What is the significance of the number e and Euler's formula?

In summary, the number e is the limit of the expression (1+\frac{1}{n})^{n} as n grows beyond bound, and it was originally shown to look like that by Euler. Proofs of this are somewhat tricky, but it can be done. The complex exponential was originally shown to look like that by Euler, who used a power series representation of the exponential.
  • #36
have you studied linear algebra? an "eigenvector" for a linear operator T is a vector v such that Tv is a scalar multiple of v. These vectors provide the most natural coordinate system appropriate to the operator T. If one wants to solve an equation like TX = Y, for X, it is easy to do if Y is expanded in terms of eigenvectors of T.

The functions e^ax provide the eigenvectors for the linear operator D (differentiation). Using them, one gets the most natural expansion of a smooth function, its Fourier series. this makes it easy to solve differential equations like Df = g, if one can expand g in a Fourier series.
 
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  • #37
I will point out, again, that there is nothing all that magical about e. Any exponential function akx is an eigenfunction of the derivative operator.
 
  • #38
well there is something special about the eigenvalue 1. or is your point that we should say "fixed points" of the operator D, to characterize ce^x?

i.e. e^x is the unique solution of the primordial ode: Dy = y, y(0) = 1.
 
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