Proposition 2.1.   Let \(k^2 = 1 - b^2/a^2\). Then the identities

\[\begin{aligned} F(k) & = a F_S(a, b) \\ E(k) & = \frac{1}{a} E_S(a, b). \end{aligned}\]

are true.

Proof. Let us prove the second equation with the first following similarly. Substituting, we get

\[\begin{aligned} E(k) & = \int_0^{\pi/2} \sqrt{1 - k^2 \sin^2 \theta} d \theta \\ & = \int_0^{\pi/2} \sqrt{1 - (1-b^2/a^2) \sin^2 \theta} d \theta \\ & = \int_0^{\pi/2} \sqrt{\cos^2 \theta + b^2/a^2 \sin^2 \theta} d \theta \\ & = \frac{1}{a} \int_0^{\pi/2} \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta} d \theta \\ & = \frac{1}{a} E_S(a, b), \end{aligned}\]

and hence our result. \(\blacksquare\)