# Page:Philosophical Transactions - Volume 145.djvu/181

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 (${\displaystyle \mu }$). Thus we may operate on any of the integrals we have obtained by such a symbol as ${\textstyle \mathrm {F} \left({\frac {d}{d\mu }}\right)}$, where ${\displaystyle \mathrm {F} }$ 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 ${\displaystyle \iint ..\mathrm {P} \varepsilon ^{\mu {\text{Q}}}{\text{F}}({\text{Q}})dv\ldots d\theta \ldots }$expressed in a series of single integrals, where the integrations are performed with respect to (${\displaystyle \mu }$), and (${\displaystyle \mu }$) may be taken between any limits. But (${\displaystyle \mu }$) must in no case pass through zero, as the definite integrals, on which we operate with respect to (${\displaystyle \mu }$), cannot be found for that value of ${\displaystyle \mu }$ 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

${\displaystyle 1+{\frac {x}{\beta }}+{\frac {x^{2}}{\beta (\beta +1).1.2}}+{\frac {x^{3}}{\beta (\beta +1)(\beta +2).1.2.3}}+\mathrm {\&c} ,}$

where (${\textstyle \beta }$) is an integer. The following integral is known:

${\displaystyle \int _{0}^{\pi }d\theta \ \varepsilon ^{a\cos \theta }\cos(a\sin \theta )\varepsilon ^{(\beta -1)^{i\theta }};}$
${\displaystyle \therefore \ {\frac {1}{\Gamma \beta }}={\frac {1}{\pi a^{\beta -1}}}\int _{-\pi }^{\pi }d\theta \ \varepsilon ^{a\cos \theta }\cos(a\sin \theta )\varepsilon ^{(\beta -1)^{i\theta }}\varepsilon {\frac {x\varepsilon ^{i\theta }}{a}}.}$

Hence we find for the sum of the above series,

${\displaystyle {\frac {\Gamma \beta }{\pi a^{\beta -1}}}\int _{-\pi }^{\pi }d\theta \varepsilon ^{a\cos \theta }\cos(a\sin \theta )\varepsilon ^{(\beta -1)i\theta }}$

Next let us consider the same series when (${\displaystyle \beta }$) is a fraction. We have

${\displaystyle {\frac {\Gamma (\beta -1)\Gamma (n+1)}{\Gamma (\beta +n)}}=\int _{0}{1}dv\ v^{n}(1-v)^{\beta -2};}$
${\displaystyle \therefore \ {\frac {\Gamma \beta }{\Gamma (\beta +n)}}={\frac {\beta -1}{\pi a^{n}}}\int _{0}^{1}\int _{-\pi }^{\pi }d\theta dv(1-v)^{\beta -2}\varepsilon ^{a\cos \theta }\cos(a\sin \theta )\varepsilon ^{ni\theta },}$

except for ${\displaystyle n=0}$, when

${\displaystyle {\frac {2\Gamma \beta }{\Gamma \beta }}={\frac {\beta -1}{\pi }}\int _{0}^{1}\int _{-\pi }^{\pi }d\theta dv(1-v)^{\beta -2}\varepsilon ^{a\cos \theta }\cos(a\sin \theta );}$

and we find for the sum of the series,

${\displaystyle {\frac {\beta -1}{\pi }}\int _{0}^{1}\int _{-\pi }^{\pi }(1-v)^{\beta -2}\varepsilon ^{a\cos \theta }\cos(a\sin \theta )\varepsilon {\frac {vx\varepsilon ^{i\theta }}{a}}d\theta dv-1.}$