162
mr. w.h.l. russell on the theory of definite integrals.
We may extend this process, by performing operations with respect to the quantity (
). Thus we may operate on any of the integrals we have obtained by such a symbol as
, where
is any rational function; and if it is an entire function, we have merely differentiations to perform. If it is a rational fraction, and the factors of the denominator are real and unequal, we may decompose it into simple rational fractions, each of which may, in its turn, be transformed into a simple integral. If we apply this operation to any of the results we have obtained, we immediately have a definite integral
expressed in a series of single integrals, where the integrations are performed with respect to (
), and (
) may be taken between any limits. But (
) must in no case pass through zero, as the definite integrals, on which we operate with respect to (
), cannot be found for that value of
by the processes we have been investigating. There are many other operations of a similar nature, which it is easy to imagine.
I am now come to the second part of this memoir, the investigation of those new methods of summation, and of the definite integrals corresponding to them, to which I have before alluded. Let us consider the series
where (
) is an integer. The following integral is known:

Hence we find for the sum of the above series,
Next let us consider the same series when (
) is a fraction. We have

except for
, when
and we find for the sum of the series,