1.4. Polar Form for Complex Numbers 23

Other Functions. If we fix a branch of the complex log function, we can define

a number of other complex functions which are extensions of familiar real functions.

We mention some of these briefly.

An nth root function can be defined by setting 01/n = 0 and

(1.4.5)

z1/n

=

e(1/n) log z

if z = 0.

A special case of this is the square root function defined by

√

0 = 0 and

(1.4.6)

√

z =

e(1/2) log z

if z = 0.

Note that the nth root function, as defined by (1.4.5), is giving only one of the

nth roots of z. Which one is determined by the branch of the log function that is

used. The other nth roots are obtained by multiplying this one by the nth roots

of unity (Exercise 1.4.9). For example, given one square root of z, the other is

obtained by multipling it by −1.

Example 1.4.9. If

√

z is defined using the principal branch of the log function,

analyze the behavior of

√

z near the cut line for this branch.

Solution: For the principal branch of the log function the interval I is (−π, π]

and so the cut line is the line defined by θ = π in polar coordinates – that is, it is

the negative real axis. If z = r

eiθ

is just above the negative real axis, then log z is

nearly log r + iπ and

√

z =

e(1/2) log z

is nearly i

√

r. On the other hand, if z is just below the negative axis, then log z is

nearly log r − iπ, (1/2) log z is close to (log r − iπ)/2 and

√

z is close to −i

√

r. In

other words, as z crosses the cut line,

√

z jumps from one square root of z to its

negative, which is the other square root of z.

Example 1.4.10. Analyze the function

√

z2 + 1, where the square root function

is defined, as above, using the principal branch of the log function.

Solution: The cut line for

√

z is the same as for log – the half-line of negative

reals. The function

√

z jumps from values on the positive imaginary axis to their

negatives as z crosses this line in the counterclockwise direction. The number

z2

+1

crosses the negative real half-line in the counterclockwise direction as z crosses either

(i, i∞) or (−i∞, −i) in the counterclockwise direction. Thus, these two half-lines

on the imaginary axes are where discontinuities of

√

z2 + 1 occur. As either of

these half-lines is crossed in the counterclockwise direction, a typical value of this

function jumps from a number it with t 0 to −it (see Figure 1.4.4).

Raising a complex number to a complex power is another function that can be

defined using a branch of the log function. If z = 0 and a is any complex number,

then we set

(1.4.7)

za

=

ea log z

.

Here, we are thinking of a as being fixed and z is the independent variable of the

function. If we want the exponent to be the variable, we would write

(1.4.8)

az

=

ez log a

.