There is no relation between e and π

[the_pulp]

Well, I obviously know the Euler's relation:

e^πi+1=0

But what I finally understood is that this is not a relation between those numbers. It is just the result of the function exp(z) when you extend the exponential function to the complex numbers and you supose analyticity (I don´t know if that is well said).

In other words, there is no simple calculation that I can make to the number "e" (I mean, something to put in the calculator) that will give as a result the number π.

The question would be: Is this last statement true?

I hope it isn´t because it was more romantic when I thought that e and pi where linked like the flowers and the bees or like Jack and Rose!

Thanks in advance for you usual help!

Ps: I also thought about the relation:

∫_-∞^+∞e^-x2/2dx=√π

But that comes from squaring the integral, transforming it in a doble integral and noting the rotational symmetry of the integrand. Everything that has rotational symmetry will give something times π, that's not properly related to the number e. It felt like cheating. I know that this is not precisely stated but I hope that you'll be able to capture the spirit of my question and to help me find an answer about whether we can say that e and π are related or not.

 

[mfb]

There is no formal definition of a "relation" between numbers. You can get pi from e by adding pi-e to it. Is that a relation?
The two numbers frequently appear in the same formulas.

 

[the_pulp]

In that case it feels like the admiration around eulers relation is exaggerated. I mean, if it would be written: exp(πi)+1=0, mathematically it will also be as important but there would not be as many t-shirts or tatoos with it. But it would be a more sincere description of the relation (it does not look as a relation about the numbers, but as a characteristic related to the exponential function).

Ps: I know that perhaps these things are ill posed and perhaps this question could be in the limit of what is admissible in this forum. I just want you to know that I have thought long enough about these ideas before I encouraged to share this grey question with you. I googled about it a lot and I did not find any source with this rational / way of approaching to this relation. I'm sorry in advanced in case that it is effectively not admissible.

 

[romsofia]

$e^{ix}=i\sin x+\cos x$ is where the relation comes from. Solve right side for pi. Are you asking why people think the equation is beautiful? That's their personal opinion.

There are a lot of things in math that make people go "Wow!", but doesn't seem a big deal to me (This is one). But other things make me go "Wow!" but other people think it's not a big deal. One of those things it that you can describe traffic flow as a partial differential equation! When I first learned this I thought it was very cool, so much so that I told everyone I knew if I had the chance. No one else thought it was a big deal..

 

[Charles Link]

There is an integral that comes up in studying the Planck blackbody function: $$ \int\limits_{0}^{+\infty} \frac{x^3}{e^x-1} \, dx=\frac{\pi^4}{15} $$ This is perhaps much better than the Gaussian integral relation, because it doesn't generate the $2 \pi$ artificially.

 

[the_pulp]

eix=isinx+cosx" role="presentation" style="display: inline-block; line-height: 0; font-size: 18.08px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; font-family: "PT Sans", san-serif; position: relative;">eix=isinx+cosxeix=isinx+cosx is where the relation comes from.

Yes I know where it comes from (I mean, the different "calculus like" approximations to this relation -whether we use taylor series or desirable differentiation properties or analyticity restrictions). Thanks for your answer and for sharing your thoughts.

 

[the_pulp]

There is an integral that comes up in studying the Planck blackbody function: $$ \int\limits_{0}^{+\infty} \frac{x^3}{e^x-1} \, dx=\frac{\pi^4}{15} $$ This is perhaps much better than the Gaussian integral relation, because it doesn't generate the $2 \pi$ artificially.

That looks promising Charles, I will give it a try!

Thanks

 

[the_pulp]

Hi, I've tried something different that could give a sincere picture of the relation. For big "n":

(1+1/n)ⁿ=e
(1+1/n⋅i)^n⋅π=-1

Those are two algorithms that can be applied with 18th century, pre calculus, mathematics. So, combining the two, we get, for big "n":

((e^1/n-1)⋅i+1)^n⋅π=-1

And there we have an actual "relation" between the numbers that can be calculated with sums and multiplications (the most difficult part is the roots that can be calculated iterating).

It also can be stated in not many words. Something like "the amount of steps needed to reach '-1' going through a circle (each step is perpendicular to my position) is pi times the amount of steps needed to reach 'e' through a straight line (each step is parallel to my position)". Well, that was a statement about geometry, if we replace i with its mattrix representation:

[0 1
-1 0]

The relation between e and pi is just the same and we can avoid using the complex numbers. The complex numbers can enter in the equation in that simple sentence when we note that:

1+1/n⋅i

could be seen as a factor that, when multiplies something, it applies a perpendicular transport of a small step in complex representation.

I'm sorry I posted a question that I finally answered myself. I thought a lot (years perhaps) about it, it's a coincidence that I found some kind of an answer a day after my question was posted. Anyway, thanks as always for your help and interest and I hope at least it helped someone to find the same light as myself about this subject. Any other thought or insight is obviously very welcomed.