1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.5 • INVERSE TRIGONOMETRIC AND HYPERBOLIC F UNCTIONS 189

We can find the derivatives of any branch of these functions by using the
chain rule:

_:!.. arcsin z =

1

1 , (5-46)

dz (1-z2)>

d - 1

dz arccos z = i , and

(1-z^2 )•

d 1

dz arctan z = 1 + zZ.

We derive Equations (5-45) and (5-46) and leave the others as exercises. If

we take a particular branch of the multivalued function, w = arcsin z, we have

z = sin w = -1 ( e•w. - e-•w. )
2i '
which we can also write as

eiw -2iz - e-iw = 0.

Multiplying both sides of this equation by eiw gives (eiw)

2

• 2izeiw - 1 = O,
  which is a quadratic equation in terms of eiw. Using the quadratic equation to
  solve for eiw, we obtain
  I

iw 2iz + ( 4 - 4z^2 )'. ( 2 ) t

e =
2

=iz+ l-z,

where the square root is a multivalued function. Taking the logarithm of both
sides of this last equation leads to the desired result:

w = arcsinz = -ilog [iz + (1- z^2 ) ' ] ,

where the multivalued logarithm is used. To construct a specific branch of
a.resin z, we must first select a branch of the square root and then select a branch
of the logarithm.
We get the derivative of w = arcsin z by starting with the equation sin w = z
and differentiating both sides, using the chain rule:

d. d

-s1nw = - z·

dz dz '

~sinwdw = 1·

dw dz '

dw 1

• = --
  dz cosw

When t he principal value is used, w = Arc sin z = - iLog [iz + ( 1 - z^2 ) ~]

maps the upper half-plane { z : Im ( z) > 0} onto a portion of the upper half-

plane {w: Im(w) > O} that lies in the vertical strip {w: 2" < Re(w) < ~}.

The image of a rectangular grid in the z plane is a "spider web" in the w plane,

as Figure 5.10 shows.