190 CHAPTER 5 • ELEMENTARY FUNCTIONS
10x- 10 -3 --0 ..... - 2 2 4 6 8 10
Figure 5.10 A rectangular grid is mapped onto a spider web by w = a.resin z. - EXAMPLE 5.11 The values of arcsin v'2 are given by
arcsin J2 = -ilog [iv'2 + ( 1 - ( v12)2)! ] = -ilog (iv12 ± i). (5-47)
Using straightforward techniques, we simplify this equation and obtaina.rcsinv'2 = -ilog[(v12±l)i)= i [in ( J2 ± 1) + i (i + 2mr)]
= i + 2mr -iln ( J2 ± 1), where n is a.n integer.
We observe tha.t( )(J2-1)(v'2+ 1) 1 ( )
In J2- 1 = In v'2+l =In v'2+l =- In J2+1
and then writea.resin J2 = i + 2mr ± i In ( J2 + 1) , where n is an integer.
- EXAMPLE 5.12 Suppose that we make specific choices in Equation (5-47)
1
by selecting + i a.s the value of the square root (1 - ( v'2)
2
]2
and using the
principal value of the logarithm. With J (z) = Arcsinz, The result isf ( v12) = arcsin J2 = -iLog ( i J2 + i) = i -i In ( J2 + 1) ,