Is Euler's formula false?
I was reading about Euler's formula, as I'm being introduced to Laplace transforms, for engineering, and it came up in class. I played around with it a bit and arrived at an obvious contradiction; here it goes:
e^ix = cos(x) + i sin(x)
let x = pi
=> e^(i*pi) = -1 (Since cos(pi) = -1 and sin(pi) = 0)
=> e^(2*i*pi) = 1 (Squaring both sides)
=> 2*i*pi = 0 (Natural log both sides)
=> -4*pi^2 = 0 (Squaring both sides)
Obviously a contradiction, which most likely means I made a mistake somewhere, but can’t figure out what it is!
Please let me know what I did wrong!
Comments
The natural log function is not a single-valued function when extended to the complex plane (including negative numbers). Ln(z) = ln(|z|) + i[arg(z) +], k = 0, ±1, ±2, ...
The only way to make this single-valued is to use a Riemann surface for the domain.
Otherwise, you need to be wary that you may be using different branches on the two sides of your equation, which is what caused your problem.
Your last line is redundant anyway.
The line 2*i*pi = 0 is already wrong since neither i nor pi is zero.
It is taking the natural log that causes the contradiction.
Strictly speaking, you cannot take a log of a complex number.
log z, where z is complex, is a many valued function and I believe (from distant memory) that it is replaced by a function called Log z (the capital 'L' is important).
Log z is quite different from your supposed log z; Log z is a many valued function, but I hesitate to go any further as it has long receded from my memory.
Consult a textbook on complex variables. Maybe Wikipedia can help.