Page:Philosophical Transactions - Volume 145.djvu/192

This page has been proofread, but needs to be validated.

mr. w.h.l. russell on the theory of definite integrals.

173

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—

$1+{\frac {x}{a^{2}+2^{2}}}+{\frac {x^{2}}{(a^{2}+2^{2})(a^{2}+4^{2})}}+{\frac {x^{3}}{(a^{2}+2^{2})(a^{2}+4^{2})\ldots (a^{2}+2^{2}n^{2})}}+\mathrm {\&c.,}$

we know that

\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\varepsilon ^{a\theta }(\cos \theta )^{n}={\frac {1.2.4\ldots 2n}{(a^{2}+2^{2})(a^{2}+4^{2})\ldots (a^{2}+2^{2}n^{2})}}\cdot {\frac {\varepsilon ^{\frac {a\pi }{2}}-\varepsilon ^{-{\frac {a\pi }{2}}}}{a}}.

Hence by substitution the above series becomes

${\begin{aligned}&{\frac {a}{\varepsilon ^{\frac {a\pi }{2}}-\varepsilon ^{-{\frac {a\pi }{2}}}}}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}d\theta \varepsilon ^{a\theta }\left\{1+{\frac {x\cos ^{2}\theta }{1.2}}+{\frac {x^{2}\cos ^{4}\theta }{1.2.3.4}}+\mathrm {\&c.} \right\}\\&={\frac {a}{2\left(\varepsilon ^{\frac {a\pi }{2}}-\varepsilon ^{-{\frac {a\pi }{2}}}\right)}}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}d\theta \varepsilon ^{a\theta }\{\varepsilon ^{{\sqrt {x}}\cos \theta }+\varepsilon ^{-{\sqrt {x}}\cos \theta }\}.\end{aligned}}$

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,' Cauchy 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.

Since

\int _{0}^{\infty }\varepsilon ^{-ax^{2}}\cos 2xdx={\frac {\sqrt {\pi }}{2{\sqrt {a}}}}\varepsilon ^{-{\frac {1}{a}}}

$\therefore \ \varepsilon ^{-{\frac {1}{a}}}={\frac {\sqrt {a}}{2{\sqrt {\pi }}}}\int _{-\infty }^{\infty }\varepsilon ^{-{\frac {ax^{3}}{4}}}\cos xdx.$

Again, since

\int _{-\infty }^{\infty }\varepsilon ^{x-ax^{2}}={\frac {\sqrt {\pi }}{\sqrt {a}}}\varepsilon ^{\frac {1}{4a}},

we find

\varepsilon ^{\frac {1}{a}}={\frac {\sqrt {a}}{2{\sqrt {\pi }}}}\int _{-\infty }^{\infty }\varepsilon ^{x-{\frac {ax^{2}}{4}}}dx,

whence we have

\varepsilon ^{\frac {1}{a}}-\varepsilon ^{-{\frac {1}{a}}}={\frac {\sqrt {a}}{2{\sqrt {\pi }}}}\int _{-\infty }^{\infty }dx(\varepsilon ^{x}-\cos x)\varepsilon ^{-{\frac {ax^{2}}{4}}}.