  A powerful approach to solve the Coulombic quantum three-body problem is
proposed. The approach is exponentially convergent and more efficient than the
Hyperspherical Coordinate(HC) method and the Correlation Function
Hyperspherical Harmonic(CFHH) method. This approach is numerically competitive
with the variational methods, such as that using the Hylleraas-type basis
functions. Numerical comparisons are made to demonstrate them, by calculating
the non-relativistic and infinite-nuclear-mass limit of the ground state energy
of the helium atom. The exponentially convergency of this approach is due to
the full matching between the analytical structure of the basis functions that
I use and the true wave function. This full matching was not reached by almost
any other methods. For example, the variational method using the Hylleraas-type
basis does not reflects the logarithmic singularity of the true wave function
at the origin as predicted by Bartlett and Fock. Two important approaches are
proposed in this work to reach this full matching: the coordinate
transformation method and the asymptotic series method. Besides these, this
work makes use of the least square method to substitute complicated numerical
integrations in solving the Schr\"{o}dinger equation, without much loss of
accuracy; this method is routinely used by people to fit a theoretical curve
with discrete experimental data, but I use it here to simplify the computation.
