#37
Oct104, 12:37 PM

P: n/a

Perhaps we could compare with [tex]\pi^{ie}[/tex]...
Is i ever used in the power of other numbers than e? Any use to to doing this? 



#38
Oct104, 10:09 PM

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P: 9,421

Look. pi^(ie) is just e^(ie ln(pi)), so NO number is ever used as an exponent for bases other than e. That is to say, e is the universal base for all exponents.
i.e. [(haha) perhaps I should say in russian: "tau yest" instead of "id est"]\\anyway: for any a, we have a^b = e^(bln(a)). Let's start with e^i. Note that since e^(it) = cos(t) + isin(t), that then e^i = cos(1)+i sin(1), not at all an interesting or elementary number at least not to me. Thus in the same spirit, pi^(ie) = e^(ie ln(pi)) = cos(e ln(pi)) + i sin(e ln(pi)), apparently a similarly ugly number. If anyone can give a nice interpretation of this number, my hat is off, and I would enjoy seeing it. 



#39
Oct204, 05:12 AM

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PF Gold
P: 12,016





#40
Jul1707, 07:24 PM

P: 3

Maybe this is getting off on a tangent (horrible pun intended ), but I'm exploring how similar the graphs of tan(x) and [tex]e^{x}\pi^{x}[/tex] are... this may weave back to the earlier observations about path integrals from mathwonk:




#41
Jul1707, 10:51 PM

P: 1,636

I actually attempted to prove Euler's identity using a different way. This was discussed recently. http://www.physicsforums.com/showthr...r%27s+identity 



#42
Jul1807, 05:38 AM

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Thanks
PF Gold
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#43
Jul1807, 10:10 AM

P: 1,060

Hello all.
Just a bit of trivia but in someway I feel descriptive of the power, beauty and simplicity of the formula. I once saw it referred to as A Mathematical Poem. Matheinste. 



#44
Jul1807, 10:26 AM

P: 640

I alwyas thought that this equation was much more beautiful and mysterious:
[tex]i^2+j^2+k^2=i j k = 1[/tex] 



#45
Jul1807, 10:40 AM

P: 2,265

[tex] e^{i \pi} + 1 = 0 [/tex] ? 



#46
Jul1807, 12:16 PM

P: 1,060

Your equation is certainly mysterious. When I find out what it means it might also be beautiful. Mateinste 



#47
Jul1807, 12:58 PM

P: 372

It's just the defining equation of quaternions.
The algebra of it is a lot more interesting... 



#48
Jul1807, 01:32 PM

P: 3

This sort of rediscovery isn't too unusual... there's Wile's 1994 rediscovery of Fermat's last theorem from 1637only 357 years later, but he didn't use the internets [sic] . 



#49
Jul1807, 02:38 PM

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exponentiation is a homomorphism from a ddition to multiplication, and surjects onto the positive reals for sure, and neither the value nor the derivative is ever zero. hence i claim it is a topological covering map from the complexes to the non zero complkexes, hence must wrap around the origin, and be periodic for some value. i.e. it must take the values 1 and 1 infinitely often.
it remains only to find the value x such that e^x = 1. that follows from trig (eulers style via power series expansions of sin, cos). i think this is a new answer to an old question. 



#50
Jul1807, 02:57 PM

P: 3





#51
Jul1907, 12:57 PM

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im just saying there is essentially no other way to map the plane onto the punctured plane with derivative non zero, except to go around and around, so it has to hit the same point more than once, i.e. it hits 1 in finitely often, and also 1.
i see it like a spiral staircase. 



#52
Nov1009, 06:23 AM

P: 1

Beautiful and useful too, phasors for AC circuit analysis are based on Euler's identity.



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