1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions

and the principal branch is defined by

1 i - z

w =Arc cot z =^1 / 2 7r - ---: Log-. -

2z z +z

305

(z "/= ±i) (5.25-15)

On this branch we have ArccotO = %7r, Arccotl =^1 / 4 7r and so on.

Similarly, from

tanhw = -itaniw = z

we obtain

w. = tan h_^1 z = k 7rZ. - 1 L - og --1 - z

2 1 + z

(z "/= ±1)

where the principal branch is defined to be

1 1 1-z

w = Tanh- z = - - Log --

2 1 +z

(z "/= ±1)

Also, from

coth w = i cot i w = z

we find that

w = cot h-1 z = - - 2 + 1 ( k l) 7rZ. + -1 L og --^1 + z

2. 2 1-z

(z "/= ±1)

and the principal branch is defined to be

w = Coth-^1 z = - -1. 7rZ + -^1 Log --l+z

2 2 1-z

(z "/= ±1)

(5.25-16)

(5.25-17)

(5.25-18)

(5.25-19)

The Riemann surfaces for these four functions have the same structure

as the Riemann surface for logz, except for the location of the branch

points. In the case of log z the branch points are at z = 0 and z = oo,

while in the case of arc tan z and arc cot z they are at z = ±i, and for the

case of tanh-^1 z and coth-^1 z the branch points are z = ±1.

The reader may observe that all of the four inverse functions above

result by composition of a bilinear function with the logarithm. Thus for

w = arc tan z it suffices to take infinitely many copies of the z-plane (or

of the z-sphere) cut from -i to +i via oo, stack them with Gk+l on top

of Gk, and identify the right edge of the cut in Gk with the left edge of

the cut in Gk+l·

Note For a general abstract treatment of the theory of Riemann surfaces,

we refer the reader to Springer [17] and Ahlfors and Sario [2].

Exercises 5.5

1. Construct the Riemann surface for each of the following functions.