Complete elliptic integrals

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Problem 1.1.33 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 1.1 "Elementary Calculus"

Show that

F(k) = \int _{0}^{\pi /2} \frac{dx}{\sqrt{1-k\cos ^{2}x}}

$0 \leq k < 1$ , is an increasing function of $k$ .

Problem 3725 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.1 "Proper Integrals Depending on a Parameter"

Find the complete elliptic integrals

E(k)=\int _{0}^{\frac{\pi }{2}}\sqrt{1-k^{2}\sin ^{2}\varphi }\, d\varphi

and

F(k)=\int _{0}^{\frac{\pi }{2}}\dfrac {d\varphi }{\sqrt{1-k^{2}\sin ^{2}\varphi }} \quad (0<k<1)

and express their derivatives in terms of the functions $E(k)$ and $F(k)$ .

Prove that $E(k)$ satisfies the differential equation

E''(k)+\dfrac {1}{k}E'(k)+\dfrac {E(k)}{1-k^{2}}=0 .

Problem 3739 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.1 "Proper Integrals Depending on a Parameter"

(a)(b)

Let $F(k)$ and $E(k)$ be complete elliptic integrals (see Problem 3725). Prove the formulas:

(a)

$\displaystyle \int_{0}^{k}F(k)k\, dk=E(k)-k_{1}^{2}F(k)$ ;

(b)

$\displaystyle \int_{0}^{k}E(k)k\, dk=\dfrac {1}{3}\left[(1+k^{2})E(k)-k_{1}^{2}F(k)\right]$ ,

where $k_{1}^{2}=1-k^{2}$ .

Problem 4651 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.5 "Integration by Means of Series"

Express, as a series in powers of the eccentricity, the length of the contour of the ellipse $x = a\cos t$ , $y = b\sin t$ .

Problem 4652 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.5 "Integration by Means of Series"

The speed of a heavy point that has slid down some curve from height $h$ to height $y$ equals $\sqrt{2g(h-y)}$ . Therefore the period of oscillation of a pendulum of length $l$ , swinging with maximum deflection angle $\alpha$ , is expressed by the formula

T = 4\sqrt{\frac{l}{2g}} \int _{0}^{\alpha } \frac{d\varphi }{\sqrt{\cos \varphi -\cos \alpha }} = 4\sqrt{\frac{l}{2g}} \int _{0}^{\alpha } \frac{d\varphi }{\sqrt{2\left(\sin ^{2}\frac{\alpha }{2}-\sin ^{2}\frac{\varphi }{2}\right)}} .

From this, after the substitution $\sin \dfrac {\varphi }{2} = \sin \dfrac {\alpha }{2}\sin \psi$ , one obtains:

T = 4\sqrt{\frac{l}{g}} \int _{0}^{\pi /2} \frac{d\psi }{\sqrt{1-\sigma ^{2}\sin ^{2}\psi }}; \quad \sigma = \sin \frac{\alpha }{2} .

Expanding in powers of $\sigma$ and integrating term by term, obtain the formula:

T = 2\pi \sqrt{\frac{l}{g}} \left(1+\left(\frac{1}{2}\right)^{2}\sin ^{2}\frac{\alpha }{2}+\left(\frac{1 \cdot 3}{2 \cdot 4}\right)^{2}\sin ^{4}\frac{\alpha }{2}+\left(\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^{2}\sin ^{6}\frac{\alpha }{2}+\ldots \right) .

Problem 1.90 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Transformations of Sequences into Functions. Theorem of Cesàro"

If $0<k<1$ and if $k$ converges to $1$ , then

\int _0^1 \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}} \sim \frac{1}{2} \log \frac{1}{1-k} . \qquad \text{[See Problem 202 of Part II.]}

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.