## The power series for sin and cos

There are many ways to *derive* the power series of and using the machinery of Taylor series etc., but below is another elementary way of demonstrating that the well-known power series expansions are the right ones. The argument below is from Tristan Needham’s *Visual Complex Analysis*, which I’m reproducing without looking at the book just to convince myself that I’ve internalized it correctly.

So: let

We will take the following two for granted (both can be proved with some effort):

- Both power series are convergent.
- The power series can be differentiated term-wise.

As suggested by (2) above, the first thing we observe is that and .

So firstly:

which means that is a constant and does not vary with . Putting shows that and , so for all .

Secondly, define the angle as a function of , by . (To be precise, this defines up to a multiple of , i.e. modulo .)

Differentiating the left-hand side of this definition gives

(where means )

while differentiating the right-hand side gives

The necessary equality of the two tells us that , which along with the initial condition that says , gives (or to be precise, ).

In other words, we have shown that the power series and satisfy and therefore and for some . The observation that (or our earlier observation that for all ) gives , thereby showing that and .

So much for and . Just as an aside, observe that if we take to be a symbol satisfying , then

the right hand side of which looks very much like the result of “substituting” in the known (real) power series

(which itself can be proved using the term-wise differentiation above and the defining property , say).

So this is one heuristic justification for us to *define* .

Or, if we *define* as the result of substituting in the real power series for , this *proves* that .

[…] had started writing a series of posts leading up to an understanding of the exponential function (here, here, here), but it seems to have got abandoned. Consider this one a contribution to that […]

The idea of logarithms, and the first appearance of e | The Lumber RoomWed, 2013-11-27 at 10:52:56