Some Gauss-Type Formulae for the Evaluation of Cauchy Principal Values of Integrals
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Numer. Math. t9, 419--424 (t972) 9 by Springer-Verlag 1972
Some Gauss-Type Formulae for the Evaluation of Cauchy Principal Values of Integrals
D. t3. Hunter School of Mathematics, University of Bradford, Yorkshire, England
Received June 28, 197t Abstract. A number of formulae are derived for the numerical evaluation of inte- 1 grals of the form fg(x)dx, where g(x) possesses one or more simple poles in the
--1 interval (--t, 1). The formulae are based on Gauss-Legendre quadrature.
1. Introduction This paper is concerned with the numerical evaluation of integrals of the form 1 I= f g(x)dx (t)
--1 where g (z) is a function of a complex variable z, analytic in some region containing the interval E--i, al except at a finite number of simple poles z 1, z~ ..... zt within the interval (--t, a), with residues Q1, Q2 ..... ~t. The more general integral b a c g(x)dx (a, b finite) may be reduced to the form (t) by means of the standard
transformation x = 89 [(a + b) + (b -- a) t]. A number of methods for evaluating integrals of this type have been de- scribed--see, e.g., Longman [51, Stewart [8]. A brief survey and further refer- ences are given in Davis and Rabinowitz [31. Recently Piessens [7] has shown that, when g(x)=](x)/x, l(x) being analytic, I may be evaluated accurately by a Gauss-Legendre formula of even order. The formulae to be derived below generalise Piessens' results.
2. Derivation of the Formulae We take as our starting-point the expression
G. = ~ H,g (x,) (2) r=I where x 1, x, ..... x~ are the zeros of the Legendre polynomial P,, (x), and
1 " ~(x)dx
H, = . (x-- x,)Pd (x,)
--1
--2 - (n+t) ~+l(x,) Pd (x,)
(3) 420 D.B. Hunter: Thus G~ is the n-point Gauss-Legendre formula for estimating the integral 1 f g (x)dx when g (x) possesses no singularities in the interval [--1, tl, (see, e.g.,
--1 Davis and Rabinowitz [3J, Section 2.7). We shall modify this formula to cope
with the evaluation of Cauchy principal values.
Two cases must be considered. Case (a). None of the poles z, coincides with a zero x, of P~ (x).
Following McNamee [6J, we consider the contour integral t f g(z) dz 2 :,i J (z- x) p~ (z) C
where C is a closed contour containing the interval E--I, 1]. In addition to the poles z,, z 2 ..... z t within that interval, C may contain further simple poles zt+ 1 ..... z., of g(z) with residues ~t+, ..... ~m; apart from those, g(z) must be
analytic within and on C. Provided x does not coincide with a zero of P. (z) or a pole of g(z), we obtain, on applying Cauchy's residue theorem to the above integral, and rearranging terms, the equation
2 2 q'Pn(x) P~(x) I g(z)dz P~(x)g (x,) + g (x) = (, -- x,) ~' (x,) (x -- z,) ~ (z,) + -2~-C .! (z -- x) P~ (z) r=l r=l C
Now integrate g (x) along a path L consisting of the interval [--1, 1 J, with a small semi-circular indentation of radius e to carry it above each of the poles z, (r =t, 2 ..... t). Letting e-+0, this gives
1 1 m
fg(x)dx=G"+ ~ 1 ~(z,)-~ J + f p.(x)ux f . (4)
.~ P~(x)dx g(z)dz
= X -- Z r --1 --1 L C
By Neumann's formula for the Legendre function of the second kind, Q~ (z) (see, e.g., Abramowitz and Stegun [t], Eq. (8.8.3)), we have, if zr [--t, 1],
1 f (z--x)-*P~(x)dx = 2Q. (z). (5)
--1
Further, if zE(--l, t), it is easy to deduce that 1 f (Z -- %)-1Pn (x) d x = 2Qn (Z), (6) --1
the convention for values of Q~(z) on the cut (--1, 1) being that of Abramowitz and Stegun [t J, Eq. (8.3.4), namely,
Q.(z) =89 [Q~(z+Oi) +Q.(z -Oi)], (-1
Using these results, we can now rewrite (4) in the form 1 f g(x)dx=G'~ + E. (8)
--1 Evaluation of Cauchy Principal Values 421 where
and G'. = G.-- R., (9)
R n ---- 2 ~, o,Q. (z,)/P~ (z,) (! 0)
r=l
E. -- ~i 3 P.(z) " C
Thus E. is the error in using G'. as an approximation for I. The properties of E. have been investigated by several authors, including McNamee [6], Stenger [9j, Chawla and Jain ~2] and Kambo [4]. McNamee obtains a formula equivalent to 00), but taking account only of those poles of g (z) which lie outside the interval [--t, t]. Formally, the poles within that interval are treated in exactly the same way as those outside it in (10), but Eq. (7) indicates that, in fact, Q.(z) is dis- continuous along the cut (--t, 1).
Case (b). Some poles of g (z) coincide with zeros of P. (z).
The above equations must be modified if any of the poles of g (z) coincide with zeros of P. (z)--for example, it is clear that Eq. (2) then breaks down. Of course, this situation may be avoided by merely altering the value of n; there is, however, some interest in seeing what modifications are necessary, and in at least one case, described below, a useful formula is obtained. Consider a pole z~, say, which coincides with a zero x s of P,, (z) (on suitably numbering the poles z, and zeros x.). It can be readily shown that the residue of g(z)/(z--x)P,,(z) at x~ is then
