ABSTRACT
The functions f(z) and g(z) are holomorphic in a domain containing the annular sector S: a < phz < /3, JzJ 2 a, and
where f,Z # 49,. The remainder terms associated with these expansions are denoted by
same notation. 12.2 By using the identity
and a similar identity for f(z)En(z), we may verify that the residual term R,(z) of (2.06) is given by
where
912 Error Bounds for the Asymptotic Solutions in the General Case 283
To construct a new integral equation equivalent to (2.07), we first seek a differential equation which approximates the given equation
w"+ fow'+gow = 0, when Jz( is large. The most obvious choice is
but this cannot be solved in terms of elementary functions, in general. We apply the result of Exercise 1.2 of Chapter 6, determining the functions p and
q in such a way that the expansions of the coefficients of dW/dz and Win powers of z-I match the expansions of f(z) and g(z), respectively, as far as the terms in z-I. Obviously the choice is not unique; for simplicity we take
f~ 9, constant q ' f o + ~ ~ z2 '
choosing the constant in the second relation to makep a perfect square. Thus
where
Wt(z) = (1 + :)-'12eAlzzwl , W2 (z) = (1 + :)- 112eA272, (12.07) satisfy the differential equation
in which
42 = +fi&-2) - P2(+f;-90) ' Cc1Cc2 + 4@1+Cc2k (1 2.09) and
Clearly (12.08) has the desired matching with (12.05).
