Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESyBC 2C 1xC 3AFigure 24.8A and B.Some alternative paths for the integral of a function f(z) betweenThe question of when such an integral exists will not be pursued, except to statethat a sufficient condition is that dx/dt and dy/dt are continuous.◮Evaluate the complex integral of f(z) =z −1 along the circle |z| = R, starting and finishingat z = R.The path C 1 is parameterised as follows (figure 24.9(a)):z(t) =R cos t + iR sin t, 0 ≤ t ≤ 2π,whilst f(z) isgivenbyf(z) = 1x + iy = x − iyx 2 + y . 2Thus the real and imaginary parts of f(z) areu =xx 2 + y = R cos t and v = −y2 R 2 x 2 + y = − R sin t .2 R 2Hence, using expression (24.34),∫ ∫12π∫C 1z dz = cos t2π( − sin t0 R(−R sin t) dt − 0 R∫ 2π∫cos t2π( − sin t+ i0 RR cos tdt+ i 0 R=0+0+iπ + iπ =2πi. ◭)R cos tdt)(−R sin t) dt (24.35)With a bit of experience, the reader may be able to evaluate integrals likethe LHS of (24.35) directly without having to write them as four separate realintegrals. In the present case,∫C 1∫dz 2πz = −R sin t + iR cos tdt =0 R cos t + iR sin t846∫ 2π0idt=2πi. (24.36)