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

Let us first consider the series whose general term is

${\frac {\alpha (\alpha +1)\ldots (\alpha +n-1)}{\beta (\beta +1)\ldots (\beta +n-1)}}\cdot {\frac {\alpha '(\alpha '+1)\ldots (\alpha '+n-1)}{\beta '(\beta '+1)\ldots (\beta '+n-1)}}\cdot {\frac {x^{n}}{1.2.3\ldots n}}.$

Its sum will be found to be

${\frac {\Gamma \beta }{\Gamma \alpha \Gamma (\beta -\alpha )}}\cdot {\frac {\Gamma \beta '}{\Gamma \alpha '\Gamma (\beta '-\alpha ')}}\ldots \int _{0}^{1}\!\int _{0}^{1}\ldots \,v^{\alpha -1}z^{\alpha '-1}\ldots (1-v)^{\beta -\alpha -1}(1-z)^{\beta '-\alpha '-1}\ldots \varepsilon ^{xvz\ldots }dvdz.$

Next, if we consider the series, whose general term is

${\frac {1}{\beta (\beta +1).\ldots (\beta +n-1)\beta '(\beta '+1)\ldots (\beta +n-1)}}\cdot {\frac {x^{n}}{1.2.3\ldots \,n}},$

we find for the sum

${\frac {\Gamma \beta .\varepsilon }{2\pi }}\cdot {\frac {\Gamma \beta '.\varepsilon }{2\pi }}\ldots \int _{-\infty }^{\infty }\!\int _{-\infty }^{\infty }dz\ dz'\ldots {\frac {\varepsilon ^{i(z+z'\ldots )}}{(1+iz)^{\beta }(1+iz')^{\beta '\ldots }}}\varepsilon ^{\frac {x}{(1+iz)(1+iz').\ldots }}.$

We may easily reduce this to a possible form by putting ${\textstyle z=\tan \theta ,z'=\tan \theta '}$, &c. If the series to be summed is of the nature of both the kinds of series we have been discussing, we must combine the two methods of summation together.

Now consider the following differential equation:

$u+\varphi (D)\varepsilon ^{r\omega }u=0,\mathrm {where} \ \varepsilon ^{\omega }=x.$

This equation can always be satisfied when the factors in the denominator of ${\textstyle \varphi (\mathrm {D} )}$ are real and unequal by a series of the form

$u=1+{\frac {\alpha \beta \gamma ..}{\alpha '\beta '\gamma '..}}x+{\frac {\alpha (\alpha +1)\beta (\beta +1)\gamma (\gamma +1)\ldots }{\alpha '(\alpha '+1)\beta '(\beta '+1)\gamma '(\gamma '+1)..}}{\frac {x^{2}}{1.2}}+\mathrm {\&c} .$

We shall suppose that the number of the quantities ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$ &c. is always less than the number of the quantities ${\textstyle \alpha '}$, ${\textstyle \beta '}$,${\textstyle \gamma '}$ &c., and, for the present, that the magnitude of ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$ &c. is always less than that of ${\textstyle \alpha '}$, ${\textstyle \beta '}$ &c., each to each. Then the sum of this series by means of definite integrals can always be found by the preceding theorems. Now Professor Boole has given, in the memoir I have before mentioned, the conditions which are necessary in order that the equation ${\textstyle u+\varphi (\mathrm {D} )\varepsilon ^{r\omega }u=0}$ may be integrable in finite terms, which are therefore the conditions that the sum of the above series, and consequently the value of any definite integral equivalent to it, may be found in finite terms. I shall now give some instances of the evaluation of definite integrals by the application of these principles Let us consider the symbolical equation

$u-{\frac {\mu ^{2}\varepsilon ^{2\theta }u}{(\mathrm {D} -1)(\mathrm {D} -2)}}=0,\ \mathrm {where} \ \varepsilon ^{\theta }=x,$

and assume for its solution

$v-{\frac {\mu ^{2}\varepsilon ^{2\theta }v}{(\mathrm {D-1} )(\mathrm {D-2} )}}=0,\mathrm {so\ that} \ u=(\mathrm {D} -2)v$

whence

$v=\mathrm {C} _{1}x\varepsilon ^{\mu x}+\mathrm {C} _{2}x\varepsilon ^{-\mu x}.$

Hence,

$u=\mathrm {C} _{1}(\mu x^{2}-x)\varepsilon ^{\mu x}+\mathrm {C} _{2}(\mu x^{2}+x)\varepsilon ^{-\mu x};$