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

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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
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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.

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/
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Associated Topics:
High School Imaginary/Complex Numbers

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