1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

486 CHAPTER ll • APPLICATIONS OF HARMONIC FUNCTIONS

the angle of the trailing edge of the modified Joukowski airfoil A , forms an angle

of 0.15.,,. radians. Hint: The image of the circle Co is the line Lo, then two rays

Ro. 1 and R.i. 2 , and then two arcs Ao, 1 and Ao, 2 in the respective Z, W, and w

planes. The image of the circle C 1 is the circle Ki. then the "cardioid-like" curve

H 1 , and then the modified Joukowski airfoil A 1 •

11.9 T he Schwarz- Christoffel Transformation

To proceed further, we must review the rotational effect of a conformal mapping

w = f (z) at a point zo. If the contour C has the parameterization z (t) =

x (t) + iy (t), then a vector r tangent to C at the point zo is

r = z' (t-0) = x' (to)+ iy' (to).

T he image of C is a contour K given byw =tt(x(t) ,y(t))+iv(x(t) ,y(t)), and

a vector T tangent to Kat the point wo = f(zo) is

T = w' (zo) = f' (zo) z' (to).

If the angle of inclination of r is f3 = Arg z' ( t 0 ) , then the angle of inclination of

Tis

ArgT = Arg[f' (zo) z' (to)] = Arg f' (zo) + (3.

Hence the angle of inclination of the tangent r to C at z 0 is rotated through the

angle Arg f' (zo) to obtain the angle of inclination of the tangent T to K at the

point wo.

Many applications involving conformal mappings require the construction of

a one-to-one conformal mapping from the upper half-plane Im(z) > 0 onto a

domain G in the w plane where the boundary consists of straight-line segments.

Let's consider the case where G is the interior of a polygon P with vertices

w 1 , w2,... , Wn specified in the positive sense (counterclockwise). We want to

find a function w = f (z) with the property

Wk=f(xk),

Wn=f(oo),

for k = 1, 2, ... , n - 1, and

where X1 < X2 < · · · < Xn- 1 < oo.

(11-38)

Two German mathematicians, Herman Amandus Schwarz (1843-1921) and El-

win Bruno Christoffel (1829-1900), independently discovered a method for find-

ing f, which we present as Theorem 11.6.