| F(z) − A1ζ (z − a) − A2ℜ (z − a) − ... + | Am+1 | (−1)m ℜm−1 (z − a) |
| m! |
will not be infinite at z = a. Adding to this a sum of further terms of the same form, one for each of the poles in a parallelogram of periods, we obtain, since the sum of the residues A is zero, a doubly periodic function without poles, that is, a constant; this gives the expression of F(z) referred to. The indefinite integral ∫F(z)dz can then be expressed in terms of z, functions ℜ(z − a) and their differential coefficients, functions ζ(z − a) and functions logσ(z − a).
§ 15. Potential Functions. Conformal Representation in General.—Consider a circle of radius a lying within the region of existence of a single valued monogenic function, u + iv, of the complex variable z, = x + iy, the origin z = 0 being the centre of this circle. If z = rE(iφ) = r(cosφ + i sinφ) be an internal point of this circle we have
| u + iv = | 1 | ∫ | (U + iV) | dt, |
| 2πi | t − z |
where U + iV is the value of the function at a point of the circumference and t = aE(iθ); this is the same as
| u + iv = | 1 | ∫ | (U + iV) [1 − (r/a) E (iθ − iφ)] | dθ. |
| 2π | 1 + (r/a)² − 2(r/a) cos (θ − φ) |
If in the above formula we replace z by the external point (a²/r) E(iφ) the corresponding contour integral will vanish, so that also
| 0 = | 1 | ∫ | (U + iV) [(r/a)² − (r/a) E (iθ − iφ)] | dθ; |
| 2π | 1 + (r/a)² − 2(r/a) cos (θ − φ) |
hence by subtraction we have
| u = | 1 | ∫ | U(a² − r²) | dθ, |
| 2π | a² + r² − 2ar cos (θ − φ) |