Euler's Formula

The beautiful and perhaps mysterious formula of Euler which is the subject of this section is


Several questions might immediately come to mind.

What does an exponential function have to do with trigonometric functions? At the pre-calculus level we are familiar with tex2html_wrap_inline16 as a function which increases rapidly as x grows, and with the oscillatory nature of the trigonometric functions.

How do we make sense of raising a real number to an imaginary power? Our rules of arithmetic have only told us how to extend addition and multiplication from the real numbers to the complex numbers.

We will eventually give a complete and airtight answer to these questions in the section on complex functions(link), but we can get acquainted with Euler's formula and strip away some of its mystery by extrapolating a few simple properties of the real function tex2html_wrap_inline16 .

First, if tex2html_wrap_inline22 then the equation which we obtain by replacing i with -i should also be true. After all -i is as good a square root of -1 as i.

So we should also have tex2html_wrap_inline34 . And if we multiply these together,


so that tex2html_wrap_inline38 So we know at least that


for some angle tex2html_wrap_inline28 . But does tex2html_wrap_inline30 ?

Recall that the exponential function tex2html_wrap_inline32 is well approximated by the linear function 1+x when x is very small. (Try it on a calculator.) We will now assume that the function tex2html_wrap_inline38 retains this property for complex z with small modulus.


so in particular


And now we use two more approximations which are very good for small values of tex2html_wrap_inline46 . (try it)




These approximations become increasingly good as tex2html_wrap_inline52 decreases and it is clear that Euler's formula holds when tex2html_wrap_inline54 .

Putting all this together we boldly claim:


Notice that by de Moivre's formula from the previous section, this means that


And since any angle can be written as an integer multiplied by something ``sufficiently small!'', our claim is bold indeed.

What we should notice at this stage, is that if we want to extend the function tex2html_wrap_inline32 to complex values of x in a way consistent with what we already know about the function, then it is very reasonable to expect oscillatory behavior from tex2html_wrap_inline64 and perhaps also reasonable to accept Euler's lovely formula.

We will soon analyze and understand the formula completely.

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Author: Michael O'Neill

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