Hello. You all will soon know of me.

Feb 2020
62
1
St Louis
In that case, how is that equation related to other uses of $\pi$ without using any geometrical concept?
That is a great question. I have no idea as I have never gave much thought to the subject.

I would think that a special use of the Gamma function could be used somehow.
a convergent improper integral used in instances where both endpoints approach their limits. That would define an analytic continuation of the integral function to a meromorphic function that is holomorphic in the whole complex plane except for non-positive integers, where the function has simple poles.
hmmm.. that is interesting.

As to "without using any geometrical concept', it would seem to contradict analytic geometry as no axiomatic theory may escape using undefined elements. In Set Theory that underlies much of mathematics and, in particular, analytic geometry, the most fundamental notion of set remains undefined.
 
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skipjack

Forum Staff
Dec 2006
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The complex plane is a geometrical concept, so your suggestion would directly relate the equation to geometry.
 
Feb 2020
62
1
St Louis
The complex plane is a geometrical concept, so your suggestion would directly relate the equation to geometry.
I said I had no idea.
My statement was that it could be some special (i.e. I don't know what) use of it.


And I see what you did.....
your question was a corollary, in regards to the equation and uses of π that do not involve geometrical concepts.
this proposition which is incidentally proved, while proving another proposition, all the while it was presented and used more casually to refer to something which naturally or incidentally accompanied something else.

The equation (expression) is related to geometry as much than saying that the 2 was a radius and the 4 was the sides of a polygon. The numbers can be derived in many ways; it just so happens that .5! can be obtained by analytic continuation.


You are clever.
 

skipjack

Forum Staff
Dec 2006
21,481
2,470
Why did you say originally that the equation "has nothing to do with geometry"?
 
Feb 2020
62
1
St Louis
And this is another reason I hate pi. The crown jewel of mathematics. If it is spoken about in a bad light, you are labeled as a blasphemer and its worshipers will crucify you.

I have a question. is there anything that you don't like about π?
Please don't say that it can't be used more. It has already been used enough as a shortcut in math for the sole purpose of convenience.

"thunder" the math gods just read that last part.
 

topsquark

Math Team
May 2013
2,532
1,052
The Astral plane
And this is another reason I hate pi. The crown jewel of mathematics. If it is spoken about in a bad light, you are labeled as a blasphemer and its worshipers will crucify you.
What's there to hate? And who are those that will speak of \(\displaystyle \pi\) in a "bad light" and who worships it?

It's a number for Heaven's sake. Fine, if you want to like the Euler-Mascheroni constant instead then you go for it. And d*mn any of those nasty old \(\displaystyle \pi\) worshippers.

-Dan
 
Oct 2009
942
367
What's there to hate? And who are those that will speak of \(\displaystyle \pi\) in a "bad light" and who worships it?

It's a number for Heaven's sake. Fine, if you want to like the Euler-Mascheroni constant instead then you go for it. And d*mn any of those nasty old \(\displaystyle \pi\) worshippers.

-Dan
Personally, I'm a golden ratio guy. Screw every single other number!
 
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