Remembering that our aim is to show that E(x)=ex, we can see how to go a large step of the way using Corollary 7.8:
In other words, we need to know that E(x)
. In fact we shall do more than this. We shall show that E(x)
is differentiable, and, of course, differentiable functions are continuous.
Proof of Theorem 7.9
What does it mean to differentiate a complex function?
For the proof of this result, just check that the proof of
works perfectly well for complex differentiation.
In view of the ideas we discussed above
, Corollary 7.11 allows us to conclude:
Now we have a power series, which is perfectly well defined for all complex numbers, and we know that it agree with ex
for all values x
for which ex
has so far been defined. We use this to now extend our definition of exponentiation to the whole complex plane:
The payoff from this extension of our definition, together with all that we know about power series (specifically Theorem 7.9) will be:
The ability to obtain new insights into existing functions: we shall show that general power functions, logarithmic function, and exponential functions are all differentiable.
Constructions for a whole new class of important functions: the trigonometric functions.
We shall do all of this in the next two classes. We conclude this class with a summary of our two most important results for
, rewritten in our new notation:
: Corollary 7.8