Page:Philosophical Transactions

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

171

Hence we have

{\frac {\Gamma {\tfrac {1}{2}}\cdot \varepsilon }{2\pi }}\cdot {\frac {\Gamma .1\varepsilon }{2\pi }}\int _{-\infty }^{\infty }\!\int _{-\infty }^{\infty }{\frac {\varepsilon ^{i(z+z')}dzdz'}{(1+iz)^{\frac {1}{2}}(1+iz)}}\varepsilon ^{\frac {\mu }{(1+iz)(1+iz').2^{2}}}

={\frac {1}{\pi }}\int _{-\pi }^{\pi }d\theta \varepsilon ^{\alpha \cos \theta }\cos(\alpha \sin \theta )\varepsilon ^{{\frac {\epsilon ^{2i\theta }}{\alpha ^{2}}}\cdot \mu }-1.

Hence

\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\!\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\varepsilon ^{{\frac {\mu }{4}}\cos \theta \cos \varphi \cos(\theta +\varphi )}\cos ^{\frac {-3}{2}}\theta \cos ^{-}1\varphi .d\theta d\varphi ,

$\cos \left({\tfrac {\mu }{4}}\cos \theta \cos \varphi \sin(\theta +\varphi )+{\frac {\theta }{2}}+\varphi -(\tan \theta +\tan \varphi )\right)$

$={\frac {4{\sqrt {\pi }}}{\varepsilon ^{2}}}\int _{-\pi }^{\pi }d\theta \varepsilon ^{\alpha \cos \theta }\cos \alpha \sin \theta .\varepsilon ^{\frac {\mu \cos 2\theta }{\alpha ^{2}}}\cos {\frac {\mu \sin 2\theta }{\alpha ^{2}}}-{\frac {4\pi ^{\frac {3}{2}}}{\varepsilon ^{2}}}$

But we may effect these reductions systematically by means of the following proposition due to M. Smaasen:—

If	$a_{0}+a_{1}\ x+a_{2}\ x^{2}+a_{3}\ x^{3}+\mathrm {\&c.} =\varphi _{1}(x),$
and	$b_{0}+b_{1}\ x+b_{2}\ x^{2}+b_{3}\ x^{3}+\mathrm {\&c.} =\varphi _{2}(x),$
then	$a_{0}\ b_{0}+a_{1}\ b_{1}\ x+a_{2}\ b_{2}\ x^{2}+\mathrm {\&c.}$

={\frac {1}{2\pi }}\int _{0}^{\pi }d\theta \{(\varphi _{1}(x\varepsilon ^{i\theta })+\varphi _{1}(x\varepsilon ^{-i\theta }))(\varphi _{2}(\varepsilon ^{i\theta })+\varphi _{2}(\varepsilon ^{-i\theta }))\}.

M. Smaasen has also proved in the same paper, that if the sums of the three series

{\begin{aligned}&a_{0}+a_{1}\ x+a_{2}\ x^{2}+a_{3}\ x^{3}+\mathrm {\&c.} \\&b_{0}+b_{1}\ x+b_{2}\ x^{2}+b_{3}\ x^{3}+\mathrm {\&c.} \\&c_{0}+c_{1}\ x+c_{2}\ x^{2}+c_{3}\ x^{3}+\mathrm {\&c.} \end{aligned}}

are known, we may determine the sum of the series

a_{0}\ b_{0}\ c_{0}+a_{1}\ b_{1}\ c_{1}x+a_{2}\ b_{2}\ c_{2}\ x^{2}+\mathrm {\&c.}

by means of a double integral, but we shall not want this in what follows.

Now

consequently

Now

\varepsilon ^{\sqrt {x\varepsilon ^{i\theta }}}+\varepsilon ^{-{\sqrt {x\varepsilon ^{i\theta }}}}+\varepsilon ^{-{\sqrt {x\varepsilon ^{-i\theta }}}}+\varepsilon ^{-{\sqrt {x\varepsilon ^{-i\theta }}}}

=2\varepsilon ^{{\sqrt {x}}\cos {\frac {\theta }{2}}}\cos \left({\sqrt {x}}\sin {\tfrac {\theta }{2}}\right)+2\varepsilon ^{-{\sqrt {x}}\cos {\frac {\theta }{2}}}\cos \left({\sqrt {x}}\sin {\tfrac {\theta }{2}}\right);

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