Can the complete elliptic integrals of the third kind to be expressed in series?

Solution 1:

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}

Solution 2:

It is just the Taylor series in two dimensions. For example, you could write $$\Pi (\eta |\kappa )=\sum_{m=0}^{n} P_m(\kappa)\, \eta^m+O(\eta^{n+1})$$ and $$P_m(\kappa)=\sum_{p=0}^{q} Q_p\, \kappa^p+O(\kappa^{q+1})$$ For example, using $n=q=4$

$$\Pi (\eta |\kappa )=\left(\frac{\pi }{2}+\frac{\pi \kappa }{8}+\frac{9 \pi \kappa ^2}{128}+\frac{25 \pi \kappa ^3}{512}+\frac{1225 \pi \kappa ^4}{32768}+O\left(\kappa ^5\right)\right)+$$ $$\eta \left(\frac{\pi }{4}+\frac{3 \pi \kappa }{32}+\frac{15 \pi \kappa ^2}{256}+\frac{175 \pi \kappa ^3}{4096}+\frac{2205 \pi \kappa ^4}{65536}+O\left(\kappa ^5\right)\right)+$$ $$\eta ^2 \left(\frac{3 \pi }{16}+\frac{5 \pi \kappa }{64}+\frac{105 \pi \kappa ^2}{2048}+\frac{315 \pi \kappa ^3}{8192}+\frac{8085 \pi \kappa ^4}{262144}+O\left(\kappa ^5\right)\right)+$$ $$\eta ^3 \left(\frac{5 \pi }{32}+\frac{35 \pi \kappa }{512}+\frac{189 \pi \kappa ^2}{4096}+\frac{1155 \pi \kappa ^3}{32768}+\frac{15015 \pi \kappa ^4}{524288}+O\left(\kappa ^5\right)\right)+$$ $$\eta ^4 \left(\frac{35 \pi }{256}+\frac{63 \pi \kappa }{1024}+\frac{693 \pi \kappa ^2}{16384}+\frac{2145 \pi \kappa ^3}{65536}+\frac{225225 \pi \kappa ^4}{8388608}+O\left(\kappa ^5\right)\right)+O\left(\eta ^5\right)$$ For example, suing the above for $\eta=\frac 12$ and $\kappa =\frac 13$ would give $2.46720$ while the exact value is $2.49525$.