- 16

- 0

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

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

Last edited: