Page:Philosophical Transactions - Volume 145.djvu/192

Rh same factorials, so that we can deduce the value of many definite integrals from one series.

I shall now give an example of the summation of a factorial series of a somewhat different nature.

Consider the series&mdash;

Hence by substitution the above series becomes

There are other series of an analogous nature which may be summed in a similar manner: the object of introducing the above summation in this paper, is to point out the use of the integral $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\varepsilon^{a\theta}(\cos\theta)^n$$, when impossible factors occur in the denominators of the successive terms of a factorial series.

In the 'Exercices de Mathématiques,' has proved that if $$z$$ be a quantity of the form $$\zeta(\cos\varphi+i\sin\varphi)$$, and $$z\varphi(z)$$ continually approach zero as $$\zeta$$ indefinitely increases whatever be $$\varphi$$, then the residue of $$\varphi(z)$$ is equal to zero, the limits of $$\zeta$$ being 0 and ($$\infty$$), and those of $$\varphi$$, $$\pi$$ and $$-\pi$$. From this theorem he deduces the sums of certain series, which I shall presently consider; but must first give certain results which will be useful in the sequel.