eix = cos(x) + isin(x)

Here is a proof of a trigonometric identity commonly used in representing complex numbers; it provides a simple relationship between polar and exponential form. So here goes...

We know the following Taylor Series:

              x2     x3     x4   
ex = 1 + x + ---- + ---- + ---- + ... ...
              2!     3!     4!

              x3     x5
sin(x) = x - ---- + ---- - ... ...
              3!     5!

              x4     x6
cos(x) = 1 - ---- + ---- - ... ...
              4!     6!

And we know that the imaginary number i represents the square root of -1, so that i2 = -1.

So if we raise e to the power ix:

                (ix)2     (ix)3     (ix)4 
eix = 1 + ix + ------- + ------- + ------- + ... ...
                 2!        3!        4!

Seperating out the powers gives:

                i2x2     i3x3      i4x4 
eix = 1 + ix + ------ + ------ + ------ + ... ...
                 2!       3!       4!

Evaluating the powers of i simplifies things a bit:

                x2     ix3     x4   
eix = 1 + ix - ---- - ----- + ---- + ... ...
                2!      3!     4!

And finally, seperate the odd- and even- powered terms, and factorize out the i, to give:

      /      x2     x4            \        /      x3     x5         \
eix = | 1 - ---- + ---- - ... ... |  +  i | x - ---- + ---- ... ... |
      \      2!     4!           /        \      3!     5!         /

Which, given the Taylor Identities given above, is equal to:

eix = cos(x) + isin(x)

This has been an essay in the craft of the <PRE> tag.

Not so much a proof as a Good Idea for a definition.

Say ariels has been a good noder and has attended all his first year calculus classes. If it's a good course, he's seen exp(x)=ex, sin(x) and cos(x) all defined for real numbers x. And he's seen the power series expansions for these functions.

None of which is relevant to the problem of defining exp(x) for complex x. He could equally well end up in some 2nd year course in which the TA categorically stated "exp(x) is not defined for nonreal x", or (worse) "exp(i)=17 [, and exp(x) for complex x doesn't follow any of the rules for exponentiation]".

The point is that if we define eix=cos(x) + i sin(x) then all the power series stuff comes out nicely. Also all the laws of exponentiation are kept, which is nice (because it means we don't have to discover and prove new ones, and because it saves us having to special-case them for real numbers x). So it makes sense to define eix in this manner -- it's "natural"!

Of course the truth is a bit more sinister than the above light and airy prose. We can show exp(x) is an analytic function on the real numbers (its power series converges to its value). And there's a theorem of complex analysis which states that an analytic function (known in complex circles as a holomorphic function!) is defined by its values on any set with a condensation point; R is such a set, so we know (a priori!) that we'll be able to extend exp(x) in a unique fashion in the vicinity of the real number line.

We also know that the power series expansion of exp(x) won't change (wherever it converges). So we can indeed perform the above manipulations and prove (without the quotes!) that exp(ix)=cos(x)+isin(x) for small enough x. But the radius of convergence for the power series of exp(x) is infinite, so the above identity follows for all x.

As a side effect, all the exponentiation laws (e.g. (eix)n=einx) follow immediately. It all works because it has to; we had a good deal less choice in defining exp(x) well for complex x.

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