this gives
|ƒ(x) − {c0μ0 + c1μ1x + ... + cmμmxm}| ⋜ εLM/2π,
where the quantities μ0, μ1, μ2, ... are the coefficients in the expansion of ƒ(x) about the origin.
If then an arbitrary finite region be constructed of the kind explained, excluding the barriers joining the singular points of ƒ(x) to x = ∞, it is possible, corresponding to an arbitrary real positive number σ, to determine a number m, and a polynomial Q(x), of order m, such that for all interior points of this region
|ƒ(x) − Q(x)| < σ.
Hence as before, within this region ƒ(x) can be represented by a series of polynomials, converging uniformly; when ƒ(x) is not a single valued function the series represents one branch of the function.
The same result can be obtained without the use of Cauchy’s integral. We explain briefly the character of the proof. If a monogenic function of t, φ(t) be capable of expression as a power series in t − x about a point x, for |t − x| ⋜ ρ, and for all points of this circle |φ(t)| < g, we know that |φ(n)(x)| < gρ−n(n!). Hence, taking |z| < 1⁄3ρ, and, for any assigned positive integer μ, taking m so that for n > m we have (μ + n)μ < (3⁄2)n, we have
| | | φ(μ + n)(x)·zn | | < | φ(μ + n)(x) | (μ + n)μ |z|n < | g | ( | 3 | ) n ( | ρ | ) n < | g | , |
| n! | (μ + n)! | ρμ + n | 2 | 3 | ρμ 2n |
and therefore
| φμ (x + z) = Σ mn=0 | φ(μ + n) (x) | zn + εμ, |
| n! |