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.
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 .
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 .