There is another way that I have found to be convincing to students but not to all profs.. Here it is:
Set f[x] = e^(a x) (cos[b x] + i Sin[b x])
Calculate using standard formulas
f'[x] = (a + i b) f[x]
So f'[x] = k f[x] just like the real exponential.
Another place to see Euler's formula come to life in a very fulfilling way is in the Feynman Lectures .(Addison-Wesley)
At 7:53 PM -0500 3/1/01, Bill wrote: > > To derive Euler's formula, > > > > e^(ix) = cos x + i sin x, > > > > I used: "If f '(x) = k f(x), then f(x) = e^(kx)." Is this right? If so, > > how may I prove it? The converse is trivial, but I don't find the above > > statement equally trivial. > > > > > >I'm sorry, but I don't under how your derivation obtains Euler's formula >as a result. The proof of your statement procedes along standard >lines for solving seperable first-order differential equations. More >trivially, >split the equation with one side containing the dependent variable and >the first derivative of said variable and the other side containing >everything >else. Take the definite integral of both sides and play around with the >appropiate constants and you're done. > >The traditional proof involves taking the Taylor series of the sin and cos >functions, >comparing these with the Taylor series of the exponential function, >extending the >domain of the Taylor series to complex numbers, and, depending on the >context >in which Euler's formula is taught, either asserting the existence of the >equality by >doing some handwaving or using the techniques of analysis to prove >equivalence. > >I believe that this proof is the most satisfactory and elegant; I'm sure >that one could >find a convulted geometric proof of the identity or a more sophistocated >one, but >the standard proof so integrally ties in with the idea that generally >speaking, >analytic functions ARE their Taylor series expansions. > >In the end, it's an incredible formula regardless of how one proves it. > >-Bill > > >__________________________________________ >NetZero - Defenders of the Free World >Get your FREE Internet Access and Email at >http://www.netzero.net/download/index.html
------------------------------------------------------------------ Jerry Uhl firstname.lastname@example.org Professor of Mathematics, University of Illinois at Urbana-Champaign Member, Mathematical Sciences Education Board of National Research Council Calculus&Mathematica, Vector Calculus&Mathematica, DiffEq&Mathematica, Matrices,Geometry&Mathematica, NetMath