reduction of elliptic integrals to standard form
Any integral of the form , where is a rational function and is a polynomial of degree 3 or 4 can be expressed as a linear combination of elementary functions and elliptic integrals of the first, second, and third kinds.
To begin, we will assume that has no repeated roots. Were this not the case, we could simply pull the repeated factor out of the radical and be left with a polynomial of degree of 1 or 2 inside the square root and express the integral in terms of inverse trigonometric functions.
Make a change of variables . By choosing the coefficients suitably, one can cast P into either Jacobi’s normal form or Weierstrass’ normal form .
for suitable polynomials . We can rationalize the denominator like so:
The rational functions and appearing in the foregoing equation are defined like so:
Since may be expressed in terms of elementary functions, we shall focus our attention on the remaining piece, , which we shall write as , where .. Because we may decompose into partial fractions, it suffices to consider the following cases, which we shall all and :
These identities will allow us to express ’s and ’s with large in terms of ones with smaller ’s.
At this point, it is convenient to employ the specific form of the polynominal . We will first conside the Weierstrass normal form and then the Jacobi normal form.
Substituting into our identities and collecting terms, we find
Note that there are some cases which can be integrated in elementary terms. Namely, suppose that the power is odd:
Then we may make a change of variables to obtain
which may be integrated using elementary functions.
Next, we derive some identities using integration by parts. Since
By colecting terms, this identity may be rewritten as follows:
By repeated use of this identity, we may express any integral of the form as the sum of a linear combination of and and the product of a polyomial and .
Likewise, we can use integration by parts to simplify integrals of the form
Will finish later — saving in case of computer crash.
|Title||reduction of elliptic integrals to standard form|
|Date of creation||2014-02-01 18:13:38|
|Last modified on||2014-02-01 18:13:38|
|Last modified by||rspuzio (6075)|