Accession Number : AD0614810

Title :   INDEFINITE INTEGRATION BY RESIDUES, II,

Corporate Author : WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER

Personal Author(s) : Boas,R. P. ,Jr. ; Schoenfeld,Lowell

Report Date : FEB 1965

Pagination or Media Count : 19

Abstract : If F(z) is holomorphic in the extended complex plane except for a finite number of singularities, and if F(z) is holomorphic on the open arc ((cos a + i sin a) ... (cos b + i sin b)) of the unit circle except for simple poles, and if F(z) is holomorphic at cos a + i sin a and at cos b + i sin b, and if a < b < a + 2 pi and u = (b-a)/2, then the Cauchy principal value integral with respect to v of F(exp iv) from a to b is shown to equal i (R + s). R = the sum of the residues of (F(z)/z) log (exp iu (z-exp ia)/(z-exp ib)) for z in the extended plane but not on the closed arc from exp(ia) to exp(ib). S = the sum of the residues of (F(z)/z) log (exp iu (z-exp ia)/ (exp ib-z)) for z on (exp ia ... exp ib).

Descriptors :   (*NUMERICAL INTEGRATION, COMPLEX VARIABLES), (*COMPLEX VARIABLES, NUMERICAL INTEGRATION), INTEGRALS, FUNCTIONS(MATHEMATICS)

Distribution Statement : APPROVED FOR PUBLIC RELEASE