**
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 2p*i.*
6) =1 if and only if there is a *k*
in
Z
such that *z=*2pk*i.*
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=2k**pi*
or Z.
By the periodicity of
, it is enough to consider
Suppose, to get a contradiction, that
Ï
Z.
Then Put
a+*i*b so that a= and b= since
. Nowexpanding 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
*w*C
-{0}. Put q=. Then
By 6) there exists yR
such that q=. So *w=* with *
*>0. So there exists *x*R
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 **Princ****ipal
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. |