## A Conversation for The Euler Equation

### ln -x

dysprosia Started conversation Feb 9, 2002

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

bobdylan_mcfc Posted Jul 24, 2009

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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