Friday, 28 June 2013

complex analysis - Is the infinite product of 1times1times1timesdots=i?

So I woke up this morning and I was thinking about the infinite product 1×1×1×, and what it equals. I came to the conclusion that it equals i. Alternatively stated,



i(1)=i



Here's how I reached this:



i(1)=eln(i(1))=eiln(1)=eiiπ=eiπi1




Now, here's where I'm a little hesitant. I want to say that, from ζ(0)=12, we can conclude that



eiπi1=e12iπ=i.



I have been told before that the sum i1 is not actually 12, but I'm not really sure why. It would seem that if this is the case, then my product would in fact not be i. Though, I must say that i sort of makes sense, because multiplying complex numbers is essentially rotating them, and so rotating by 180 every time will get you 180+180+180+... is the same as 180(1+1+1+...) which is (if my premise is right) 180(12)=90. 90 degrees on the complex plane turns out to be i.



So my question is, is there a hole in my logic? I know what not accounting for ζ(0)=12, the sum 1+1+1+... is divergent, but taking that into account, can I say with confidence that 1×1×1×=i?

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