Proof of e^(ix) = cos(x) + isin(x)

Date: 04/07/97 at 21:07:52
From: Walter Graf
Subject: Proof of e^(ix) = cos(x) + isin(x)

In the equation e^(iPi) - 1 = 0, the proof is to evaluate 

    e^(ix) = cos(x) + isin(x) for x = Pi.

I would like to see a rigorous proof of of the the above equation.

Thank you,
Walter Graf 

Date: 04/08/97 at 05:16:59
From: Doctor Mitteldorf
Subject: Re: Proof of e^(ix) = cos(x) + isin(x)

Dear Walter,

This is called the Euler equation, and it's not something you can 
prove rigorously.  It's a definition, and I'd like to convince you 
that it's the only sensible definition, of how to compute imaginary 
exponentials.

I can think of three approaches to verifying the Euler equation, 
but unfortunately one of them is all calculus, one uses calculus 
explicitly, and only the third is free of calculus.  I'm just guessing 
from your age that you may not have studied calculus yet.

You can verify that the Euler equation makes a sensible definition 
by expanding the two sides as Taylor series in x.  You can also 
differentiate both sides and see that the answer is self-consistent. 
Thirdly, you can use the formula for cos(2x) and sin(2x) to show that 
the right side has the property you expect from an exponential, so 
that e^i(2x) = (e^ix)^2.

So start with choice 3.  You have the formulas 

  cos(2x) = cos^2(x) - sin^2(x) and
  sin(2x) = 2 sin(x) cos(x)

You'd also want to demand that e^i(2x) = (e^ix)^2.  That means that
your new definition of e^ix is behaving like an exponential.  See if 
you can put these together to show that 

  e^i(2x) = cos (2x) + i sin(2x).

The Taylor expansion is something you can appreciate without calculus, 
although its roots are in calculus.  It's a series expression for a 
function. You may have run across the following infinite series 
representations of cos and sin and e^x.  In fact, this is the most 
straightforward way to compute the value of sin(x) or e^x for any 
given x.

  cos(x) = 1 - x^2/2!   + x^4/4!   - x^6/6! + ...
  sin(x) = x - x^3/3!  + x^5/5!  - x^7/7! + ...
  e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...

The ! in these equations means factorial.  In other words, 
4! = 4*3*2*1.

See if you can use these infinite series expressions to verify the 
Euler equation.

After you complete these two projects, I'm hoping you'll find the 
Euler equation very plausible.  If you still want more proof, write 
back again...

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/