COMPUTATION OF ELLIPTIC INTEGRALS USING CONTINUED FRACTIONS

V. E. KORNILOV

(Presented by the Department of Higher Mathematics of the Tomsk Polytechnic Institute)

In this article elliptic integrals are represented in the form of a sum of arctangents.

An estimate of the remainder term is obtained for \(\operatorname{Re}(\kappa^{2}z^{2})<1\).

In what follows the abbreviated notation is used

\[ \kappa^{2}z^{2}\equiv v;\quad (a)_k=a(a+1)\ldots(a+k-1),\quad (a)_0=1. \tag{1} \]

1. With respect to the even convergents of the following power function [1]

\[ (1-v)^{-\frac12} = \frac{ \displaystyle \sum_{m=0}^{n-1} C_{n-1}^{m}\, \frac{(0.5-n)_m}{(1-2n)_m}\,(-v)^m }{ \displaystyle \sum_{m=0}^{n} C_{n}^{m}\, \frac{(0.5-n)_m}{(1-2n)_m}\,(-v)^m } +r_{2n}(v) = \]

\[ = \frac{p_{2n}(v)}{q_{2n}(v)}+r_{2n}(v),\quad n=1,2,\ldots \tag{2} \]

we shall prove the theorem.

Theorem. The even convergent fraction (2) is represented by the following sum of elementary fractions:

\[ \frac{p_{2n}(v)}{q_{2n}(v)} = \frac{1}{n}\sum_{m=1}^{n} \frac{1}{1-v\sin^{2}\dfrac{2m\pi-\pi}{4n}}, \quad n=1,2,\ldots \tag{3} \]

Proof. It is known [2] that the denominators of the convergents (2) are connected with Chebyshev polynomials of the first kind, namely:

\[ q_{2n}(v)=T_n(1-2v)=\cos[n\arccos(1-2v)]. \tag{4} \]

According to equality (4), it is not difficult to compute that

\[ q_{2n}(v)= \prod_{m=1}^{n} \left(1-v\sin^{2}\frac{2m\pi-\pi}{4n}\right), \quad n=1,2,\ldots \tag{5} \]

To prove identity (3), let

\[ \beta_m=-\sin^{2}\frac{2m\pi-\pi}{4n}, \]

then suppose that

\[ \frac{1:n}{1+v\beta_1}+\cdots+\frac{1:n}{1+v\beta_n} =\frac{p_{2n}(v)}{q_{2n}(v)}. \tag{6} \]

The first coefficient of the numerator \(p_{2n}(v)\) is equal to one, and also the sum of the \(n\) terms, each of which is equal to \(1:n\), is equal to one. In view of equalities (2) and (6), for the roots of the polynomial \(q_{2n}(v)\) the equalities

\[ - C_n^1\,\frac{0.5-n}{1-2n}=\beta_1+\cdots+\beta_n;\quad C_n^m(-1)^m\frac{(0.5-n)_m}{(1-2n)_m}= \]

\[ =\beta_1\ldots\beta_m+\cdots+\beta_{n-m+1}\ldots\beta_n;\quad m=2,\ldots,n-1. \tag{7} \]

hold. On the basis of equalities (7) we compute the coefficients of \(v^m\) for the numerator \(p_{2n}(v)\); here, on the left-hand side of the equality, we have \(nC_{n-1}^m\) terms, and obtain

\[ \frac{1}{n}\left(\beta_2\ldots\beta_{n-1}+\cdots+\beta_{n-m+1}\ldots\beta_n\right) +\cdots+ \frac{1}{n}\left(\beta_1\ldots\beta_n+\cdots+\beta_{n-m}\ldots\beta_{n-1}\right)= \]

\[ = C_{n-1}^m(-1)^m\frac{(0.5-n)_m}{(1-2n)_m}; \]

\[ m=1,\ldots,n-1. \]

On the basis of the last equality and equality (2), the theorem is proved. In view of equalities (2) and (3) we have

\[ (1-v)^{-\frac12} =\frac{1}{n}\sum_{m=1}^{n} \frac{1}{1-v\sin^2\frac{2m\pi-\pi}{4n}} +r_{2n}(v). \tag{8} \]

1. The elliptic integrals of the first, second, and third kinds ([3], p. 82) \((z=\sin\varphi,\ h=n,\ \kappa^2<1)\), after replacing the factor \((1-v)^{-1/2}\) of the integrands by the elementary fractions (8) and computing the resulting sum of integrals, are equal to the following sums of arctangents and remainder terms:

\[ F(\varphi,\kappa) =\frac{1}{n}\sum_{m=1}^{n} \frac{\arctan(\alpha_m\tan\varphi)}{\alpha_m} +\rho_{2n}(v); \]

\[ \kappa^2<1,\quad n=1,2,\ldots \tag{9} \]

\[ E(\varphi,\kappa) =2n\varphi-\frac{1}{n}\sum_{m=1}^{n} \frac{1}{\alpha_m}\arctan(\alpha_m\tan\varphi)\times \]

\[ \times \cot^2\frac{2m\pi-\pi}{4n} +R_{2n}(v);\quad n=1,2,\ldots \tag{10} \]

\[ \Pi(\varphi,h,\kappa) = \frac{p_{2n}(-\kappa^2:h)}{q_{2n}(-\kappa^2:h)} \cdot \frac{\arctan\!\left(\sqrt{1+h}\tan\varphi\right)}{\sqrt{1+h}} + \]

\[ +\frac{1}{n}\sum_{m=1}^{n} \frac{(1-\alpha_m)^2\arctan(\alpha_m\tan\varphi)} {(h+1-\alpha_m^2)\alpha_m} +\sigma_{2n}(v);\quad h<-\kappa^2, \tag{11} \]

where

\[ \left\{ \begin{aligned} a_m^2&=1-\kappa^2\sin^2\frac{2\pi m-\pi}{4n},\qquad \rho_{2n}(v)=\int_0^z \frac{r_{2n}(v)\,dz}{\sqrt{1-z^2}},\\ R_{2n}(v)&=\int_0^z \frac{(1-v)r_{2n}(v)\,dz}{\sqrt{1-z^2}},\qquad \sigma_{2n}(v)=\int_0^z \frac{r_{2n}(v)\,dz}{(1+hz^2)\sqrt{1-z^2}}. \end{aligned} \right. \tag{12} \]

If, for the elliptic integrals, the factor \((1-u)^{-1/2}\) of the integrands is replaced by the elementary fractions (8), and the resulting sum of integrals is evaluated, then the elliptic integrals are represented in the form of the following sums of a logarithmic function and remainder terms:

\[ \left\{ \begin{aligned} F(\varphi,\kappa) &=\frac{1}{2n}\sum_{m=1}^{n}\frac{1}{\beta_m} \ln\frac{1+\beta_m\sin\varphi}{1-\beta_m\sin\varphi} +\rho_{2n+1}(u),\\ &\hspace{3.7cm} n=1,2,\ldots; \end{aligned} \right. \tag{13} \]

\[ \left\{ \begin{aligned} E(\varphi,\kappa) &=\frac{1}{2n}\sum_{m=1}^{n}\frac{1-\kappa^2}{\beta_m^3} \sin^2\frac{2\pi m-\pi}{n}\cdot \ln\frac{1+\beta_m\sin\varphi}{1-\beta_m\sin\varphi}\\ &\quad+\frac{p_{2n}(1-\kappa^{-2})}{q_{2n}(1-\kappa^{-2})}\sin\varphi +R_{2n+1}(u),\qquad n=1,2,\ldots; \end{aligned} \right. \tag{14} \]

\[ \left\{ \begin{aligned} \Pi(\varphi,h,\kappa) &=\frac{\sqrt h}{\kappa^2+h}\cdot \frac{p_{2n}(x_1)}{q_{2n}(x_1)} \operatorname{arctg}(\sqrt h\,\sin\varphi)\\ &\quad+\frac{1}{2n}\sum_{m=1}^{n} \frac{\beta_m}{\beta_m^2+h} \ln\frac{1+\beta_m\sin\varphi}{1-\beta_m\sin\varphi} +\sigma_{2n+1}(u),\\ &\hspace{0.4cm} h\ge 0,\quad x_1=(\kappa^2-1):(\kappa^2+h),\quad n=1,2,\ldots; \end{aligned} \right. \tag{15} \]

where

\[ \left\{ \begin{aligned} u&=\frac{(1-\kappa^2)z^2}{1-\kappa^2z^2},\qquad \beta_m^2=\kappa^2+(1-\kappa^2)\sin^2\frac{2\pi m-\pi}{4n},\\ \rho_{2n+1}(u)&=\int_0^z \frac{r_{2n}(u)\,dz}{1-v},\qquad R_{2n+1}(u)=\int_0^z r_{2n}(u)\,dz,\\ \sigma_{2n+1}(u)&=\int_0^z \frac{r_{2n}(u)\,dz}{(1+hz^2)(1-v)}. \end{aligned} \right. \tag{16} \]

1. Let us prove a theorem on an inequality for the moduli of polynomials.

Theorem. If the roots of the polynomial \(Q_{2\kappa+2}(z)\) separate the roots of the polynomial \(Q_{2\kappa}(z)\), i.e.

\[ \left\{ \begin{aligned} Q_{2\kappa}(z)&=(z+\beta_1)\ldots(z+\beta_\kappa),\\ Q_{2\kappa+2}(z)&=(z+\alpha_1)\ldots(z+\alpha_{\kappa+1}),\\ \beta_0&=\alpha_1>\beta_1>\alpha_2>\ldots>\alpha_\kappa>\beta_\kappa>\alpha_{\kappa+1}=\beta_{\kappa+1}>0, \end{aligned} \right. \tag{17} \]

then for the moduli of these polynomials the inequality holds

\[ \left\{ \begin{aligned} |Q_{2\kappa+2}(z)|&\ge |Q_{2\kappa}(z)|\,|z+\alpha|,\qquad \kappa=1,2,\ldots;\\ \text{where } \operatorname{Re}z&\ge 0,\quad \alpha_1\ldots\alpha_{\kappa+1}=\beta_1\ldots\beta_\kappa\alpha,\quad \alpha_1>\alpha>\alpha_{\kappa+1}. \end{aligned} \right. \tag{18} \]

Proof. By direct calculation we verify that inequality (18) is valid in the case \(\kappa=1\).

Let \(\beta_{l-1}>\alpha>\beta_l,\ l=1,\ldots,\kappa+1\); then, on the basis of inequalities (17) and (18) for \(\kappa=1\) and of equalities (17) and (18), we obtain the following inequalities:

\[ \left\{ \begin{aligned} &|z+\alpha_1|\,|z+\alpha_2|\ge |z+\beta_1|\left|z+\frac{\alpha_1\alpha_2}{\beta_1}\right|,\\ &\left(\alpha_1>\frac{\alpha_1\alpha_2}{\beta_1}>\alpha_2\right);\\ &\ldots\\ &\left|z+\frac{\beta_l\ldots\beta_\kappa}{\alpha_{l+2}\ldots\alpha_{\kappa+1}}\right| |z+\alpha_{l+2}| \ge |z+\beta_l| \left|z+\frac{\beta_{l+1}\ldots\beta_\kappa}{\alpha_{l+3}\ldots\alpha_{\kappa+1}}\right|,\\ &\left( \frac{\beta_l\ldots\beta_\kappa}{\alpha_{l+2}\ldots\alpha_{\kappa+1}} > \frac{\beta_{l+1}\ldots\beta_\kappa}{\alpha_{l+3}\ldots\alpha_{\kappa+1}} > \beta_{l+1} \right);\\ &\ldots\\ &\left|z+\frac{\beta_{\kappa-1}\beta_\kappa}{\alpha_{\kappa+1}}\right| |z+\alpha_{\kappa+1}| \ge |z+\beta_{\kappa-1}|\,|z+\beta_\kappa|,\\ &\left( \frac{\beta_{\kappa-1}\beta_\kappa}{\alpha_{\kappa+1}} > \beta_{\kappa-1}>\beta_\kappa>\alpha_{\kappa+1} \right). \end{aligned} \right. \tag{19} \]

Multiplying the left- and right-hand sides of inequalities (19) and canceling both sides of the resulting inequality by equal factors, we obtain the required inequality (18), as was to be proved.

1. On the basis of the formula ([1], p. 23 (11)) we obtain:

\[ q_{2n}(v)=(0.5-n)_n!:\bigl[(1-2n)_n\bigr]\times \]

\[ \times F(-n,\ 0.5-n;\ 0.5;\ 1-v) = \frac{(0.5-n)_n}{(1-2n)_n}\,Q_{2n}(v). \tag{20} \]

In view of ([4], pp. 28, 34), inequality (18), and equality (20), we transform the remainder term \(|r_{2n}(v)|\):

\[ |r_{2n}(v)|= \]

\[ = \left| \sum_{\kappa=n}^{\infty} \frac{2v^{2\kappa}} {\left[ \sum_{m=0}^{\kappa} C_\kappa^m \frac{(0.5-\kappa)_m}{(0.5)_m} (v-1)^m \right] \left[ \sum_{m=0}^{\kappa+1} C_{\kappa+1}^m \frac{(-0.5-\kappa)_m}{(0.5)_m} (v-1)^m \right]} \right| < \]

\[ < \frac{2|v|^{2n}}{|Q_{2n}(v)Q_{2n+2}(v)|} \sum_{\kappa=0}^{\infty}\frac{|v|^{2\kappa}}{|2-v|^{2\kappa}}, \]

i.e.,

\[ |r_{2n}(v)| < \frac{|v|^{2n}|2-v|} {|Q_{2n}(v)|^2\bigl(|2-v|-|v|\bigr)} =f(v),\qquad \operatorname{Re}v<1. \tag{21} \]

In view of inequality (21) and equalities (12) and (16), we finally obtain:

\[ \left\{ \begin{aligned} |\sigma_{2n}(v)|&\le |\rho_{2n}(v)|<f(v)\delta(z),\\ |\sigma_{2n+1}(u)|&\le |\rho_{2n+1}(u)|<f(u)\Theta(z), \end{aligned} \right. \qquad \operatorname{Re}(hz^2)\ge 0; \tag{22} \]

\[ \left\{ \begin{aligned} |R_{2n}(v)|&<f(v)(1+|v|)\delta(z),\\ |R_{2n+1}(u)|&<f(u)|z|; \end{aligned} \right. \tag{23} \]

\[ \left\{ \begin{aligned} |\sigma_{2n}(v)|&< f(v)\,\delta(z)\,|1+h z^2|^{-1},\\ |\sigma_{2n+1}(u)|&< f(u)\,\Theta(z)\,|1-h z^2|^{-1}, \end{aligned} \right. \qquad -1<\operatorname{Re}(h z^2)<0. \tag{24} \]

For the estimates (22)—(24), the factors \(\delta(z)\) and \(\Theta(z)\) are as follows:

\[ \delta(z)= \begin{cases} \arcsin |z|, & \text{for } |z|\leqslant 1,\\[6pt] \dfrac{|z|}{|\sin(\arg z)|}, & \text{for } |z|>1, \end{cases} \qquad \operatorname{Re} v<1; \tag{25} \]

\[ \Theta(z)= \begin{cases} \dfrac{1}{2k}\ln\dfrac{1+|kz|}{1-|kz|}, & \text{for } |kz|<1,\\[8pt] \dfrac{|z|}{|\sin(\arg z)|}, & \text{for } |kz|>1, \end{cases} \qquad \operatorname{Re} u<1. \tag{26} \]

References

1. V. E. Kornilov. A system of Padé approximations for power functions. Publ. TPI, vol. 154, pp. 20—23, Tomsk, 1967.

2. V. L. Danilov, A. N. Ivanova et al. Mathematical Analysis (functions, limits, series, continued fractions). Moscow, Fizmatgiz, 1961.

3. I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. Moscow—Leningrad, Gostekhizdat, 1948.

4. T. I. Stieltjes. Investigations on Continued Fractions. Moscow, ONTI, 1936.