This document shows that the denominator of the definition of shift_phi
in the code (p2*Sigma) is equal to the denominator of equation (11) in
Phys. Rev., D54, 1403­1416

Lines starting with '>' are commands for sagemath, line starting with
>> are sagemath output, everything else is comments.


Variable definition:
> Sigma,a,r,m,R,Delta,st2,ct2=var('Sigma,a,r,m,R,Delta,st2,ct2')

This is p2*Sigma from the code, st2 being sin^2(theta) = rho^2/R^2:

> code_denom=(a^2+r^2)*Sigma+2*m*a^2*r*st2

Now substitute Sigma, with ct2 being cos^2(theta) = z^2/R^2:

> code_denom=code_denom.substitute(Sigma=r^2+a^2*ct2)

and use that st2+ct2=1:

> code_denom=code_denom.substitute(ct2=1-st2)

Now look at the denominator in the paper:

> paper_denom=(r^2+a^2)^2-a^2*st2*Delta

Substitute Delta:

> paper_denom=paper_denom.substitute(Delta=r^2-2*m*r+a^2)

And look at the difference between code_denom and paper_denom:

> (code_denom-paper_denom).expand()
>> 0

qed

And here is the notebook as textblock:

sage: Sigma,a,r,m,R,Delta,st2,ct2=var('Sigma,a,r,m,R,Delta,st2,ct2')
sage: code_denom=(a^2+r^2)*Sigma+2*m*a^2*r*st2
sage: code_denom=code_denom.substitute(Sigma=r^2+a^2*ct2)
sage: code_denom=code_denom.substitute(ct2=1-st2)
sage: paper_denom=(r^2+a^2)^2-a^2*st2*Delta
sage: paper_denom=paper_denom.substitute(Delta=r^2-2*m*r+a^2)
sage: (code_denom-paper_denom).expand()
0