FUNCTIONS
OF
A
COMPLEX VARlABLE
139
realy
I
imagy
,
0
0
,
Figure
6.4
The
Plots
of
the
Real
and
Imaginary
Parts
of
&)
as
Functions
of
n
and
y
6.2.2
Extending
the
Domain
of
Elementary
Functions
The elementary functions of real variables can be extended to deal with complex
numbers. One example
of
this,
the complex exponential function,
was
already intro-
duced in the previous section. We defined that function by extending its Taylor series
expansion to cover complex quantities. Series expansions
of
complex functions are
covered in much greater detail later in
this
chapter.
Trigonometric Functions
We can
perform
a similar extension with the sine func-
tion. The Taylor series of
sin(x)
expanded around zero
is
(6.22)
where
x
is a real variable. The
complex
sine function can be defined by extending
this series to complex values,
where
g
is
now a complex quantity.
This
series converges for all
g.
Similarly, the
complex cosine function has the series
z2
z4
z6
cosz
=
1
-
=
+
L
-
L
+
...
-
2!
4!
6!
(6.24)
which
also
converges for all values of
g.
to all complex quantities:
With these extended definitions, Euler's equation (Equation
6.8)
easily generalizes
e'g
=
cosz
-
+
ising. (6.25)
This
allows the derivation
of
the commonly used expressions for the
sine
and cosine
of complex variables:
,iz
-
e-iz
2i
ei<
+
e-ig
2,
sing
=
cosg
=
(6.26)
(6.27)
