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In-Depth Information
has a radius of convergence R, 0 £ R £•and a disk of convergence. The proofs are
the same. An important fact is
E.2.5. Theorem.
If R is the radius of convergence of the series (E.3), then f(
z
) is an
analytic function for all
z
, |
z
| < R. The derivative of f(
z
) can be obtained by term-wise
differentiation and the resulting series has the same radius of convergence.
Proof.
See [Ahlf66].
E.2.6. Corollary.
The function f(
z
) in (E.3) is infinitely differentiable for all
z
, |
z
| <
R, where R is its radius of convergence.
We can now use series to define some standard functions.
The
exponential function
e
z
is defined by
Definition.
2
n
zz
z
z
e
=
1
+
+
+ ◊◊◊+
+ ◊◊◊
.
12
!
!
n
!
Definition.
The
sine function
sin
z
and
cosine function
cos
z
are defined by
iz
-
iz
iz
-
iz
ee
+
ee
-
sin
z
=
and
cos
z
=
.
2
2
i
It is easy to show that the complex sine and cosine functions have series repre-
sentations like their real cousins:
3
5
2
4
zz
zz
zz
sin
=
-
+
- ◊◊◊
and
cos
z
=
1
-
+
- ◊◊◊
35
!
!
24
!
!
Definition.
A function f(t) defined on
R
or
C
is said to be
periodic of period T
if
f(t + T) = f (t).
One can check that the sine and cosine functions are periodic of period 2p. Finally,
the definitions also give the famous
Euler formula
e
i
q
=
cos
q
+
i
sin .
q
Any root of the equation
z
n
Definition.
= 1 is called an
nth root of unity
.
It should be clear from the above that there are n nth roots of unity. In fact, they
are
2
n-
1
1
,
ww w
,
,...,
,
where