ƒ = Σ Ap(t − a)−p,
and differing, outside R or upon the boundary of R, from ƒ, in the fact that while ƒ is infinite at t = a, F is infinite only at t = b. By a succession of steps of this kind we thus have the theorem that, given a rational function of t whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in R0 arbitrarily little from the former, whose poles are all at the point c.
Now any monogenic function ƒ(t) whose region of definition includes C and the interior of R can be represented at all points z in R0 by
| ƒ(z) = | 1 | ∫ | ƒ(t)dt | , |
| 2πi | t − z |
where the path of integration is C. This integral is the limit of a sum
| S = | 1 | Σ | ƒ(ti) (ti+1 − ti) | , |
| 2πi | ti − z |
where the points ti are upon C; and the proof we have given of the existence of the limit shows that the sum S converges to ƒ(z) uniformly in regard to z, when z is in R0, so that we can suppose, when the subdivision of C into intervals ti+1 − ti, has been carried sufficiently far, that
|S − ƒ(z)| < ε,
for all points z of R0, where ε is arbitrary and agreed upon beforehand. The function S is, however, a rational function of z with poles upon C, that is external to R0. We can thus find a rational function differing arbitrarily little from S, and therefore arbitrarily little from ƒ(z), for all points z of R0, with poles at arbitrary positions outside R0 which can be reached by finite continuous curves lying outside R from the points of C.
In particular, to take the simplest case, if C0, C be simple closed polygons, and Γ be a path to which C approximates by taking the number of sides of C continually greater, we can find a rational function differing arbitrarily little from ƒ(z) for all points of R0 whose poles are at one finite point c external to Γ. By a transformation of the form t − c = r−1, with the appropriate change in the rational function, we can suppose this point c to be at infinity, in which case the rational function becomes a polynomial. Suppose ε1, ε2, ... to be an indefinitely continued sequence of real positive numbers, converging to zero, and Pr to be the polynomial such that, within C0, |Pr − ƒ(z)| < εr; then the infinite series of polynomials