote Doctor of Philosophy domain Edmond Darrell Cashwell elementary functions ential equation exponent find the asymptotic follows formula 19 greater than 1/2 higher order independent of u^z independent solutions induction infinite series integration In 16 JJL plane kernel monotone decreasing Omitting the calculations ORDINARY DIFFERENTIAL EQUATION pair of independent pair of solutions PARAMETER IS SINGULAR path of integration point zQ pole of second Prom range region regular singular point related equation relation 23 right of 20 schlicht second order Second Solution solution independent solution of equation subregions suitably large constant take zQ transformation 13 ug(z uniformly bounded University of Wisconsin vanishes or becomes Wisconsin in partial