1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

420 CHAPTER 10 • CONFORMAL MAPPING

y

2

(a)

w =sinz

-----+

x

t = Arcsin w

~

Figure 1 0.17 The transformation w = sin z.

for -; < x < j. We rewrite them as

. u d v

smx = -COS 1 l b an cosx = SI "nhb.

v

u

(b)

We now eliminate x from the equations by squaring and using the trigonometric

identity sin^2 x + cos^2 x = 1. The result is the single equation

u2 v2

--+--=l.

cosh^2 b sinh^2 b

(10-24)

The curve given by Equation ( 10 -24) is identified as an ellipse in the uv plane

that passes through the points (± coshb,O) and (0,±sinhb) and has foci at the

points (± 1,0). Therefore, if b > 0, then v = cosxsinhb > 0, and the image of

the horizontal segment is the portion of the ellipse given by Equation ( 10 -24)

that lies in the upper half-plane Im ( w) > 0. If b < 0, then it is the portion that

lies in t he lower half-plane. The images of several segments are shown in Figure

10. 17 (b). •

10.4.1 The Complex Arcsine Function

We now develop explicit formulas for the real and imaginary parts of the principal

value of the arcsine function w = f (z) = Arcsinz. We use this mapping to solve

problems involving steady temperatures and ideal fluid flow in Section 11.7. The

mapping is found by solving t he equation

x + iy = sinw = sinucosh v + icosusinhv (10-25)