A Conversation for The Euler Equation

ln -x

Post 1

dysprosia

Ln of negative numbers is multiple valued:
e^(i*pi)=-1
But
e^(i*3pi)=-1
And
e^(i*(2k+1)*pi)=-1.
Therefore
ln -1=i*pi,i*3*pi,i*5*pi...etc


ln -x

Post 2

bobdylan_mcfc

Yes, but in fact this is true of any number. Since e^(2.pi.i) = 1, we can add 2.pi.i to any log. In fact you can't define the logarithm as a continuous function from the complex plane without zero* to the complex plane. Somewhere you must have a discontinuity where the value changes by 2.pi.i. This "problem" is a motivating example for the beautiful theory of Riemann surfaces.

*There is no number y such that e^y=0, and no sensible way to define log(zero)


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ln -x

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