4. METHOD OF SOLUTION
For (1+x**2) > 20 Beta**2, the calculation is based upon the asymptotic expansion:
Psi+(i*Phi) = 1/(1-ix)*(1-t+3t**2-3.5t**3+3.5+7t**4---)
with: t = 1/(2z**2) z = (1-ix)/Beta
The half-plane (Beta,x) is split in several parts, and use is made of PADE approximants.
For 1 + x**2 < 20 Beta**2, the calculation is based upon the relation with the erf function:
Psi + i*Phi = SQRT(Pi)/Beta*(e**(z**2))*(1-erf(z))
(z = (1-ix)/Beta, and erf(z) being calculated from its analytic expansion:
erf(z) = 2/SQRT(Pi)*z*e**(-z**2)*(1+z**2/3+z**4/(3*5)
PADE approximants are used to compute the expansion and e**z**2.