Youth researches 1983
Subject area physics

A novel method for the determination of the periodic time of the mathematical pendulum with large amplitudes and a discussion of its applicability to other selected plane oscillations

Compiled by Boris Haase (18),
Theodor-Heuss-Gymnasium in Göttingen

Abstract of the present paper

The period of mass points moving on curved oscillation paths in the vertical plane — for example the mathematical pendulum and the cycloidal pendulum — is calculated here by a method intended to be simpler than the customary approaches.

Most such periods depend on the amplitude whenever the mass point is displaced substantially from the equilibrium position, whose lowest point serves as the reference state. This amplitude dependence usually complicates the calculation.

In order to compute a period independently of amplitude, one needs, apart from the mass, only the directional parameter, that is, the quotient of restoring force and corresponding amplitude or path of displacement.

In the present paper this idea is extended and applied to periods that do in fact depend on amplitude.

In part, differences from the conventional methods arise — in the case of the mathematical pendulum up to about one per cent at an amplitude of 90°. In part, however, this is not so, as can be seen from the cycloidal pendulum. No references could be found for the elliptical pendulum, which is likewise treated.

In the experimental section, the calculated period for the mathematical pendulum could be confirmed within the limits imposed by sources of error and measurement accuracy.

The author concludes that the independently developed method should be usable in physics.

Table of contents

1. Introduction

1.1. Reason and basic idea

The period of the mathematical pendulum for large amplitudes has hitherto been determined by rather difficult methods from infinitesimal calculus and series expansion. The new method developed in this paper for selected plane oscillations is intended, on the one hand, to simplify the calculation and, on the other hand, to allow a comparison with the older method in two examples with respect to accuracy.

The basic idea is to simulate the pendulum oscillation in question by a rectilinear harmonic oscillation, that is, by a spring pendulum or oscillator.

1.2. Presuppositions

All selected oscillations start from a mass point that swings in the vertical plane on a massless string about a fixed point. The pendulum string may be displaced from the rest position by at most 90°, since the maximally restoring force must act at the initial height, where no additional acceleration is to arise.

Otherwise a free fall would have to be prevented by using rods instead of strings. Yet rods could not, as required, lie along the evolutes of the oscillation paths. Even with such contact, the pendulum length in the rest position is assumed for reasons of simplification.

The radius of curvature of the oscillation path must either remain constant from the rest position onward or decrease continuously, so that free fall is prevented in this respect as well.

Frictional and damping effects are neglected. Otherwise further calculations would be necessary for the period, which would go beyond the scope of this paper.

2. Main part

2.1. Theoretical solution of the problem

2.1.1. General solution

The period of a harmonic oscillation is given by:

\[(a)\;T = 2\pi \sqrt{\frac{m}{D}}.\]

By the principle of conservation of energy one has:

\[(b)\;\frac12 m v^2 = \frac12 D s^2 = mgh.\]

During an oscillation, the potential energy \(W_{pot}\) and the kinetic energy \(W_{kin}\) satisfy

\[(c)\;W_{pot}+W_{kin}=W_{pot_0}.\]

Now \(W_{pot_0}\) is to be expressed by the spring energy \(W_{Sp}\), in order to determine a directional parameter \(D(\varphi_0)\) depending on the amplitude. This is then inserted into formula (a) to determine the period.

For this purpose an equivalent motion of the mass \(m\) is considered under a force \(F\) along the displacement path \(s\). One has:

\[(d)\;D = F/s \quad\text{(Hooke’s law)}\qquad\text{and}\qquad (e)\;F = ma = ml\,\ddot\varphi\quad\text{(Newton’s law).}\]

Furthermore, we always refer to the directional parameter \(D\) of the plane string pendulum for minimal amplitude. There, for the restoring force \(F_R\), which acts tangentially to the oscillation path while the normal force is compensated by the string tension, one has:

\[(f)\;F_R = mg\sin\varphi = -ml\,\ddot\varphi.\]

Hence

\[(g)\;\ddot\varphi = \sin\varphi\,\frac{g}{l}.\]

For very small amplitudes, \(\sin\varphi\) may be replaced by \(\varphi\), and the conditions for harmonic oscillation apply. Thus

\[(h)\;\ddot\varphi = \omega^2\varphi\]

and from (g) one obtains

\[(i)\;\omega = \sqrt{\frac{g}{l}}.\]

Therefore,

\[(j)\;D = m\omega^2 = m\frac{g}{l}.\]

For larger amplitudes, the directional parameter must be the product of the harmonic directional parameter and a function value \(f(\varphi_0)\) of the amplitude angle:

\[(k)\;D(\varphi_0) = \left(m\frac{g}{l}\right)f(\varphi_0).\]

However, the function value \(f(\varphi_0)\) is not defined under the harmonic form of the energy theorem, so it must cancel if it is introduced. Writing \(s=xl\), one obtains

\[(l)\;\frac12\frac{\frac{f(\varphi_0)mxg}{xl}(xl)^2}{f(\varphi_0)} = mgh.\]

In this way, the unchanged displacement path \(s\) can still be obtained from the energy theorem if the function value \(f(\varphi_0)\) is assigned to the restoring force \(F_R\). Thus from

\[\frac12\left(m\frac{g}{l}\right)s^2 = mgh\]

one gets

\[(m)\;s = \sqrt{2lh}.\]

The restoring force \(F_R\) must therefore be determined independently of the energy theorem. It is fixed instead by the initial oscillation as the tangentially acting and hence maximally restoring force at the initial height of the oscillation path. From

\[(n)\;F_R = (-)ma,\]

one obtains with (a) and (d):

\[(o)\;T = 2\pi\sqrt{\frac{\sqrt{2lh}}{a}}.\]

Geometrically, the displacement path \(s\) in (m) is the chord of a circle of radius \(l\). Since this circle is the osculating circle of the oscillation path at the equilibrium position, the chord must lead from a point of the circular arc at the initial height \(h\) to the equilibrium position.

The mathematical pendulum is the simplest form of oscillation path, since it coincides with the osculating circle and has a point evolute. It is therefore treated first.

2.1.2. The mathematical pendulum

From Figure 1, one obtains for the displacement path \(s\):

\[s = 2\sin\left(\frac{\varphi_0}{2}\right)l.\]

Using (f), the maximal restoring force is

\[F_R = (-)mg\sin(\varphi_0).\]

Hence, by (a) and (d), the period is

\[T = 2\pi\sqrt{\frac{l}{\cos\left(\frac{\varphi_0}{2}\right)g}}.\]

2.1.3. The cycloidal pendulum

The ordinary cycloid has the parametric equations

\[x = a(\Phi-\sin\Phi), \qquad y = a(\cos\Phi-1).\]

Its radius of curvature is

\[\rho = 4a\sin\left(\frac{\phi}{2}\right).\]

Since the angles \(\Phi\) increase from the initial height \(h\) to the equilibrium position with \(\Phi=\pi\), and after passing the equilibrium position the sine decreases from \(\Phi/2\) towards the turning point, the conditions for calculating the period are fulfilled.

The initial height is

\[h = 2a+y = a(\cos\Phi_0+1).\]

Substituting this into (m), with \(l=4a\), yields

\[s = l\sqrt{\frac{\cos\phi_0+1}{2}} = l\cos\left(\frac{\phi_0}{2}\right).\]

Since the tangential acceleration \(b_t\) acts vertically to the radius of curvature \(\rho\), one obtains

\[b_t = \alpha\cos\left(\frac{\phi_0}{2}\right)g.\]

If \(\Phi_0=0\), then \(b_t=g\), and if \(\Phi_0=\pi\), then \(b_t=0\). Therefore \(\alpha=1\), so that

\[b_t = \cos\left(\frac{\phi_0}{2}\right)g.\]

Using (o), the period is therefore

\[T = 2\pi\sqrt{\frac{l}{g}}.\]

This had already been proved centuries ago by Huygens.

2.1.4. The elliptical pendulum

The ellipse has equation

\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.\]

By differentiating the radius of curvature \(\rho\), one obtains

\[\rho = \frac{(a^4-a^2x^2+b^2x^2)^{3/2}}{a^4b}.\]

From the equilibrium position, where \(x=0\), the coordinate \(x\) increases towards the turning points, and since \(a^2x^2 > b^2x^2\), the radius of curvature decreases continuously. Thus, at the equilibrium position,

\[\rho = l = \frac{a^2}{b}.\]

Writing

\[y = -b\sqrt{1-\frac{x^2}{a^2}},\]

the initial height is

\[h = b+y = b\left(1-\sqrt{1-\frac{x^2}{a^2}}\right).\]

Applying (m), one obtains for the displacement path

\[s = \sqrt{2a\left(a-\sqrt{a^2-x^2}\right)}.\]

Hence the displacement path is independent of the semi-minor axis of the ellipse.

The tangential acceleration is

\[b_t = \cos\beta\,g = \frac{bxg}{\sqrt{a^4-a^2x^2+b^2x^2}}.\]

Applying (o), one obtains for the period

\[T = 2\pi\sqrt{\frac{\sqrt{a^4-a^2x^2+b^2x^2}\,\sqrt{2a\left(a-\sqrt{a^2-x^2}\right)}}{bxg}}.\]

Although this formula is not defined at \(x=0\), for very small \(x\) one has

\[T = 2\pi\sqrt{\frac{a^2}{bg}} = 2\pi\sqrt{\frac{l}{g}}.\]

For \(x=a\), the following statement results:

The period on an ellipse with maximal amplitude is independent of the semi-minor axis.

2.2. Experiment on the mathematical pendulum

For reasons of time, the experimental part was limited to the mathematical pendulum.

Otherwise, not only would the construction of evolutes have been very laborious, but additional sources of error would also have arisen in the cycloidal or elliptical oscillation. One example is the difficulty of making the pendulum follow the evolute exactly, since after only a few oscillations it already leaves the vertical plane. Moreover, impulse and frictional forces arise when the motion begins and in the bearing itself, so that these pendula tend to become mathematical pendula with large amplitudes.

The latter may therefore be regarded as representative of the whole.

2.2.1. Experimental setup

Two one-metre rods are connected to a third rod by four clamps and are passed symmetrically through two ceiling hooks. At the centre a protractor and the support for the pendulum thread of length 1.60 m are additionally attached.

Parallel to this, an air track is placed on the bench in such a way that its centre lies vertically beneath the centre of the protractor. A trolley of mass 0.2 kg is placed at the centre and held by two springs so that, between the clamp at the end of the bench and the trolley, a directional parameter of 2 N/m is obtained.

A stopwatch is placed within easy reach with respect to the initial height of both pendulum and trolley.

The experiment is shown at the beginning; all instruments used are also listed in the appendix.

2.2.2. Experimental procedure and results

First, the independence of the period from the mass of the pendulum was checked by attaching once a sphere of approximately 5 kg and on another occasion the trolley loaded to 1 kg to the pendulum thread. The difference in the measured period amounted, over ten oscillations, to only a few hundredths of a second.

In the simulation of the mathematical pendulum by the harmonic oscillator, the time difference per oscillation amounted to 0.6 s for ten oscillations at a maximal amplitude of 60° of the pendulum.

For the mass of the harmonic oscillator, the formula used was

\[m = \frac{2lD}{\cos\left(\frac{\varphi_0}{2}\right)g}.\]

This can be obtained directly from the periods of the two systems.

Here the directional parameter \(D_0\) for the oscillator is twice the spring constant of one of the two identical springs used. Hence the period of the oscillator is

\[T = 2\pi\sqrt{\frac{m}{2D}}.\]

The calculated period of the mathematical pendulum is

\[T = 2\pi\sqrt{\frac{l}{\cos\left(\frac{\varphi_0}{2}\right)g}}.\]

The damping of the pendulum at amplitude 60° amounted to 1°, whereas the oscillator showed a damping of 0.01 m at an amplitude of 0.9 m.

The appendix contains the measured values for amplitudes from 0° to 90° in Table 1.

2.3. Calculation of errors

Hitherto, the period of the mathematical pendulum was given by the elliptic integral

\[\int_0^{\pi/2}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}\]

with

\[k = \sin\left(\frac{\varphi_0}{2}\right),\]

which yields

\[T = 2\pi\sqrt{\frac{l}{g}\left(1+\left(\frac12\right)^2k^2+\left(\frac{1\cdot3}{2\cdot4}\right)^2k^4+\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^2k^6+\cdots\right)}.\]

For direct comparison with the new period formula

\[T = 2\pi\sqrt{\frac{l}{\cos\left(\frac{\varphi_0}{2}\right)g}},\]

the function value

\[f(\varphi_0) = \sqrt{\frac{1}{\cos\left(\frac{\varphi_0}{2}\right)}}\]

is expanded by means of the series, valid for \(|x|<1\),

\[(1+x)^p = 1+px+\frac{p(p-1)x^2}{1\cdot2}+\frac{p(p-1)(p-2)x^3}{1\cdot2\cdot3}+\cdots\]

and the relation

\[f(\varphi_0)=\frac{1}{\sqrt[4]{1-\sin^2\left(\frac{\varphi_0}{2}\right)}}\]

with respect to

\[x=-\sin^2\left(\frac{\varphi_0}{2}\right)=-k^2\]

to the comparison series

\[1+\left(\frac{k}{2}\right)^2+\left(\frac{1\cdot5}{1\cdot2}\right)^2\left(\frac{k}{2}\right)^4+\left(\frac{1\cdot5\cdot9}{1\cdot2\cdot3}\right)^2\left(\frac{k}{2}\right)^6+\cdots\; .\]

Thus the two series agree up to the first squared term.

If one divides the old series by the new one and subtracts 1, one obtains the series of the relative error, where the former has the smaller modulus:

\[\frac1{64}k^4 + \frac1{64}k^6 + \frac{231}{16384}k^8 + \cdots\; .\]

If one divides conversely, the relative error is larger:

\[\frac1{64}k^4 + \frac1{64}k^6 + \frac{235}{16384}k^8 + \cdots\; .\]

The absolute error is given by subtracting the old series from the new one:

\[\frac1{64}k^4 + \frac5{256}k^6 + \frac{335}{16384}k^8 + \cdots\; .\]

The appendix records the values for selected angles from 0° to 90° for all these series.

2.4. Discussion

Here the proportionalities arising from the formula

\[T = 2\pi\sqrt{\frac{\sqrt{2lh}}{a}}\]

must be considered. First, the displacement path \(s\) is not proportional to the acceleration \(a\) of the restoring force \(F_R\); rather, one must take into account the function value \(f(\varphi_0)\) contained in \(a\).

The length of the oscillation path is adapted via the displacement path. If the initial height \(h\) is very small, the distance to be traversed in the \(y\)-direction becomes smaller, and therefore the period becomes smaller as well.

If the oscillation path is also very flat, the small initial height is opposed by a large pendulum length, so that a kind of mean value for the displacement path results, as can be seen especially clearly for maximal amplitudes of the elliptical pendulum.

For strongly curved oscillation paths and large initial heights, the period is relatively larger than in the mathematical pendulum.

Very close to the equilibrium position, the numerical values of displacement path and acceleration cancel in such a way that the displacement path becomes the pendulum length \(l\) and the acceleration becomes the gravitational acceleration \(g\).

Strictly speaking, the period is not defined exactly at the equilibrium position itself, because there both the initial height \(h\) and the acceleration \(a\) vanish.

The greater the acceleration \(a\), the smaller the period; conversely, if the acceleration is smaller, the period is larger.

These insights could be confirmed fairly well by the experiment on the mathematical pendulum.

3. Conclusion

Looking back, one can say that the actual line of thought was not entirely identical with the one presented here for reasons of clarity.

The problem of the mathematical pendulum was solved first, and only afterwards was a more general solution set up. All periods were calculated without prior knowledge of the corresponding periods found in the literature, which was consulted only several weeks later, when the experiment took place.

The method of calculation could indeed be substantially simplified, and the error margin of 0.01 or about one per cent with respect to the older procedure need not be exceeded.

The elliptical pendulum could not be compared with references, since no corresponding literature could be found.

Most likely, however, the error margin shifts upward, since the mathematical pendulum as a special case of the elliptical pendulum — with equal semi-axes — already deviates by scarcely one per cent at maximal amplitude.

By contrast, the period of the cycloidal pendulum agrees with the references.

3.1. Critical appraisal of the paper

Without doubt, the problem of evolutes is not yet completely resolved. If one no longer presupposes the pendulum length in the equilibrium position, one may again have to resort to infinitesimal calculus and series expansion in order to achieve higher precision.

On the one hand, the mathematical pendulum, and on the other hand the cycloidal pendulum, point in opposite directions. In the first case, a distinct evolute is missing despite deviation; in the second, there is no deviation despite the presence of an evolute.

The new method must therefore be regarded as independent.

Unfortunately, measurements of periods other than that of the mathematical pendulum are lacking in this paper. Such measurements would clarify the situation more precisely. They would, however, impose very high demands on measurement accuracy and on the execution of the experiments.

All in all, the method developed here requires some adjustment in one’s way of thinking, but, given the good results obtained, it should be applicable in physics.

4. Appendix

Figures

All constructions were carried out with the aid of reference (9).

Auxiliary calculations

The general formula for the radius of curvature is

\[\rho = \frac{(\dot x^2+\dot y^2)^{3/2}}{\dot x\ddot y-\dot y\ddot x}.\]

For the ordinary cycloid with parametric equations

\[x = a(\Phi-\sin\Phi), \qquad y = a(\cos\Phi-1),\]

one has

\[(\dot x^2+\dot y^2)^{3/2} = \left(2a^2-2\cos\phi\,a^2\right)^{3/2}\]

and

\[\dot x\ddot y-\dot y\ddot x=(a-\cos\phi\,a)(-\cos\phi\,a)-(-\sin\phi\,a)\sin\phi\,a = a^2-\cos\phi\,a^2.\]

Hence

\[\rho = 2a\sqrt{2(1-\cos\phi)} = 4a\sin\left(\frac{\phi}{2}\right).\]

If, for the ellipse, one temporarily writes \(x=\sin(\varphi)a\) and

\[y=-b\sqrt{1-\frac{x^2}{a^2}} = -\cos\varphi\,b,\]

then

\[\dot x\ddot y-\dot y\ddot x = \cos\varphi\,a(\cos\varphi\,b)-\sin\varphi\,b(-\sin\varphi\,a)=ab.\]

Thus the radius of curvature is

\[\rho = \frac{(\cos^2\varphi\,a^2+\sin^2\varphi\,b^2)^{3/2}}{ab}.\]

Reversing the transformation and multiplying by \(a^3\), one obtains

\[\rho = \frac{(a^4-a^2x^2+b^2x^2)^{3/2}}{a^4b}.\]

The evolute of the ordinary cycloid is again a cycloid with the same \(a\).

For the evolute of the ellipse, however, one has

\[\xi = \frac{(a^2-b^2)\cos^3 t}{a}, \qquad \eta = \frac{(a^2-b^2)\sin^3 t}{b}\]

with \(x=a\cos t\) and \(y=b\sin t\).

To calculate the tangential acceleration \(b_t\) of the ellipse, the following preliminary considerations are useful:

\[\tan\gamma = \frac{\sqrt{a^2-x^2}}{x} = \frac{q}{\sqrt{a^2-x^2}}, \qquad q = \frac{a^2-x^2}{x}.\]

Then

\[\cos\beta = \frac{y}{\sqrt{q^2+y^2}} = \frac{bx}{\sqrt{a^4-a^2x^2+b^2x^2}}.\]

To transform the period formula of the elliptical pendulum into that of the mathematical pendulum, one sets \(x=\sin(\varphi_0)a\) and \(a=b=1\). Then

\[T = 2\pi\sqrt{\frac{a^2\sqrt{2\left(1-\sqrt{1-\frac{x^2}{a^2}}\right)}}{xg}} = 2\pi\sqrt{\frac{l\sqrt{2(1-\cos\varphi_0)}}{\sin\varphi_0\,g}},\]

and hence finally

\[T = 2\pi\sqrt{\frac{l}{\cos\left(\frac{\varphi_0}{2}\right)g}}.\]

Experimental apparatus

as well as various additional weights.

Tables

Table 1

\(T_{exp}\) = experimentally determined period (mean values)
\(T_{thn}\) = theoretical period according to the new method
\(T_{tha}\) = theoretical period according to the old method
\(\Delta T_{an}\) = absolute difference between \(T_{exp}\) and \(T_{thn}\), with \(T_{thn}\) as reference value
\(\Delta T_{aa}\) = absolute difference between \(T_{exp}\) and \(T_{tha}\), with \(T_{tha}\) as reference value

Here, \(l=1.60\,\mathrm{m}\) and \(g=9.80665\,\mathrm{m/s^2}\).

Table 2

\(f(T_0)_n\) = function value of the period of the mathematical pendulum by which the period of the cycloidal pendulum must be multiplied, according to the new method
\(f(T_0)_a\) = the same according to the old method
\(\Delta f(T_0)_{rkl}\) = smaller relative difference of the two
\(\Delta f(T_0)_{rgr}\) = larger relative difference of the two
\(\Delta f(T_0)_a\) = absolute difference of the two

All values \(\Delta f(T_0)_x\) are given in per cent.

5. Bibliography

Postscript

Today I am, of course, in a position to calculate the correct periods and to state the error of my approximation formula.

One has

\[mgh_0 = mgh + \frac12 m\frac{ds^2}{dt^2}.\]

For \(x=x(t)\) and \(y=y(t)\), this yields

\[T = \sqrt{\frac{8}{g}}\int_0^{s_0}\sqrt{\frac{1}{h_0-h}}\,ds = \sqrt{\frac{8}{g}}\int_0^{x_0}\sqrt{\frac{1+(y’)^2}{y_0-y}}\,dx = \sqrt{\frac{8}{g}}\int_0^{t_0}\sqrt{\frac{\dot x^2+\dot y^2}{y_0-y}}\,dt.\]

The approximation formula is

\[T=2\pi\sqrt{\frac{\sqrt{2lh}}{a}} =2\pi\sqrt{\frac{\sqrt{2\left(1+y'(0)^2\right)^{3/2}y_0\left(1+y_0’^2\right)}}{gy_0’\sqrt{y”(0)}}} =2\pi\sqrt{\frac{\sqrt{2\left(\dot x(0)^2+\dot y(0)^2\right)^{3/2}y_0\left(\dot x_0^2+\dot y_0^2\right)}}{g\dot y_0\sqrt{\dot x(0)\ddot y(0)-\dot y(0)\ddot x(0)}}}.\]

For \(x=a\sin\varphi\), \(y=b-b\cos\varphi\),

\[ds = \sqrt{dx^2+dy^2} = \sqrt{a^2\cos^2\varphi+b^2\sin^2\varphi}\,d\varphi,\qquad k=\sin\left(\frac{\varphi_0}{2}\right),\qquad \varepsilon^2 = \frac{a^2-b^2}{a^2}.\]

With \(\varepsilon^2<1\), the approximation formula becomes

\[T = \frac{2\pi a\sqrt[4]{1-\varepsilon^2\sin^2\varphi_0}}{\sqrt{bg\cos\left(\frac{\varphi_0}{2}\right)}}\]

with series expansion

\[T = \frac{2\pi a}{\sqrt{bg}}\left\{1+\frac14 k^2-\varepsilon^2k^2+\frac{5}{32}k^4+\frac34\varepsilon^2k^4-\frac32\varepsilon^4k^4+\mathcal O(k^6)\right\}.\]

For the elliptical pendulum one has exactly

\[T = \sqrt{\frac{8a^2}{bg}}\int_0^{\varphi_0}\sqrt{\frac{1-\varepsilon^2\sin^2\varphi}{\cos\varphi-\cos\varphi_0}}\,d\varphi,\]

hence the series expansion

\[T = \frac{2\pi a}{\sqrt{bg}}\left\{1+\frac14 k^2-\varepsilon^2k^2+\frac{9}{64}k^4+\frac38\varepsilon^2k^4-\frac34\varepsilon^4k^4+\mathcal O(k^6)\right\}.\]

Using the quadratically convergent arithmetic–geometric mean \(M\left(1,\cos\left(\frac{\varphi_0}{2}\right)\right)\), one obtains a still more accurate formula, and in the case of the mathematical pendulum (where \(a=b\), so \(\varepsilon=0\)) an exact one:

\[T=\frac{2\pi a}{\sqrt{bg}}\left\{\frac{1}{M\left(1,\cos\left(\frac{\varphi_0}{2}\right)\right)}-\varepsilon^2k^2+\frac38\left(\varepsilon^2-2\varepsilon^4\right)k^4+\frac{5}{64}\left(\varepsilon^2+12\varepsilon^4-16\varepsilon^6\right)k^6+\mathcal O\big(\varepsilon^2k^8\big)\right\}.\]

With \(e:=\varepsilon^2\), the expression in braces is

\[g(k,e)=\frac{2}{\pi}\int_0^{\pi/2}\sqrt{\frac{1}{1-k^2\sin^2x}-4e\,k^2\sin^2x}\,dx.\]

Upon setting \(t:=k^2\sin^2x\), one obtains immediately

\[g(k,e)=\sum_{n\ge 0} c_{2n}(e)k^{2n},\qquad c_{2n+1}(e)=0.\]

Moreover,

\[\sqrt{\frac{1}{1-t}-4et}=\frac{\sqrt{1-4et+4et^2}}{\sqrt{1-t}} = \left(\sum_{m\ge0} b_m(e)t^m\right)\left(\sum_{j\ge0} a_j t^j\right),\]

where

\[a_j=\frac{\binom{2j}{j}}{4^j}\qquad\text{(the series of }(1-t)^{-1/2}\text{)}\]

and \(b_m(e)\) are the coefficients of the square-root series

\[\sqrt{1-4et+4et^2}=\sum_{m\ge0} b_m(e)t^m.\]

The coefficient of \(t^n\) in the product is

\[A_n(e)=\sum_{j=0}^n a_j\,b_{n-j}(e),\]

and since

\[\int_0^{\pi/2}\sin^{2n}x\,dx=\frac{\pi}{2}\frac{\binom{2n}{n}}{4^n},\]

one obtains the recurrence formula

\[c_{2n}(e)=\frac{\binom{2n}{n}}{4^n}A_n(e)=\frac{\binom{2n}{n}}{4^n}\sum_{j=0}^n \frac{\binom{2j}{j}}{4^j}\,b_{n-j}(e),\qquad c_{2n+1}(e)=0.\]

For \(b_m(e)\) there is the convenient recursion law, obtained from \((\sum b_m t^m)^2=1-4et+4et^2\):

\[b_0=1,\qquad b_m(e)=\frac{r_m-\sum_{p=1}^{m-1} b_p(e)b_{m-p}(e)}{2}\quad(m\ge1),\]

with

\[r_1=-4e,\qquad r_2=4e,\qquad r_m=0\quad(m\ge3).\]

The coefficients \(a_j\) likewise satisfy the simple recursion

\[a_0=1,\qquad a_j=\frac{2j-1}{2j}\,a_{j-1}.\]

Hence the coefficients \(c_{2n}(e)\), and therefore the series for \(g\), can be generated systematically: first compute \(b_m\) by recursion, then \(A_n=\sum a_jb_{n-j}\), and finally \(c_{2n}=\frac{\binom{2n}{n}}{4^n}A_n\).

© 1983-2025 by Boris Haase

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