Integrating the inverse of the square of a cubic polynomial


The required integral has the form:

First, consider a cubic polynomial of the form:

then according to the methodology detailed in [ 42 ] define the following variables :

Now, three cases must be considered :

  1. y 2 N ˆ’ h 2 > 0;

  2. y 2 N ˆ’ h 2 = 0;

  3. y 2 N ˆ’ h 2 < 0.

y 2 N ˆ’ h 2 > 0 . Here, the cubic polynomial has only one real root, given by:

Hence:

where ¼ = + x 1 and = .

Applying this to (A.2.) the integral takes the form:

where:

Further:

where:

and so:

where · = ½ ¼ and = .

y 2 N ˆ’ h 2 = 0 . For this case there are three real roots, with a repeated root. Again from the methodology in [ 42 ], the three roots are:

Substituting into (A.2.) we have:

where:

y 2 N ˆ’ h 2 < 0 . Here we have three distinct real roots. These roots are:

where [7] :

Again, substituting into (A.2.), we evaluate the integral as:

where:

and

[7] As defined in (A.3.d), h = 2 d 3 .




Interest Rate Modelling
Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net