(notes by Roberto Bigoni)

The ellipse *ε* with semiaxes *a* and *b*
(*a* > 0; *b* > 0; *a* ≥ *b*)
in a rectangular planar Cartesian coordinate system *Oxy* with origin *O*
at its center of symmetry and cartesian axes superimposed on its axes, has equation

The equation of the arc of *ε* contained in the first quadrant is

then

The length *dl* of an infinitesimal segment of this arc is given by

Since

where *e* is the eccentricity of *ε*, we we have

The length *q* of the arc is given by the integral

If we let

we obtain

Ultimately, the length of *ε* is expressed by

The integral at the right side of (1.3) is said **complete elliptic integral of second kind**:

With a further change of variable we can simplify its expression. In fact if we set we obtain

The integrand can be expanded as a power of a binomial

so

Since for *j* ≥ 1,

where the double exclamation mark represents the double factorial of its argument, we obtain

From (1.4) e (2.4) the length of the ellipse with semi-major axis *a* and eccentricity *e* is

Similarly we can get the series expansion of the integral

said **complete elliptic integral of first kind**.

The binomial series expansion of the integrand gives

so

Since

we have

By a Landen's transformation the complete elliptic integral of first kind can be expressed in terms of the arithmetic-geometric_mean

Using this identity we can approximate the value of *K(e)* with a quite fast and accurate algorithm.
Here we show a Javascript code that calculates also *E(e)*. The JavaScript function is derived from a page of
A. C. M. de Queiroz.

function ellipticInt() { var a, b, a1, b1, amb, E, i, k, kk, IK, IE; k = parseFloat(document.getElementById("input_e").value); kk = k*k; a = 1; b = Math.sqrt(1-kk); E = 1-kk/2; i = 1; do { a1 = (a+b)/2; b1 = Math.sqrt(a*b); amb = a-b; E -= i*amb*amb/4; i *= 2; a = a1; b = b1; } while (Math.abs(a-b)>1e-15); IK = Math.PI/(2*a); IE = E*IK; document.getElementById("agm").value = a; document.getElementById("intK").value = IK; document.getElementById("intE").value = IE; }

To calculate *K(e)* and *E(e)* *Mathematica (Wolfram)* has the functions
**EllipticK[m]** e **EllipticE[m]**; due to different notation, the argument *m* of these functions must be the square of
*e*. Try WolframAlpha.

An **ideal pendulum** is a physical system that consists of a point mass *m* at one end of an
inextensible and massless bar with length *l*. The bar can rotate without friction around the other end O.

The system is located in a field of constant gravitational acceleration with intensity *g* and its
gravitational potential energy is assumed to be 0 when the mass is vertically below O.

Initially the bar is at rest and forms an angle *θ _{0}* with respect to the vertical through O.

and its total energy is

When the bar is released, the mass begins to fall and *θ* to diminish. During the fall
the potential energy is

and the kinetic energy is

Since the mass moves on a circular path

therefore

Applying the principle of conservation of mechanical energy we get

e simplifying

From (5.1) we get

If *T* is the period of the pendulum, ie the duration of one complete swing, the duration of the descent
from *θ _{0}* to 0 is

If we set

we obtain

Moreover

Therefore from (5.2) we have

The integral in in (5.3) is a complete elliptic integral of first kind and from (3.2) has value

In conclusion, the period of the pendulum is

If *θ _{0}* is very small, the powers of the sine of its half are physically negligible
because they are below the sensitivity of the instruments and the (5.4) simply becomes

last revision: May 2018