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Definition.
The exponential function,
denoted by  , is defined by
=
Proposition.
1)
 =
2)
≠0
3)
is differentiable on
C
and ()′=
4) The
restriction  of
to
R
is a positive strictly increasing function such that
 and
5) There
exists a positive number
p
such that  =0. The function
is periodic with periodic 2pi.
6) =1 if and only if there is a k
in
Z
such that z=2pki.
7) Let
 C:
|z|=1}. The mapping
 from
R
to  is indeed onto.
8) For
every nonzero complex number w, there exists z C
such that w=.
Proof.
1)
 = where
 = = = 
Thus
=  =.
2)
By 1)  = . So
≠0.
3)
()′=()′= ==.
4)
 In
addition,
since  is strictly increasing on
R.
Moreover,   for
x>0
and
so .
Furthermore, 
5)
Let
 R.
For all  C,
 In particular,
 and
so  So Let
 and
 denote
 and
 respectively. Then
 and thus
 Moreover,
 Thus
 It follows that
 and
Now
 since each series is convergent.
That is,
 .
It follows
that   and
 .
In
particular, 
 is an alternating series whose general
term decreases in modulus to 0. So
 But
 So 0Î[, ] and therefore , by the Intermediate
Value Theorem, there exists at least one
 such that
 Let
 be the smallest
such that
Then
 and
 for all
 Since
 is increasing in
 . So for all
  In particular,
 Thus, since
 ,
 =1. That is,
 =1 since
Now
Finally,
is periodic with period
 since
and
 is the smallest positive number p
with the property
6)
(Ü)
If  for some integer k, then
(Þ)
Suppose  Then
 So
 Thus
=1 and so x=0. It follows that
z=iy. We need to show that iy=2kpi
or  Z.
By the periodicity of
 , it is enough to consider
 Suppose, to get a contradiction, that
 Ï
Z.
Then  Put
 a+ib so that a= and b= since
 . Now expanding 1= in terms of a and b leads to a
contradiction.
7) Let
 There are four cases to consider
depending on the four quadrants that z may lie in.
Case 1.
 Here
 that is,
 Using the
Intermediate Value
Theorem there is
 such that
 so
 Here
 This implies that
 and
Case 2.
 Here
 and
 . Then
 and
 By case 1 there is
 such that
 So
 with
Case 3.
 We have
 Thus
 and
 . By case 1 the result follows.
Case 4.
 We have
 Thus
 and
 By case 1 the result follows.
8)
Let
wC
-{0}. Put q= . Then
 By 6) there exists yR
such that q=. So w= with
>0. So there exists xR
such that  . Therefore,
Definitions.
We have just seen that the
mapping from
R
to
C:
|z|=1} is onto. Thus given
 , the set
 R: is nonempty. Each element of this
subset of the real line is called an argument of z
and is denoted by arg z.
The value
of arg z in [ is called the Principal
Determination of arg z
and is
denoted by Arg z.
The
logarithmic Function
We have
seen that the function
 is a strictly increasing function from
R
onto (0, . Thus it has an inverse, denoted by
  R.
Now let
C
- {0}. We have seen that there exists w
C
such that  Let
w=x+iy.
Then
 So
 . Thus
 Moreover,
 .
Definitions.
The logarithmic
function,
denoted by
 , is defined by
Now the function
 is called the Principal
determination or principal branch of logz.
Definitions. Euler’s equation
suggests the following definitions
 .
Since
 and
 are entire functions, so are
 and
 . Using the definition, we see that
most of the real-variable properties hold for complex
variables like . However, not all real-variable
properties hold as the next example shows:
Example.
Solve
Using the
definition we have
 =3. Let
 . Then
 or
 Then
 . Consequently,
 i.
Definitions.
As in real
analysis we define the following hyperbolic functions
of a complex variable by
 .
Using the
above definitions it is quite easy to verify that
We have
the following results:
Theorem.
1) If
  then z must be real.
2)
If  then z must be real.
Proof.
Let
 We show that
=
Now
Let
 in particular. Then
 But
 Since
 it follows that
 and consequently
The proof
of 2) is similar where we take
 in particular. |
