Advanced Calculus fi..

'Chapter 8 Functions of a Complex Variable 61 7of the quarter-plane. Accordingly, u satisfies all conditions. If we return to the xyplane, u becomes a function of x and y which solves the given boundary value problem.This can be shown to be the only bounded solution, provided h is piecewisecontinuous.PROBLEMS1. Prove thatf (z) = ao zeIa + - + const (a real)( z:la)maps D: lzl > 1 in a one-to-one conformal manner on a slit domain Dl (this is the mostgeneral such map). Interpret f as a complex velocity potential.2. Show that the vectorcan be interpreted as the velocity of an irrotational, incompressible flow past the obstaclebounded by the circle x2 + y2 = 1. Find the complex velocity potential and the streamfunction, and plot several stream lines.3. Let w = F(z) be analytic in the domain D outside the simple closed curve C and have apole of first order at oo. Let F(z) be continuous in D plus C and let Im [F(z)] = conston C. Show that F(z) maps D in a one-to-one fashion on a silt-domain. [Hint: Show thatthe argument principle of Section 8.17 here takes the form: the increase in arg F(z) asz traces C in the negative direction is 2n times (No - N,), where No and N, are thenumbers of zeros and poles of F(z) in D plus the point z = oo. Show that the increase inarg [F(z) - wo] is 0 for wo not on the image of C. Hence No - N, = 0; but N, = 1, sothat No = 1.14. Let U(x, y) be biharmonic for x2 + y2 -= 1. Show that U can be expanded in a Taylorseries in this domain, so that U is analytic in x and y.5. Find the equilibrium temperature distribution T in the half-strip: 0 < x < 1, y > 0 if theedge x = 0 is maintained at a temperature To, the edge x = 1 is maintained at a temperatureT,, while the edge y = 0 is insulated (aT/an = 0).8.27 GENERAL FORMULASONE-TO-ONESCHWARZ-CHRISTOFFEL TRANSFORMATIONMAPPINGThe examples above have indicated the importance of conformal mapping for applications.There still remains the problem of exhibiting a reasonably large class ofexplicit mappings to meet the needs of applications. Here we give several formulasthat help in this direction.Mapping onto injinite strip with slits.chosen such that for some m,Let real constants hl, h2, . . . , h,,, h,+, behl < hZ < .-. < h,; h, > h,+l > -.. > h,+l. (8.115)