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

[mathwonk]

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.

 

[HallsofIvy]

I will point out, again, that there is nothing all that magical about e. Any exponential function a^kx is an eigenfunction of the derivative operator.

 

[mathwonk]

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.