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: $$ E(κ ,\pi/2 ) = \frac{\pi}{2}(1 - (\frac{1}{2})^2 κ^2 - (\frac{1 \cdot 3}{2 \cdot 4})^4 \frac{κ^4}{3} + (\frac{1 \cdot 3 \cdot 5 }{2 \cdot 4 \cdot 6})^6 \frac{κ^6}{5} ...) $$
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 $2.49525$.