Class PowerLawSingularity

  • All Implemented Interfaces:
    SubstitutionRule

    public class PowerLawSingularity
    extends Object
    implements SubstitutionRule
    This transformation is good for an integral which diverges at one of the end points.

    For singularity at the lower limit, we have \((x-a)^{-\gamma}\) diverging near \(x = a\), \(0 \leq \gamma < 1\). The substitution rule is \[ \int_{a}^{b}f(x)dx = \int_{0}^{(b-a)^{1-\gamma}}\frac{t^{\frac{\gamma}{1-\gamma}}f(t^\frac{1}{1-\gamma}+a)}{1-\gamma}dt, b > a \]

    For singularity at the upper limit, we have \((x-b)^{-\gamma}\) diverging near \(x = b\), \(0 \leq \gamma < 1\). \[ \int_{a}^{b}f(x)dx = \int_{0}^{(b-a)^{1-\gamma}}\frac{t^{\frac{\gamma}{1-\gamma}}f(b-t^\frac{1}{1-\gamma})}{1-\gamma}dt, b > a \] A common case is when \(\gamma = 0.5\).