Complex Variables

Date: 03/25/2003 at 05:35:21
From: Kenny Wong
Subject: Complex root for equations of trigonometry

Is there any complex root for an equation like sin(x)=3/2?
What does a^i= ? where a is a real constant.

Date: 03/25/2003 at 11:02:30
From: Doctor Jerry
Subject: Re: Complex root for equations of trigonometry

Hi Kenny,

If I enter 1.5 on my calculator (HP48GX) and press the ASIN button 
(arcsin), I find

1.57079632679 - i*0.962423650119.  

I'll try to explain this.

We must start with something - maybe  e^{i*t} = cos(t) + i*sin(t), 
where t is real.

From this we see that

e^{i*t} = cos(t) + i*sin(t)

e^{-i*t} = cos(-t) + i*sin(-t) = cos(t) - i*sin(t)

So,

e^{i*t} - e^{-i*t} = 2i*sin(t)

So,

sin(t) = [e^{i*t} - e^{-i*t}]/(2i)

Let's set

3/2=1.5=sin(t) = [e^{i*t} - e^{-i*t}]/(2i)

Multiply both sides by 2i*e^{i*t}:

3ie^{i*t} = e^{2i*t} - 1

Now let w=e^{2i*t}.  So,

3i*w = w^2 - 1 or

w^2 -(3i)w-1=0.

Solving this by the quadratic formula (and just looking at one root)

w = (3/2)i+i*sqrt(5)/2.

So,

e^{i*t} = (3/2+sqrt(5)/2)i = (3/2+sqrt(5)/2)*e^{pi*i/2}

Take natural logs of both sides:

i*t = ln(3/2+sqrt(5)/2)+pi*i/2

So,

t=-i*ln(3/2+sqrt(5)/2) + pi/2 = 1.57079632679 - i*0.962423650119.

Most of the unfamiliar ideas above are in a course called "complex 
variables," sometimes taught as a junior or senior undergraduate 
course in the U.S. and usually as a first year graduate course.

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