422 CHAPTER 10 • CONFORMAL MAPPING
v
y w = Arcsin z'(^4) ' ·~ ..,.....,
'
-~
3 '1
2
,...
. x
2 3 4 0
Figuro 10.18 The mapping w = Arcsin z.
x and y:!!
4[J(x+ 1)
2
+y 2 + V(x -1)
2
- yz l
v(x,y)=(signy)Arccosh
2
!!
2( 10 -27)where sign y = 1, if y ~ 0, and s ign y = -1, if y < O. The real function given by
Arccosh t =In (t + vt2=1') with t;:::: 1 is used in Equation (10-27).
Therefore, the mapping w = Arcsinz is a one-to-one conformal mapping of
the z plane cut along the rays x ~ -1, y = 0, and x ;::: 1, y = 0, onto the
vertical strip --i" ~ u ~ ~ in the w plane, which can be construed from Figure
10.17 if we interchange the roles of the z and ·w planes. The image of the square
0 ~ x :::;; 4, 0 ~ y ~ 4, under w = Arcsinz, is shown in Figure 10 .18. We
obtained it by plotting t he two families of curves {(u (c, t), v (c, t)) : 0:::;; t:::;; 4}
and {(u (t,c), v (t , c)) : 0:::;; t:::;; 4}, where c = ~' k = 0, 1, ... , 20. The formulas
in Equations (10-26) and (10-27) are also convenient for evaluating Arcs in z, as
shown in Example 10 .14.
- EXAMPLE 10.1 4 Find the principal value Arcsin (1 + i).
Solution Using Formulas (10-26) and (10-27), we getvs-1
u (1, 1) = Arcsin -
2- ::::: 0.666239432 and
V5+ 1
v (1, 1) = Arccosh -
2- ::::: 1.061275062.