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Solution of e^x = x
Okay, before you scream x = ∞, I'm finding the complex solution to the problem.
I'll show you my working so far, maybe you'll see something I missed.
First let x = a+bi
e^(a+bi) = a+bi
e^a * e^bi = a+bi
Applying Euler's identity
e^a*cos(b) + ie^a*sin(b) = a+bi
e^a*cos(b) = a
e^a*sin(b) = b
Simple rearranging;
[1] cos(b) = a/e^a
[2] sin(b) = b/e^a
[3] tan(b) = b/a
Using the identity;
cos^2(b) + sin^2(a) = 1
It follows that..
(a/e^a)^2 + (b/e^a)^2 = 1
[4] a^2 + b^2 = e^2a
Okay so I have these 4 equations and I still can't find solutions to any of them,
I only need to find a or b and the solution to e^x = x will follow.
Could you please help?
+I'm only a college student and haven't done much uni level maths, so go easy on me if I've missed something blindingly obvious.
Also, would analysis of the series of e^x help? (Just sprung into my mind as I was about to submit thread)
EDIT: A solution can be found using Lamberts W-function, x =~ 0.318 + 1.337i, you can delete this thread if you want
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