Monday, 16 December 2013

complex numbers - Why does this work, and why is it wrong?




I have devised a "proof" that i=0. Obviously it can't be true, but I can't see why it is wrong. I've been thinking about this for a while, and my friend and I are very confused. What is wrong with the "proof" below?



eiπ=1
e2iπ=1
2iπ=ln(1)
i=02π
i=0


Answer



Now that you have proved in the second line e2iπ=1=e0, so the complex exponential function is not injective. Thus you can not define a complex log function as its inverse function.


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