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Science Forum Index » Physics Forum » Quantum Gravity 279.0: Probable Causation/Influence...
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| OsherD... |
 Posted: Tue Jul 15, 2008 5:19 pm |
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From Osher Doctorow
The definition of Probable Causation/Influence (PI) has the forms:
1) 1 + y - x, y = P(AB), x = P(A); or 1 + y - x, y = P(B) < = P(A), x
= P(A)
The Confluent Hypergeometric Functions PHI(a, c; y) or W(a, c; y)
(respectively Regular or Irregular) immediately have relationships
such as:
2) W(a, c; y) = y^(1-c)W(1 + a - c, 2 - c; y)
3) lim [gamma(a + 1 - c) W(a, c; y/a)] = 2y^(1- c)/2 K_(c-1)(2
sqrt(y)) (limit as a --> infinity)
as Readers can see from Appendix B of Lea J. E. Bartolomeu and D. B.
Figueiredo's (Centro Brasileiro de Pesquisas Fisicas (CBPF) Rio de
Janeiro Brazil: "On certain solutions for confluent and double-
confluent Heun equations," arXiv: today's Math. Physics section (I
omitted writing down the arXiv number, which I'll try to do soon).
The Schrodinger equation with inverse 4th and 6th power potentials
leads respectively to the DCHE form (3) of their paper and its
Whittaker-Ince limit (7), describing intermolecular forces and
scattering of ions by polarisable atoms, while the equations that
govern time-dependence of K-G (Klein-Gordon) and Dirac test-fields in
some nonflat Friedmannian spacetimes are representable as CHEs &
DCHEs. In their Appendix A, they found that the normal forms of Heun
equations are helpful to find out whether given 1-dimensional
Schrodinger equation reduce to Heun equations, which are quasi-exactly
solvable (QES) potentials representative of each of the 5 Heun
equations, which are useful if we know enough about the solutions of
the equation.
The Double-Confluent Heun equation has 2 irregular points at z = 0,
infinity, with:
4) [Dzz + Q(z)]y(z) = 0, Q(z) = A + B/z + C/z^2 + D/z^3 + E/z^4, 1st
normal or Schrodinger form
Every second order ODE in the complex plane or on Riemann sphere with
4 regular singular points can be transformed into the Heun Equation
with 4 regular singular points at 0, 1, d, infinity:
5) [Dzz(w) + g/z + delta/(z - 1) + epsilon/(z - d)]Dz(w) + [abz - q]w/
[z(z-1)(z - d)] = 0
with epsilon = a + b - g - delta + 1 needed to remove singularity at
point at infinity (see "Heun's equation" Wikipedia.
An example of the Biconfluent Heun equation is the scenario with the
sextic potential V1z^6 + V2z^4 + V3z^2 + V4 + V5/z^2, and an example
of (4) has all z exponents replaced by z^(-k) with k = 0,1, 2, 3, 4 in
the different terms.
Osher Doctorow |
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