5.3 • COMPLEX EXPONENTS 175If we restrict zc to the principal branch, zC = exp [cLog (z)], then Equation
(5-28) can be written in the familiar form that you learned in calculus. That is,
for z # 0 and z not a negative real number,
-zc d = c -zc = czc -1.
dz z
We can use Identity (5-21) to define the exponential function with base b,
where b # 0 is a complex number:b' = exp [zlog (b)].If we specify a branch of the logarithm, then b' will be single-valued and we
can use the rules of differentiation to show that the resulting branch of b' is an
analytic function. The derivative of b' is then given by the familiar rule
:z b" = b' log°' (b), (5-29)
where log°'(z) is any branch of the logarithm whose branch cut does not include
the point b.-------.... EXERCISES FOR SECTION 5.3
- Find the principal value of
(a) 4i.
(b) (1 + i)"i.
I
(c) (-1);;.
(d) (1 + iv'3) ~.- Find all values of
(a) ii.
(b) (- 1)"'2.
(c) (i)~.
(d) (1 + i)^2 - i.(e) (-1)~.(f) (i)~.- Show that if z =fa 0, then z^0 has a unique value.
- For z = re'^8 =fa 0, show that the principal branch of
(a) z• is given by the equation
z^1 = e-^0 [cos(lnr) + isin (lnr)),
where r > 0 and -1T < () $ 1T.