Complex VariablesDate: 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^{i*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/ |
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