Page:Philosophical Transactions - Volume 145.djvu/178

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MR. W.H.L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS.

and we find from this the series

Whence we find, putting ${\displaystyle \mu }$ for ${\textstyle {\frac {\mu ^{2}x^{2}}{2^{2}}}}$,

{\displaystyle {\begin{aligned}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}d\theta &{\sqrt {\cos \theta }}\ \varepsilon ^{\mu \cos ^{2}\theta }\cos \left(\mu \sin \theta \cos \theta +{\frac {5\theta }{2}}-\tan \theta \right)\\&={\frac {\sqrt {\pi }}{2\mu ^{2}\varepsilon }}\left\{2\mu \varepsilon ^{2{\sqrt {\mu }}}-{\sqrt {\mu }}\ \varepsilon ^{2{\sqrt {\mu }}}+2\mu \varepsilon ^{-2{\sqrt {\mu }}}+\varepsilon ^{-2{\sqrt {\mu }}}\right\}\end{aligned}}}

Next consider the symbolical equation

${\displaystyle (\mathrm {D} -1)(\mathrm {D} -3)(\mathrm {D} -5)u-\mu ^{3}\varepsilon ^{3\omega }u=0}$, where ${\displaystyle \varepsilon ^{\omega }=x}$;

and assume as the transformed equation

${\displaystyle (\mathrm {D} -1)(\mathrm {D} -2)(\mathrm {D} -3)v-\mu ^{3}\varepsilon ^{3\omega }v=0}$.

 Then ${\displaystyle u=(\mathrm {D} -2)v,}$
 and ${\displaystyle v=\mathrm {C} _{1}x\varepsilon ^{\mu x}}$

where ${\displaystyle 1}$, ${\displaystyle \alpha }$, ${\displaystyle \beta }$ are the three cube roots of unity.

 Hence

We must determine ${\displaystyle \mathrm {C} _{1}}$, ${\displaystyle \mathrm {C} _{2}}$, ${\displaystyle \mathrm {C} _{3}}$ according to the series we have to sum.

 If

we find

 Whence ${\displaystyle \int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\!\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}d\theta d\varphi \varepsilon ^{\mu \cos \theta \cos \varphi \cos(\theta +\varphi )}\cos ^{-{\frac {1}{3}}}\theta \cos ^{\frac {1}{3}}\varphi }$