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Exponential Functions

You have seen the procedure by which a new function, the exponential function $q(t) = q_0 \exp(kt)$, was constructed from a power series just to provide a solution to the differential equation $\dot{q} = k \, q$. (There are, of course, other ways of "inventing" this delightful function, but I like my story.) You may suspect that this sort of procedure will take place again and again, as we seek compact notation for the functions that "solve" other important differential equations. Indeed it does! We have Legendre polynomials, various Bessel functions, spherical harmonics and many other "named functions" for just this purpose. But - pleasant surprise! - we can get by with just the ones we have so far for almost all of Newtonian Mechanics, provided we allow just one more little "extension" of the exponential function . . . .