where
| a0 = | 1 | ∫ Udθ, a1 = | 1 | ∫ | U cosθ | dθ, b1 = | 1 | ∫ | U sinθ | dθ, |
| 2π | π | a | π | a |
| a2 = | 1 | ∫ | U cos 2θ | dθ, b2 = | 1 | ∫ | U sin 2θ | dθ. |
| π | a² | π | a² |
In this series the terms of order n are sums, with real coefficients, of the various integral polynomials of dimension n which satisfy the equation ∂²ψ/∂x² + ∂²ψ/∂y²; the series is thus the real part of a power series in z, and is capable of differentiation and integration within its region of convergence.
Conversely we may suppose a function, P, defined for the interior of a finite region R of the plane of the real variables x, y, capable of expression about any interior point x0, y0 of this region by a power series in x − x0, y − y0, with real coefficients, these various series being obtainable from one of them by continuation. For any region R0 interior to the region specified, the radii of convergence of these power series will then have a lower limit greater than zero, and hence a finite number of these power series suffice to specify the function for all points interior to R0. Each of these series, and therefore the function, will be differentiable; suppose that at all points of R0 the function satisfies the equation
| ∂²P | + | ∂P² | = 0, |
| ∂x² | ∂y² |
we then call it a monogenic potential function. From this, save for an additive constant, there is defined another potential function by means of the equation
| Q = ∫ (x, y) ( | ∂P | dy − | ∂P | dx ). |
| ∂x | ∂y |
The functions P, Q, being given by a finite number of power series, will be single valued in R0, and P + iQ will be a monogenic function of z within R0· In drawing this inference it is supposed that the region R0 is such that every closed path drawn in it is capable of being deformed continuously to a point lying within R0, that is, is simply connected.
Suppose in particular, c being any point interior to R0, that P approaches continuously, as z approaches to the boundary of R, to the value log r, where r is the distance of c to the points of the perimeter of R. Then the function of z expressed by