Page:Philosophical Transactions - Volume 145.djvu/196

Again, we have

${\frac {1}{\varepsilon ^{\pi }-\varepsilon ^{-\pi }}}-{\frac {2^{4m+1}}{\varepsilon ^{2\pi }-\varepsilon ^{-2\pi }}}+{\frac {3^{4m+1}}{\varepsilon ^{3\pi }-\varepsilon ^{-3\pi }}}-\mathrm {\&c.} =0.$

We must transform the element $n^{4m+1}$ thus:

$n^{4m+1}={\frac {1}{\Gamma (4m+1)}}\int _{0}^{1}x^{{\frac {1}{n}}-1}\left(\log _{\varepsilon }{\frac {1}{x}}\right)^{4m}dx={\frac {1}{\Gamma (4m+1)}}\int _{0}^{1}\varepsilon ^{\frac {\log _{\varepsilon }x}{n}}{\frac {1}{x}}\left(\log _{\varepsilon }{\frac {1}{x}}\right)^{4m}dx.$

Again,

$\varepsilon ^{\frac {\log _{\varepsilon }x}{n}}={\frac {\sqrt {n}}{2{\sqrt {\pi \log _{\varepsilon }x}}}}\int _{-\infty }^{\infty }\varepsilon ^{z-{\frac {nz^{2}}{4\log _{\varepsilon }x}}}dz.$

Lastly,

${\sqrt {n}}={\frac {2n}{\sqrt {\pi }}}\int _{0}^{\infty }\varepsilon ^{-nv^{2}}dv.$

Hence, combining these integrals together, and substituting for ${\frac {1}{\varepsilon ^{n\pi }-\varepsilon ^{-n\pi }}},$ as before, we are able to transform the above series into one which can be summed by the ordinary rules. The resulting definite integral will of course be equal to zero.

Cauchy has applied the methods of the residual calculus to the determination of the sum of the series whose general term is

$(-1)^{n-1}{\frac {\varepsilon ^{n\alpha }-\varepsilon ^{-n\alpha }}{\varepsilon ^{n\pi }-\varepsilon ^{-n\pi }}}\cdot {\frac {n\cos n\alpha }{n^{4}+c^{4}}}$

in finite terms. We may transform the element ${\frac {1}{n^{4}+c^{4}}}$ thus:

${\frac {1}{n^{4}+c^{4}}}={\frac {1}{c^{2}}}\int _{0}^{\infty }\varepsilon ^{-n^{2}z^{2}}\sin c^{2}zdz.$

Again,

$\varepsilon ^{-n^{2}z^{2}}={\frac {2}{\sqrt {\pi z}}}\int _{0}^{\infty }\varepsilon ^{-{\frac {v^{2}}{z^{2}}}}\cos 2nz.$

Wherefore, combining these integrals, and transforming the other elements as before, we may find its sum by means of definite integrals. We may resolve ${\frac {n}{n^{4}+c^{4}}}$ into its partial fractions, and then find the sum of the series, which would be simpler.

The transformation of $\varepsilon ^{-n^{2}z^{2}}$ which I have used above, is due to Professor Kummer, who has applied it in the seventeenth volume of Crelle's Journal, in a paper to which I am indebted for many ideas relative to the connexion of definite integrals with series, to the expression of the series

$1+q+q^{4}+q^{9}+\mathrm {\&c.} ,$

and others of a similar nature by means of a definite integral. The integral $\int _{0}^{\infty }{\frac {\sin \mu z.dz}{\varepsilon ^{2\pi z}-1}}$ was first applied to the summation of series, whose terms involve elements of the form ${\frac {1}{\alpha -\beta x^{n}}},$ by Poisson in his Memoir on the Distribution of Electricity in two electrized spheres, which mutually act upon each other. He proves that the cal-