Letting complex algebra make our calculus lives easier

If one wants to use complex arithmetic in elementary calculus, one should, of course, verify a few things first. One might talk about elementary complex arithmetic and about complex valued functions of a real variable at an elementary level; e. g. . Then one might discuss Euler’s formula: and show that the usual laws of differentiation hold; i. e. show that and one might show that for an integer. The latter involves some dreary trigonometry but, by doing this ONCE at the outset, one is spared of having to repeat it later.

This is what I mean: suppose we encounter where is an even integer. I use an even integer power because is more challenging to evaluate when is even.

Coming up with the general formula can be left as an exercise in using the binomial theorem. But I’ll demonstrate what is going on when, say, .

So it follows reasonably easily that, for even,

So integration should be a breeze. Lets see about things like, say,

Of course these are known formulas, but their derivation is relatively simple when one uses complex expressions.