1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

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190 CHAPTER 5 • ELEMENTARY FUNCTIONS


10

x


  • 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 obtain

a.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 write

a.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 is

f ( v12) = arcsin J2 = -iLog ( i J2 + i) = i -i In ( J2 + 1) ,

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