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
e(1/n) log z
if z = 0.
A special case of this is the square root function defined by
0 = 0 and
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
is just above the negative real axis, then log z is
nearly log r + iπ and
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
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
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
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
ez log a