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).

Copywrite © 1997 R. I. 'Scibor-Marchocki last modified on Thursday 07-th of August 1997.