The Exponential, Logarithm, Trigonometric and Hyperbolic  Functions

Sessions

 

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 zC 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 0[,] 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)       () If for some integer k,  then

()  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

Z.  Then   Put a+ib 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 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.  Eulers 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.