1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 301

w = arcsinz = 1/ 2 7r =i= iLog(z + ~) + 2h (5.25-3)

To choose a principal branch for arc sin z we select that which will give

w = 0 for z = 0. This is accomplished by taking k = 0 and the positive sign

in the second term on the right of (5.25-3). Thus, denoting the principal

branch by Arcsinz, we have

w = Arcsinz = %7r + i Log(z + ~)

and introducing this notation in (5.25-3), we have

{

Arc sin z + 2k7r

arcsinz =

-Arcsinz + {2k + 1)7r

(5.25-4)

(5.25-5)

The reader will note that arc sin z is defined for all finite values of z, since
z + .../z^2 - 1 f= 0 for all z.
In a similar manner, any solution of the equation cos w = z for given
z E C is a value of the multiple-valued inverse cosine, denoted

w = arccosz or w = cos-^1 z

Since cos w = 'sin{% 7r - w ), we have

w = arccosz =^1 / 2 7r - arcsinz

= ±iLog(z + ~)+2h {5.25-6)

The principal branch of this function is defined by

w = Arccosz = -iLog(z + ~) {5.25-7)

On this branch we have ArccosO =^1 / 2 7r, Arccosl = 0, and so on.
As to the inverse hyperbolic sine, we write w = sinh-^1 z to denote
any value of w such that sinh w = z. The last equation is equivalent to
s1n iw = iz, so

w=sinh-^1 z= -iarcsiniz

= { Log{z + .Jz2+i) + 2hi

• Log{z + .Jz2+i) + {2k + l)7ri

{5.25-8)

The principal branch of the inverse hyperbolic sine is defined to be

w = Sinh-^1 z = Log(z + Jz^2 +1) (5.25-9)

For this branch we have Sinh-^1 0 = O, Sinh-^1 i = %7ri, and so on.