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

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

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

$\int _{0}^{\pi }d\theta \ \varepsilon ^{a\cos \theta }\cos(a\sin \theta )\varepsilon ^{(\beta -1)^{i\theta }};$

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

Hence we find for the sum of the above series,

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

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

${\frac {\Gamma (\beta -1)\Gamma (n+1)}{\Gamma (\beta +n)}}=\int _{0}{1}dv\ v^{n}(1-v)^{\beta -2};$

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

except for $n=0$, when

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

and we find for the sum of the series,

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