Euler's Equation: First Step

Date: 05/18/99 at 09:46:18
From: Max Shaffer
Subject: Euler's equation (e^pi*i=-1)

I've read your section on Euler's equation.

  http://mathforum.org/dr.math/faq/faq.euler.equation.html   

I have a question on the very first step: any complex number, z, 
equals cos(x)+isin(x). Why is this? Is this a definitional thing, that 
any complex number can be expressed in this fashion?

Date: 05/18/99 at 09:57:30
From: Doctor Mitteldorf
Subject: Re: Euler's equation (e^pi*i=-1)

Dear Max,

Of course it's not true that any complex number can be expressed as 
cos(t)+i*sin(t). For example, the number 2 is a complex number that 
cannot, because there's no x for which cos(t) = 2. Any complex number 
with UNIT MODULUS can be expressed as cos(t)+i*sin(t), i.e., any 
complex number that obeys |z|^2 = 1.

So starting with any z at all, you can first write it as R times z', 
where z' has unit modulus, and R is the modulus |z|. Then it is true 
that z' can be expressed as cos(t)+i*sin(t). Just let x be the angle 
whose tangent is the quotient of the imaginary part of z to the real 
part of z.  In other words, take the imaginary part of z and the real 
part, and divide them. Think in terms of the complex plane where the y 
axis represents the imaginary part and the x axis the real part. The 
angle t is formed when you connect the point x+iy to the origin. The 
length of the connection is R=|z|. Basic trig definitions give the 
imaginary part as R sin(t) and the real part as R cos(t).

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/18/99 at 10:11:05
From: Doctor Rob
Subject: Re: Euler's equation (e^pi*i=-1)

Thanks for writing to Ask Dr. Math!

I suppose you are talking about the second (or calculus) proof of
Euler's Equation. No, it is not true that any complex number can be
written in the form cos(x) + i*sin(x). That is not what is stated
there. What is meant is this: let z be a shorthand name for the
expression cos(x) + i*sin(x).

If you look at the section entitled "An Application," you will see
how to express any complex number z as r*[cos(t) + i*sin(t)], where
t and r = |z| = sqrt(a^2+b^2) are real numbers.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/