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Subject: "Fagnano's formula for ð"     Previous Topic | Next Topic
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melefthe
Member since Feb-8-07
 Feb-23-07, 06:36 AM (EST)
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"Fagnano's formula for ð"
 
   Has anyone seen the proof of Fagnano's formula for pi? It's so bizarre yet so beautiful. Ð=2ilog<(1-i)/(1+i)>. How did he think of it. He couldn't have known of Taylor series at the time. He was living in Italy while Taylor in England and there weren't any magazines at the time.


Marios Eleftheriadis, Athens, Greece.


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alexb
Charter Member
1980 posts		 Feb-25-07, 03:34 PM (EST)
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1. "RE: Fagnano's formula for ð"
In response to message #0
 
   In all honesty, I do not know. But mind you it is quite possible that the atribution has been made posthumously and mistakenly. On the other hand, Fagnano is known to have obtained some results on division of the lemniscate into equal parts, which were extended by Gauss via some sort of elliptic integrals which might have led to the formula.


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Sumudu
guest
 Mar-26-07, 05:40 AM (EST)
 
2. "RE: Fagnano's formula for ð"
In response to message #0
 
   Are Taylor series needed to prove that???

All you need is log(r*eit) = log(r) + it

(1-i)/(1+i) is just a fancy way of writing -i = e-i(pi/2)

From which the formula follows directly.

I'm not sure that it is any more special than Euler's formula for eit


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