Elliptic Integration in Maple
Integrals of the form ∫abaxbx yx ⅆx, with ax, bx, and yx polynomials, often return answers in closed form. For example, when yx has degree 1 or 2, the integral can always be returned in terms of elementary functions, such as logs, exponentials, trig, or arctrig functions. Thus, you have:
Note that the previous example evaluated the integral in an equation with the inert form on the left-hand side in order to see the integral "prettyprinted."
When y(x) has degree 3 or 4, the integral is called an elliptic integral. Only sometimes do these return answers in terms of elementary functions. In these cases, they are called pseudo-elliptic integrals.
However, in most cases, elliptic integrals return answers in terms of nonelementary functions, for example, the Legendre elliptic functions. A small variation of the previous problem gives:
Elliptic integrals appear in a variety of applications (some mentioned in the classical text by Byrd and Friedmann). For example, the length of an arc of a hyperbola x2α2−y2β2=1 measured from the vertex to any point x,y is determined by the integral
For example, when α=2 and β=1, the length from the vertex to the point 2 5,2 is given by
with the result given in terms of Elliptic functions. Numerically, this is approximated by
Definitions of F, E, Pi
The following equations were constructed to show Maple definitions of the Legendre F, E, and Pi functions.
with the variable k in 0,1. For x=1, you obtain the complete forms of these integrals. For example,
For more information, see EllipticE, EllipticF, EllipticK, and EllipticPi.
Trigonometric Forms of Elliptic Integrals
Integrals of the form ∫0θRsint,cost ysint,cost ⅆt where R is a rational function of sint and cost and y is a degree 2 polynomial in sint and cost are elliptic integrals in trigonometric form. Corresponding forms for hyperbolic trig expressions (as above) also exist. Such integrals can also be expressed in terms of Legendre's elliptic functions. For example,
In both cases you can compare the results numerically:
Why Reduce to Normal Form
Because of the possible presence of poles in the integrand, elliptic integrals are often very difficult to compute numerically with standard numerical integration methods. However, the Legendre functions can be numerically evaluated very efficiently using the AGM method. Therefore, reducing to normal form allows for easy numerical evaluation. For example,
which can be (quickly) numerically approximated by
Of course, one can always compute the numerical approximation directly by using
However, having a formula for an answer allows one to also compute to higher digits by using the same form:
which can also be done directly with evalf. However, in this case, the numerical integrator takes considerable time to come up with a result:
Why Use RootOf
and consider the problem of integrating
The classical technique of reducing such integrals to Legendre normal form involves working with the poles of the rational function, converting the rational function to partial fractions, and reducing each "smaller" part. However, the poles of ab2 and ab3 are seen to be
so all the poles involve nested radicals. Numerically, at the current setting of Digits, these are given by:
Converting say ab3 to a complete partial fraction decomposition can be accomplished by
again an expression that involves complicated nested radicals. The elliptic integration algorithm in Maple takes special care to avoid working with unnecessary radicals whenever possible. Integrals are first reduced to ones involving no repeated poles by using a classical reduction technique discovered by Hermite in the last century which introduces no new radicals during the reduction. Thus, you obtain
In addition, whenever possible, the elliptic integration algorithm in Maple works with an implicit rather than with an explicit representation of such roots. After Hermite reduction to eliminate multiple poles, a final answer is reduced to normal form by using the sum over roots functionality available in Maple. Thus, you obtain
You can evaluate this numerically by