Isolating Odd Terms
Any function, f(x), that may be expressed by Taylor series expansion is represented by a series containing integer powers of x. Such powers of x with even exponents are called even terms, and those with odd exponents are called odd terms. Since all exponents are either even or odd, all terms must be one of the two. Grouping the even terms as feven and the odd as fodd gives,
(1)
An operator to isolate odd terms must have the effect of illiminating even terms while preserving the odd ones. To find such an operator, consider the property of even terms:
(2)
or
(3)
We might remove these terms with this property by subtracting f(-x),
(4)
obtaining from eqns. 1 and 4:
(5)
By eqn. 3, the even terms cancel, leaving
(6)
Now applying the property of odd terms,
(7)
yields,
(8)
or
(9)
Finally, isolating the function's odd terms gives:
(10)
So, the operation to isolate odd terms, called Od[], is
 (11) |