A simple application of elliptic integrals of the second kind, good for the building of confidence
Finding the lengths of circular arcs is beautifully simple. The length is the product of the radius of the arc and the central angle, s = rθ, provided the angle is measured in radians (180° = π radians). The circumference of a circle is then 2πr or πd, where d = 2r is the diameter. The number π = 3.14159... is defined by this relation, and is a mysterious property of nature. It is not a rational number, meaning that it cannot be expressed exactly as the ratio of two integers, a rational number, but it can be approximated as closely as desired by a rational number. Archimedes used the approximation 22/7 = 3.1429..., which was commonly accepted in the Roman world, and is still an easily-remembered value, less than 1.2% in error. There are even better rational approximations with the inconvenience of a larger denominator, but this value is adequate for all normal daily purposes. It was suspected that π was irrational in Archimedes' time, but this is so difficult to prove that a proof was not given until 1767, by J. H. Lambert. π is also transcendental, meaning that it is not the solution of any equation with rational coefficients, which is even worse than being irrational.
The ellipse presents a more difficult, but still simple and straightforward, problem. Let us consider the ellipse defined by (x/a)2 + (y/b)2 = 1, with a > b. The major axis, along the x-axis, is "a" and the minor axis, along the y-axis, is "b." The focal distance c = √(a2 - b2), and the eccentricity is e = c/a, and 0 ≤ e ≤ 1. An eccentricity e = 0 gives a circle, while for e → 1, the ellipse approaches a line of length 2a along the x-axis. The ellipse consists of four equal arcs, the arc in the first quadrant reflected by the axes into the other quadrants.
The length s of a curve given by y = f(x) is s = ∫ ds, where ds =(dx2 + dy2)1/2, or ds = dx(1 + y'2), where y' = dy/dx. Therefore, the length of arc between x = x1 and x = x2 is s = ∫ √(1 + y'2) dx between these limits. This is an easy and familiar application of the differential and integral calculus.
The equation for the ellipse can be solved for y, y = (b/a)(a2 - x2)1/2, and this can easily be differentiated (by the rules for powers) to find y' = (-bx/a)/√(a2 - x2). This is squared and put into the integral. Now the substitution w = (x/a) results in s = a ∫ [√(1 - e2w2)/√(1 - w2)] dw, between the limits 0 and x/a. This is the result for the arc length from x = 0 to x/a in the first quadrant, beginning at the point (0,b) on the y-axis. We have the whole arc for x/a = 1, and four times this is the circumference of the ellipse.
This does not appear to be a particularly difficult integral, but all attempts to express it in terms of elementary functions (rational, exponential, logarithmic, trigonometric) fail. It must be considered as defining a new function, the elliptic integral of the second kind. Often we express it in a different form using the substitution w = sin φ, where φ is called the amplitude, and k = e the modulus of the function E(φ,k). In this case, s = a ∫ √(1 - e2sin2 φ)dφ between the limits 0 and sin-1 (x/a). The whole arc corresponds to the upper limit π/2, and E(π/2,e) = E(e), the complete elliptic integral of the second kind. All this is really painless, when you consider it. We found an integral for the arc length, defined a new function, and expressed the arc length in terms of this new function. The reason for the name "elliptic" is now clear.
By putting in the values e = 0 and e = 1, we find by easy integrations that the limits of the function are E(0) = π/2 and E(1) = 1. This gives the circumference of the circle as s = 2πr, and of the degenerate e = 1 case as 4a, both correct. Tables of the complete elliptic integral are easy to find, for example in Dwight (see References). The easiest way to find values is by using a computer program that you can write in C. Press, et. al., give programs that work excellently. The first edition used the Bulirsch algorithms, but the second edition employs Carlson's canonical forms for the integrals which are very easy to evaluate. A program like this one makes using elliptic integrals in practical cases very easy.
This elementary application should make the reader more comfortable with elliptic integrals, if they have appeared a bit frightening. They are tools for the numerical evaluation of integrals, of practical importance at least. It happens that the inverse functions, also called elliptic functions, are of great theoretical interest in complex variables, where they have many interesting properties, among them the curiosity of being doubly periodic, and of reducing to the trigonometrical functions in limiting cases. For more information on this, see Whittaker and Watson.
H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (New York: The Macmillan Co., 1961), Chapter 9 and pp 320-322. It is a disgrace that this excellent hand reference is no longer in print. There are alternatives, but none of them as good.
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge: Cambridge University Press, 1992). Far more than just recipes--a rich toolchest for numerical work that is indispensable to the scientific computer.
R. Courant, Differential and Integral Calculus, 2nd ed. (London: Blackie and Son, 1937), pp 242-244 and p 289.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge: Cambridge University Press, 1958), Chapter XXII. This is rather daunting theory, so only approach it when feeling lucky.
Composed by J. B. Calvert
Last revised 20 July 2008