INDEFINITE INTEGRATION BY RESIDUES, II,
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).
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1965
- Accession Number
- AD0614810
Entities
People
- Lowell Schoenfeld
- R. P. Boas Jr.
Organizations
- University of Wisconsin–Madison