function satisfying this equation; indeed, when r < a, we can write
| 1 | ∫ U | (a2 − r2) | dθ = | 1 | ∫ U [ 1 + 2 | r | cos (θ − φ) + 2 | r2 | cos 2(θ − φ) + ... ] dθ |
| 2π | a2 + r2 − 2ar cos (θ − φ) | 2π | a | a2 |
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
will be developable by a power series in (z − z0) about every point z0 interior to R0, and will vanish at z = c; while on the boundary of R it will be of constant modulus unity. Thus if it be plotted upon a plane of ζ the boundary of R will become a circle of radius unity with centre at ζ=0, this latter point corresponding to z=c. A closed path within R0, passing once round z=c, will lead to a closed path passing once about ζ = 0. Thus every point of the interior of R will give rise to one point of the interior of the circle. The converse is also true, but is more difficult to prove; in fact, the differential coefficient dζ/dz does not vanish for any point interior to R. This being assumed, we obtain a conformal representation of the interior of the region R upon the interior of a circle, in which the arbitrary interior point c of R corresponds to the centre of the circle, and, by utilizing the arbitrary constant arising in determining the function Q, an arbitrary point of the boundary of R corresponds to an arbitrary point of the circumference of the circle.
There thus arises the problem of the determination of a real monogenic potential function, single valued and finite within a given arbitrary region, with an assigned continuous value at all points of the boundary of the region. When the region is circular this problem is solved by the integral 1/π ∫ Udω − 1/π ∫ Udθ previously given. When the region is bounded by the outermost portions of the circumferences of two overlapping circles, it can hence be proved that the problem also has a solution; more generally, consider a finite simply connected region, whose boundary we suppose to consist of a single closed path in the sense previously explained, ABCD; joining A to C by two non-intersecting paths AEC, AFC lying within the region, so that the original region may be supposed to be generated by the overlapping regions AECD, CFAB, of which the common part is AECF; suppose now the problem of determining a single valued finite monogenic potential function for the region AECD with a given continuous boundary value can be solved, and also the same problem for the region CFAB; then it can be shown that the same problem can be solved for the original area. Taking indeed the values assigned for the original perimeter ABCD, assume arbitrarily values for the path AEC, continuous with one another and with the values at A and C; then determine the potential function for the interior of AECD; this will prescribe values for the path CFA which will be continuous at A and C with the values originally proposed for ABC; we can then determine a function for the interior of CFAB with the boundary values so prescribed. This in its turn will give values for the path AEC, so that we can determine a new function for the interior of AECD. With the values which this assumes along CFA we can then again determine a new function for the interior of CFAB. And so on. It can be shown that these functions, so alternately determined, have a limit representing such a potential function as is desired for the interior of the original region ABCD. There cannot be two functions with the given perimeter values, since their difference would be a monogenic potential function with boundary value zero, which can easily be shown to be everywhere zero. At least two other methods have been proposed for the solution of the same problem.
A particular case of the problem is that of the conformal representation of the interior of a closed polygon upon the upper half of the plane of a complex variable t. It can be shown without much difficulty that if a, b, c, ... be real values of t, and α, β, γ, ... be n real numbers, whose sum is n − 2, the integral
as t describes the real axis, describes in the plane of z a polygon of n sides with internal angles equal to απ, βπ, ..., and, a proper sign being given to the integral, points of the upper half of the plane of t give rise to interior points of the polygon. Herein the points a, b, ... of the real axis give rise to the corners of the polygon; the condition Σα = n − 2 ensures merely that the point t = ∞ does not correspond to a corner; if this condition be not regarded, an additional corner and side is introduced in the polygon. Conversely it can be shown that the conformal representation of a polygon upon the half plane can be effected in this way; for a polygon of given position of more than three sides it is necessary for this to determine the positions of all but three of a, b, c, ...; three of them may always be supposed to be at arbitrary positions, such as t = 0, t = 1, t = ∞.
As an illustration consider in the plane of z = x + iy, the portion of the imaginary axis from the origin to z = ih, where h is positive and less than unity; let C be this point z = ih; let BA be of length unity along the positive real axis, B being the origin and A the point z = 1; let DE be of length unity along the negative real axis, D being also the origin and E the point z = − 1; let EFA be a semicircle of radius unity, F being the point z = i. If we put ζ = [(z2 + h2)/(1 + h2z2)]12, with ζ = 1 when z = 1, the function is single valued within the semicircle, in the plane of z, which is slit along the imaginary axis from the origin to z = ih; if we plot the value of ζ upon another plane, as z describes the continuous curve ABCDE, ζ will describe the real axis from ζ = 1 to ζ = − 1, the point C giving ζ = 0, and the points B, D giving the points ζ = ±h. Near z = 0 the expansion of ζ is ζ − h = z2 (1 − h4 / 2h) + ..., or ζ + h = −z2 (1 − h4 / 2h) + ...; in either case an increase of 12π in the phase of z gives an increase of π in the phase of ζ − h or ζ + h. Near z = ih the expansion of ζ is ζ = (z − ih)12 [2ih/(1 − h4)]12 + ..., and an increase of 2π in the phase of z − ih also leads to an increase of π in the phase of ζ. Then as z describes the semicircle EFA, ζ also describes a semicircle of radius unity, the point z = i becoming ζ = i. There is thus a conformal representation of the interior of the slit semicircle in the z-plane, upon the interior of the whole semicircle in the ζ-plane, the function
being single valued in the latter semicircle. By means of a transformation t = (ζ + 1)2 / (ζ − 1)2, the semicircle in the plane of ζ can further be conformably represented upon the upper half of the whole plane of t.
As another illustration we may take the conformal representation of an equilateral triangle upon a half plane. Taking the elliptic function ℜ(u) for which ℜ′2(u) = 4ℜ3(u) − 4, so that, with ε = exp (23πi), we have e1 = 1, e2 = ε2, e3 = ε, the half periods may be taken to be
| 12ω = ∫ ∞1 | dt | , 12ω′ = ∫ ∞e3 | dt | = 12εω; |
| 2(t3 − 1)12 | 2(t3 − 1)12 |
drawing the equilateral triangle whose vertices are O, of argument O, A of argument ω, and B of argument ω + ω′ = −ε2ω, and the equilateral triangle whose angular points are O, B and C, of argument ω′, let E, of argument 13(2ω + ω′), and D, of argument 13(ω + 2ω′), be the centroids of these triangles respectively, and let BE, OE, AE cut OA, AB, BO in K, L, H respectively, and BD, OD, CD cut OC, BC, OB in F, G, H respectively; then if u = ξ + iη be any point of the interior of the triangle OEH and v = εu0 = ε(ξ − iη) be any point of the interior of the triangle OHD, the points respectively of the ten triangles OEK, EKA, EAL, ELB, EBH, DHB, DBG, DGC, DCF, DFO are at once seen to be given by −εv, ω + εu, ω − η2v, ω + ω′ + ε2u, ω + ω′ − v, ω + ω′ − u, ω + ω′ + εv, ω′ − εu, ω′ + ε2v, −ε2u. Further, when u is real, since the term − 2(u + mω + m′ε2ω)−3, which is the conjugate complex of −2(u + mω + m′ε2ω)3, arises in the infinite sum which expresses ℜ′(u), namely as −2(u + μω + μ′εω)−3, where μ = m − m′, μ′ = −m′, it follows that ℜ′(u) is real; in a similar way we prove that ℜ′(u) is pure imaginary when u is pure imaginary, and that ℜ′(u) = ℜ′(εu) = ℜ′(ε2u), as also that for v = εu0, ℜ′(v) is the conjugate complex of ℜ′(u). Hence it follows that the variable
