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American Mathematical Society arbitrary degree asymptotic forms asymptotic solutions asymptotio axis of imaginaries axis of reals Bessel functions boundary problem defined characteristic equation characteristic values explicit coefficients complex plane conditions in 3.8 constant denote derived differential equation 1.1 differential system 1.3 Doctor of Philosophy epsilon function equal intervals equation 2,6 equation in 1.3 equation in sector explicit form explicit to terms exponential sums forms similar fulfill the conditions gives the characteristic H^are Hence Henoe highly irregular initial terms J.J.Barron Langer in paper linear differential equation linearly independent means of relations mildly irregular obtain forms pair of solutions polynomials H positive integer Prom ranges regular boundary problem regular','mildly irregular relations 3.8 represented asymptotically roots of equation SECOND ORDER BASED sector Sj singular points Sinoe solutions u,(x strip bounded Substituting this value suoh terms containing terms in powers Thesis Submitted University of Wisconsin values in sector written zero,the