ABSTRACT
When 1 # 1, it is obvious that the right-hand side of (9.02) does not reduce to the right-hand side of (9.03) as a + 0. When 1 = I , the limiting value of the right-hand side of (9.02) is finite and nonzero, but differs by a factor of 2 from (9.03).' This abrupt change in the form of the approximation at a = 0 indicates that the accuracy of (9.02) deteriorates as a diminishes. To put this another way, the approximation (9.02) cannot be uniformly valid for arbitrarily small values of a. 9.2 We now outline a method, due to Bleistein (1966), which yields a generalized asymptotic expansion for I (a , x ) which is uniformly valid in a closed a interval containing a = 0. The underlying idea is to transform (9.01) into an integral which closely resembles the simplest kind of integral maintaining the essential features:
the relationship is free from singularity at t = a. Making the transformation (9.04), we obtain
