Can the complete elliptic integrals of the third kind which are defined by
$$ \Pi (\eta,κ)=\int_0^{\pi/2} d\theta \frac {1}{\sqrt{1−κ \sin^2\theta}} \frac{1}{1 -ηsin^2\theta }$$
to be expressed in series?
Like the fist kind:
$$ F(κ ,\pi/2 ) = \frac{\pi}{2}(1 + (\frac{1}{2})^2 κ^2 + (\frac{1 \cdot 3}{2 \cdot 4})^4 κ^4 + (\frac{1 \cdot 3 \cdot 5 }{2 \cdot 4 \cdot 6})^6 κ^6 …) $$
or the second kind:
$$ E(κ ,\pi/2 ) = \frac{\pi}{2}(1 – (\frac{1}{2})^2 κ^2 – (\frac{1 \cdot 3}{2 \cdot 4})^4 \frac{κ^4}{3} + (\frac{1 \cdot 3 \cdot 5 }{2 \cdot 4 \cdot 6})^6 \frac{κ^6}{5} …) $$
Best Answer
In Maple's notation, \begin{align} &\int_{0}^{\pi/2}\!{\frac {1}{\sqrt {1-\kappa\, \left( \sin \left( t \right) \right) ^{2}} \left( 1-\eta\, \left( \sin \left( t \right) \right) ^{2} \right) }}\,{\rm d}t={\Pi} \left( \eta,\sqrt {\kappa} \right) \\ &= {\frac {\pi}{2}{\mbox{$_1$F$_0$}({\frac{1}{2}};\,\ ;\,\eta)}}+{\frac {\pi}{2^3}{\mbox{$_2$F$_1$}(1,{\frac{3}{2}};\,2;\,\eta)}}\kappa+{\frac { 3^2\,\pi}{2^7}{\mbox{$_2$F$_1$}(1,{\frac{5}{2}};\,3;\,\eta)}}{\kappa}^{2 }+{\frac {5^2\,\pi}{2^9}{\mbox{$_2$F$_1$}(1,{\frac{7}{2}};\,4;\,\eta)}} {\kappa}^{3} \\ & +{\frac {35^2\,\pi}{2^{15}} {\mbox{$_2$F$_1$}(1,{\frac{9}{2}};\,5;\,\eta)}}{\kappa}^{4}+{\frac { 63^2\,\pi}{2^{17}}{\mbox{$_2$F$_1$}(1,{\frac{11}{2}};\,6;\,\eta)}}{ \kappa}^{5}+{\frac {231^2\,\pi}{2^{21}} {\mbox{$_2$F$_1$}(1,{\frac{13}{2}};\,7;\,\eta)}}{\kappa}^{6} \\ & +O \left( {\kappa}^{7} \right) \\ &= {\frac {\pi}{2}{\frac {1}{\sqrt {1-\eta}}}}+{\frac {\pi}{8} \left( -2 \,{\eta}^{-1}+2\,{\frac {1}{\eta\,\sqrt {1-\eta}}} \right) }\kappa+{ \frac {9\,\pi}{128} \left( -{\frac {4\,\eta+8}{3\,{\eta}^{2}}}+{\frac {8}{3\,{\eta}^{2}}{\frac {1}{\sqrt {1-\eta}}}} \right) }{\kappa}^{2} \\&+ { \frac {25\,\pi}{512} \left( -{\frac {6\,{\eta}^{2}+8\,\eta+16}{5\,{ \eta}^{3}}}+{\frac {16}{5\,{\eta}^{3}}{\frac {1}{\sqrt {1-\eta}}}} \right) }{\kappa}^{3} \\&+ {\frac {1225\,\pi}{32768} \left( -{\frac {40\,{ \eta}^{3}+48\,{\eta}^{2}+64\,\eta+128}{35\,{\eta}^{4}}}+{\frac {128}{ 35\,{\eta}^{4}}{\frac {1}{\sqrt {1-\eta}}}} \right) }{\kappa}^{4} \\&+ { \frac {3969\,\pi}{131072} \left( -{\frac {70\,{\eta}^{4}+80\,{\eta}^{3 }+96\,{\eta}^{2}+128\,\eta+256}{63\,{\eta}^{5}}}+{\frac {256}{63\,{ \eta}^{5}}{\frac {1}{\sqrt {1-\eta}}}} \right) }{\kappa}^{5} \\&+ {\frac { 53361\,\pi}{2097152} \left( -{\frac {252\,{\eta}^{5}+280\,{\eta}^{4}+ 320\,{\eta}^{3}+384\,{\eta}^{2}+512\,\eta+1024}{231\,{\eta}^{6}}}+{ \frac {1024}{231\,{\eta}^{6}}{\frac {1}{\sqrt {1-\eta}}}} \right) }{ \kappa}^{6} \\&+O \left( {\kappa}^{7} \right) \end{align}