Page:Philosophical Transactions - Volume 145.djvu/181

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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,