Page:Philosophical Transactions - Volume 145.djvu/178

${\displaystyle }$

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

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

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