1940LaSalle
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- TL;DR Summary
- Reconciling an apparent (?) paradox in an exponential
We all know from Euler's identity that exp(iπ)=-1. And from the laws of exponents, exp(2iπ)=(exp(iπ))²=1.
Further, for any real number a≠0, a⁰=1.
Then, since two things equal to the same third thing are necessarily equal, exp(2iπ)=(exp(iπ))²=a⁰=1.
Here's where I'm wondering if I've stripped one or more intellectual gears. If we now take natural logarithms, it would seem we get
ln(exp(2iπ)) = 2iπ = 0*ln (a) = 0
That would suggest at first glance that 2iπ = 0, which made me do a double take, to say the least. I realize this may well be fallacious, and I may be guilty of missing something obvious and fundamental. Help me out here (gently, if you would, please): what did I miss?
Further, for any real number a≠0, a⁰=1.
Then, since two things equal to the same third thing are necessarily equal, exp(2iπ)=(exp(iπ))²=a⁰=1.
Here's where I'm wondering if I've stripped one or more intellectual gears. If we now take natural logarithms, it would seem we get
ln(exp(2iπ)) = 2iπ = 0*ln (a) = 0
That would suggest at first glance that 2iπ = 0, which made me do a double take, to say the least. I realize this may well be fallacious, and I may be guilty of missing something obvious and fundamental. Help me out here (gently, if you would, please): what did I miss?