Some elementary functions

COMPLEX VARIABLES
which is an alternating series whose terms decrease in magnitude and which
therefore converges.
The ratio test discussed in subsection 4.3.2 may also be employed to investigate the absolute convergence of a complex power series. A series is absolutely
convergent if
|an+1 ||z|n+1
|an+1 ||z|
<1
= lim
n→∞
n→∞
|an ||z|n
|an |
lim
(24.14)
and hence the radius of convergence R of the series is given by
1
|an+1 |
= lim
.
n→∞ |an |
R
For instance, in case (i) of the previous example, we have
1
n!
1
= lim
= lim
= 0.
n→∞ (n + 1)!
n→∞ n + 1
R
Thus the series is absolutely convergent for all (finite) z, confirming the previous
result.
Before turning to particular power series, we conclude this section by stating
n
the important result§ that the power series ∞
0 an z has a sum that is an analytic
function of z inside its circle of convergence.
As a corollary to the above theorem, it may further be shown that if f(z) =
an z n then, inside the circle of convergence of the series,
f (z) =
∞
nan z n−1 .
n=0
Repeated application of this result demonstrates that any power series can be
differentiated any number of times inside its circle of convergence.
24.4 Some elementary functions
In the example at the end of the previous section it was shown that the function
exp z defined by
exp z =
∞
zn
n=0
n!
(24.15)
is convergent for all z of finite modulus and is thus, by the discussion of
the previous section, an analytic function over the whole z-plane.¶ Like its
§
For a proof see, for example, K. F. Riley, Mathematical Methods for the Physical Sciences (Cambridge: Cambridge University Press, 1974), p. 446.
¶
Functions that are analytic in the whole z-plane are usually called integral or entire functions.
832
24.4 SOME ELEMENTARY FUNCTIONS
real-variable counterpart it is called the exponential function; also like its real
counterpart it is equal to its own derivative.
The multiplication of two exponential functions results in a further exponential
function, in accordance with the corresponding result for real variables.
Show that exp z1 exp z2 = exp(z1 + z2 ).
From the series expansion (24.15) of exp z1 and a similar expansion for exp z2 , it is clear
that the coefficient of z1r z2s in the corresponding series expansion of exp z1 exp z2 is simply
1/(r!s!).
But, from (24.15) we also have
exp(z1 + z2 ) =
∞
(z1 + z2 )n
.
n!
n=0
In order to find the coefficient of z1r z2s in this expansion, we clearly have to consider the
term in which n = r + s, namely
r+s
(z1 + z2 )r+s
1
C0 z1r+s + · · · + r+s Cs z1r z2s + · · · + r+s Cr+s z2r+s .
=
(r + s)!
(r + s)!
The coefficient of z1r z2s in this is given by
r+s
Cs
(r + s)!
1
1
1
=
=
.
(r + s)!
s!r! (r + s)!
r!s!
Thus, since the corresponding coefficients on the two sides are equal, and all the series
involved are absolutely convergent for all z, we can conclude that exp z1 exp z2 = exp(z1 +
z2 ). As an extension of (24.15) we may also define the complex exponent of a real
number a > 0 by the equation
az = exp(z ln a),
(24.16)
where ln a is the natural logarithm of a. The particular case a = e and the fact
that ln e = 1 enable us to write exp z interchangeably with ez . If z is real then the
definition agrees with the familiar one.
The result for z = iy,
exp iy = cos y + i sin y,
(24.17)
has been met already in equation (3.23). Its immediate extension is
exp z = (exp x)(cos y + i sin y).
(24.18)
As z varies over the complex plane, the modulus of exp z takes all real positive
values, except that of 0. However, two values of z that differ by 2πki, for any
integer k, produce the same value of exp z, as given by (24.18), and so exp z is
periodic with period 2πi. If we denote exp z by t, then the strip −π < y ≤ π in
the z-plane corresponds to the whole of the t-plane, except for the point t = 0.
The sine, cosine, sinh and cosh functions of a complex variable are defined from
the exponential function exactly as are those for real variables. The functions
833
COMPLEX VARIABLES
derived from them (e.g. tan and tanh), the identities they satisfy and their
derivative properties are also just as for real variables. In view of this we will not
give them further attention here.
The inverse function of exp z is given by w, the solution of
exp w = z.
(24.19)
This inverse function was discussed in chapter 3, but we mention it again here
for completeness. By virtue of the discussion following (24.18), w is not uniquely
defined and is indeterminate to the extent of any integer multiple of 2πi. If we
express z as
z = r exp iθ,
where r is the (real) modulus of z and θ is its argument (−π < θ ≤ π), then
multiplying z by exp(2ikπ), where k is an integer, will result in the same complex
number z. Thus we may write
z = r exp[i(θ + 2kπ)],
where k is an integer. If we denote w in (24.19) by
w = Ln z = ln r + i(θ + 2kπ),
(24.20)
where ln r is the natural logarithm (to base e) of the real positive quantity r, then
Ln z is an infinitely multivalued function of z. Its principal value, denoted by ln z,
is obtained by taking k = 0 so that its argument lies in the range −π to π. Thus
with −π < θ ≤ π.
ln z = ln r + iθ,
(24.21)
Now that the logarithm of a complex variable has been defined, definition (24.16)
of a general power can be extended to cases other than those in which a is real
and positive. If t (= 0) and z are both complex, then the zth power of t is defined
by
tz = exp(z Ln t).
(24.22)
Since Ln t is multivalued, so is this definition. Its principal value is obtained by
giving Ln t its principal value, ln t.
If t (= 0) is complex but z is real and equal to 1/n, then (24.22) provides a
definition of the nth root of t. Because of the multivaluedness of Ln t, there will
be more than one nth root of any given t.
Show that there are exactly n distinct nth roots of t.
From (24.22) the nth roots of t are given by
t1/n = exp
834
1
Ln t .
n