Physics
Scientific paper
Apr 1956
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1956rspta.249...65o&link_type=abstract
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 249, Issue 959,
Physics
19
Scientific paper
The differential equation d2w/dz2 = left\{uzn+γ/z2+g(z)right\}w, where n is an integer (>= -1), u a parameter and γ a constant, has the formal solution w = P(z)left\{1+sums=1∞As(z)/usright\} + P'(z)/usums=0∞Bs(z)/ us, where P(z) is a solution of the equation d2P/dz2 = (uzn+γ/z2)P. The coefficients As(z) and Bs(z) are given by recurrence relations. It is shown that they are analytic at z = 0 if, and only if, the differential equation for w can be transformed into a similar equation with n = 0, γ = 0, or n = 1, γ = 0, or n = -1. The first two cases (for which P is an exponential and Airy function respectively) have been treated in a previous paper. The third case, for which P is a Bessel function of order ± (1 + 4γ )1/2, is examined in detail in the present paper. It is proved that for large positive u, solutions exist whose asymptotic expansions in Poincare's sense are given by the formal series, and that these expansions are uniformly valid with respect to the complex variable z.
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