Identities and derivatives of the complete elliptic integrals E and K

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Problem 3.13.27 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.13.2 "Problems"

Consider the complete elliptic integrals

E(k)=∫0π/21−k2sin⁡2φdφ,K(k)≡F(k)=∫0π/2dφ1−k2sin⁡2φ, E(k) = \int _{0}^{\pi / 2} \sqrt{1-k^{2} \sin ^{2} \varphi } d \varphi , \quad K(k) \equiv F(k) = \int _{0}^{\pi / 2} \frac{d \varphi }{\sqrt{1-k^{2} \sin ^{2} \varphi }},

where 0<k<10 < k < 1. Prove that:

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(1)

E′(k)=E(k)−K(k)k,K′(k)=E(k)k(1−k2)−K(k)kE^{\prime }(k) = \dfrac {E(k)-K(k)}{k}, \quad K^{\prime }(k) = \dfrac {E(k)}{k\left(1-k^{2}\right)}-\dfrac {K(k)}{k};

(2)

E′′(k)+1kE′(k)+E(k)1−k2=0E^{\prime \prime }(k)+\frac{1}{k} E^{\prime }(k)+\frac{E(k)}{1-k^{2}} = 0;

(3)

∫0ktK(t)dt=E(k)−(1−k2)K(k)\int_{0}^{k} t K(t) d t = E(k)-\left(1-k^{2}\right) K(k);

(4)

∫0ktE(t)dt=13((1+k2)E(k)−(1−k2)K(k))\int_{0}^{k} t E(t) d t = \frac{1}{3}\left(\left(1+k^{2}\right) E(k)-\left(1-k^{2}\right) K(k)\right).

References

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.