the Exponential function

The McLaurin's series for the exponential function exp(x) is

1 + x + x^2 / 2 + x^3 / 6 + ... + x^i / i! + ....

It has an infinite radius of convergence; that is, it converges for all bounded values of x.

The ratio of (n+1)-th term to the n-th term is x / (n+1) which (as it should), for any fixed x, approaches zero as n increases without bound.

It can be shown that its addition theorem is exp(x + y) = exp(x) * exp(y).

It also can be shown that the exponential function satisfieis the linear ordinary differential equation of first order, with constant coefficients:

d exp(x) / dx - exp(x) = 0.

The exponential function is singly periodic, with a modulus of periodicity of 2 pi i.

Finally, it can be shown that any one of these three properties characterizes the exponential function; that is, any one could be used as the definition, then both of the other two derived from it as theorems.

Trigonometry is the study of the various commonly-occuring sums and products of the exponential function. The hyperbolic functions are the real, while the circular functions are the purely imaginary counterparts.

While the Mclaurin's series converges for any finite complex x, the addition theorem should be employed, as an aid, for ease and speed of computation. The exponential of a complex x = a + i b (where a and b are real-complex numbers) is

exp(x) = exp(a + i b) = exp(a) (cos(b) + i sin(b))

Also, integral powers of epsilon = exp(1) should be factored out so that the argument x in the McLaurin's sesries is abs(x) <= 1 / 2. The integral powers of epsilon may be computed by judicious use of successive squaring.

We define a ^ b as exp(b ln(a)), where we have named the inverse of the exponential function as the logarithmic function ln(x). For b a natural number, it may be shown by Mathematical induction, from the addition theorem, that a ^ b is the product of a multiplied by itself b times. Thus, the present definition of a ^ b extends the successive-multiplication definition to any complex number b.

It may be shown, by l'Hospital's rule, that the limit, as n approaches zero, of (1 + n x) ^ (1 / n) is exp(x).