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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
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