μn being a properly chosen positive integer.
If it should happen that the points (c) determine a path dividing the plane into separated regions, as, for instance, if an = R(1 − n−1) exp (iπ √2·n), when (c) consists of the points of the circle |z| = R, the product expression above denotes different monogenic functions in the different regions, not continuable into one another.
§ 9. Construction of a Monogenic Function with a given Region of Existence.—A series of isolated points interior to a given region can be constructed in infinitely many ways whose limiting points are the boundary points of the region, or are boundary points of the region of such denseness that one of them is found in the neighbourhood of every point of the boundary, however small. Then the application of the last enunciated theorem gives rise to a function having no singularities in the interior of the region, but having a singularity in a boundary point in every small neighbourhood of every boundary point; this function has the given region as region of existence.
§ 10. Expression of a Monogenic Function by means of Rational Functions in a given Region.—Suppose that we have a region R0 of the plane, as previously explained, for all the interior or boundary points of which z is finite, and let its boundary points, consisting of one or more closed polygonal paths, no two of which have a point in common, be called C0. Further suppose that all the points of this region, including the boundary points, are interior points of another region R, whose boundary is denoted by C. Let z be restricted to be within or upon the boundary of C0; let a, b, ... be finite points upon C or outside R. Then when b is near enough to a, the fraction (a − b)/(z − b) is arbitrarily small for all positions of z; say
| | | a − b | | < ε, for |a − b| < η; |
| z − b |
the rational function of the complex variable t,
| 1 | [ 1 − ( | a − b | )n ], |
| t − a | t − a |
in which n is a positive integer, is not infinite at t = a, but has a pole at t = b. By taking n large enough, the value of this function, for all positions z of t belonging to R0, differs as little as may be desired from (t − a)−1. By taking a sum of terms such as
| F = Σ Ap { | 1 | [ 1 − ( | a − b | ) n ] } p, |
| t − a | t − b |
we can thus build a rational function differing, in value, in R0, as little as may be desired from a given rational function