§ 12. Expansion of a Monogenic Function in Polynomials, over a Star Region.—Now consider any monogenic function ƒ(z) of which the origin is not a singular point; joining the origin to any singular point by a straight line, let the part of this straight line, produced beyond the singular point, lying between the singular point and z = ∞, be regarded as a barrier in the plane, the portion of this straight line from the origin to the singular point being erased. Consider next any finite region of the plane, whose boundary points constitute a path of integration, in a sense previously explained, of which every point is at a finite distance greater than zero from each of the barriers before explained; we suppose this region to be such that any line joining the origin to a boundary point, when produced, does not meet the boundary again. For every point x in this region R we can then write
| 2πiƒ(x) = ∫ | ƒ(t) | ƒ(t) | , | |
| t | 1 − xt−1 |
where ƒ(x) represents a monogenic branch of the function, in case it be not everywhere single valued, and t is on the boundary of the region. Describe now another region R0 lying entirely within R, and let x be restricted to be within R0 or upon its boundary; then for any point t on the boundary of R, the points z of the plane for which zt− 1 is real and positive and equal to or greater than 1, being points for which |z| = |t| or |z| > |t|, are without the region R0, and not infinitely near to its boundary points. Taking then an arbitrary real positive ε we can determine a polynomial in xt− 1, say P(xt−1), such that for all points x in R0 we have
|(1 − xt−1)−1 − P(xt−1)| < ε;
the form of this polynomial may be taken the same for all points t on the boundary of R, and hence, if E be a proper variable quantity of modulus not greater than ε,
| | 2πiƒ(x) − ∫ | dt | ƒ(t)P(xt−1) | = | ∫ | dt | ƒ(t)E | ⋜ εLM, |
| t | t |
where L is the length of the path of integration, the boundary of R, and M is a real positive quantity such that upon this boundary |t−1 ƒ(t)| < M. If now
P (xt−1) = c0 + c1xt−1 + ... + cm xm t−m,
and
| 1 | ∫ t−r−1 ƒ(t)dt = μr, |
| 2πi |