1550251515-Classical_Complex_Analysis__Gonzalez_

304 Chapter5

Gi to another Gp by crossing the cut (-oo,-1) or (+1,+oo), or both, in

a suitable direction a certain number of times.

It is clear from formulas (5.25-6), (5.25-8) and (5.25-10) that the struc-

ture of the Riemann surfaces for arccosz, sinh-^1 z, and cosh-^1 z is the
same, mutatis mutandis, as that· of arc sin z. For instance, in the case of
sinh-^1 z, each copy of the z-plane is to be subjected to a preliminary ro-
tation of 1/irr, by letting z' = iz, thus transforming the branch points into
z' = ±i. After the mapping w' = arc sin z' is done, the w' -plane is to

be rotated -^1 / 2 rr (since w = -iw'), thus placing the vertical strips Hj of

Fig. 5.44 in a horizontal position.

Next, we proceed with a discussion of the inverse tangent function. Any

solution of the equation tan w = z for a given z E C - { i, -i} is a value of

the multiple-valued function called the inverse tangent, denoted

w = arctanz or w = tan-^1 z

From

e2iw -1

tan w = -i 2. = z

e iw + 1

we get

e2iw =^1 + iz = i - z

1 - iz i + z

Therefore,

1 i-z

w = arctanz = - log --

2i i + z

1 i -z

= ---:-Log -. - + k7r

2i i +z

(5.25-12)

Obviously, the points z = ±i, which are not in the domain of definition

of arctan z, are logarithmic branch points for this function. The principal

branch of arc tan z is defined to be

1 i-z

w = Arctanz = - Log --

2i i + z

(z-:/= ±i) (5.25-13)

For this branch we have Arctan 0 = 0, Arctan 1 = 7r / 4, and so on. If

we write

cot w = tan(1/irr - w) = z

for the inverse cotangent we derive the formula

1 1 i -z

w = arc cot z = - ( 2k + 1 )rr - ---:-Log -. -

2 2i i + z

(z-:/= ±i) (5.25-14)