Advanced Calculus fi..

Chapter 8 Functions of a Complex Variable 609vtFigure 8.38 Example 7.maps It( < 1 on the half-plane Im (zl) r 0. The point ela has as image zl = m; thepoint elB has as image zl = 0; the arc u < B < /l corresponds to the negative realaxis in the z, plane. Thus the solution is given byThis rather clumsy formula is the formula for the circle corresponding to (8.103) forthe half-plane. By use of geometry, this can be written in the much simpler form:where s is the arc length of the arc on lzl = 1 determined by the chords through e'"and z and through elB and z; see Fig. 8.38.This problem could also have been solved directly by the Poisson integral formula(8.101):u(ro, 00) = -21 - roix /.' I + ri - 210cos(Bo - 8)The integration, though awkward, can be carried out with elementary functions.dB.PROBLEMS1. Solve the following boundary value problems:a) u harmonic and bounded in the first quadrant,lim u(x,y) = 1 forO