Page:Philosophical Transactions - Volume 145.djvu/192

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—

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

we know that

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

Hence by substitution the above series becomes

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

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 ${\displaystyle \int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\varepsilon ^{a\theta }(\cos \theta )^{n}}$, 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 ${\displaystyle z}$ be a quantity of the form ${\displaystyle \zeta (\cos \varphi +i\sin \varphi )}$, and ${\displaystyle z\varphi (z)}$ continually approach zero as ${\displaystyle \zeta }$ indefinitely increases whatever be ${\displaystyle \varphi }$, then the residue of ${\displaystyle \varphi (z)}$ is equal to zero, the limits of ${\displaystyle \zeta }$ being 0 and (${\displaystyle \infty }$), and those of ${\displaystyle \varphi }$, ${\displaystyle \pi }$ and ${\displaystyle -\pi }$. 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

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

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

Again, since

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

we find

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

whence we have

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