means [ƒ(z)]2. This may be verified directly by showing, if R denote the right side of the equation, that ∂R/∂z = ∂R/∂t; this will require the use of the differential equation
and in fact we find
| ( | ∂2 | − | ∂2 | ) log [ƒ(z) + ƒ(t)] = ƒ2(z) − ƒ2(t) = ( | ∂2 | − | ∂2 | ) log [ƒ(z) − ƒ(t)]; |
| ∂z2 | dt2 | ∂z2 | dt2 |
hence it will follow that R is a function of z + t, and R is at once seen to reduce to ƒ(z) when t = 0. From this the addition equation for each of the functions sn(u), cn(u), dn(u) can be deduced at once; if s1, c1, d1, s2, c2, d2 denote respectively sn(u1), cn(u1), dn(u1), sn(u2), cn(u2), dn(u2), they can be put into the forms
| sn(u1 + u2) = (s1c2d2 + s2c1d1) / D, | cn(u1 + u2) = (c1c2 − s1s2d1d2) / D, | dn(u1 + u2) = (d1d2 − k2s1s2c1c2) / D, |
where
The introduction of the function ƒ1(z) is equivalent to the introduction of the function ℜ(z; ω, 2ω′) constructed from the periods ω, 2ω′ as was ℜ(z) from ω and ω′; denoting this function by ℜ1(z) and its differential coefficient by ℜ′1(z), we have in fact
| ƒ1(z) = 12 | ℜ′1(z) |
| ℜ1(ω′) − ℜ1(z) |
as we see at once by considering the zeros and poles and the limit of zƒ1(z) when z = 0. In terms of the function ℜ1(z) the original function ℜ(z) is expressed by
as a consideration of the poles and expansion near z = 0 will show.
A function having ω, ω′ for periods, with poles at two arbitrary points a, b and zeros at a′, b′, where a′ + b′ = a + b save for an expression mω + m′ω′, in which m, m′ are integers, is a constant multiple of
| {ℜ [z − 12(a′ + b′)] − ℜ [a′ − 12(a′ + b′)]} / {ℜ [z − 12(a + b)] − ℜ [a − 12(a + b)]}; |
if the expansion of this function near z = a be
the expansion near z = b is
as we see by remarking that if z′ − b = −(z − a) the function has the same value at z and z′; hence the differential equation satisfied by the function is easily calculated in terms of the coefficients in the expansions.
From the function ℜ(z) we can obtain another function, termed the Zeta-function; it is usually denoted by ζ(z), and defined by
| ζ(z) − | 1 | = ∫ π0 [ | 1 | − ℜ(z) ] dz = Σ′ ( | 1 | + | 1 | + | z | ), |
| z | z2 | z − Ω | Ω | Ω2 |
for which as before we have equations
| ζ(z + ω) = ζ(z) + 2πiη, ζ(z + ω′) = ζ(z) + 2πiη′, |
where 2η, 2η′ are certain constants, which in this case do not both vanish, since else ζ(z) would be a doubly periodic function with only one pole of the first order. By considering the integral
round the perimeter of a parallelogram of sides ω, ω′ containing z = 0 in its interior, we find ηω′ − η′ω = 1, so that neither of η, η′ is zero. We have ζ′(z) =−ℜ(z). From ζ(z) by means of the equation
| σ(z) | = exp { ∫ z0 [ ζ(x) − | 1 | ] dz } = Π′ [ ( 1 − | z | ) exp ( | z | + | z2 | ) ], |
| z | z | Ω | Ω | 2Ω2 |
we determine an integral function σ(z), termed the Sigma-function, having a zero of the first order at each of the points z = Ω; it can be seen to satisfy the equations
| σ(z + ω) | = −exp [2πiη(z + 12ω)], | σ(z + ω′) | = −exp [2πiη′ (z + 12ω′)]. |
| σ(z) | σ(z) |
By means of these equations, if a1 + a2 + ... + am = a′1 + a′2 + ... + a′m, it is readily shown that
| σ(z − a′1) σ(z − a′2) ... σ(z − a′m) |
| σ(z − a1) σ(z − a2) ... σ(z − am) |
is a doubly periodic function having a1, ... am as its simple poles, and a′1, ... a′m as its simple zeros. Thus the function σ(z) has the important property of enabling us to write any meromorphic doubly periodic function as a product of factors each having one zero in the parallelogram of periods; these form a generalization of the simple factors, z − a, which have the same utility for rational functions of z. We have ζ(z) = σ′(z)/σ(z).
The functions ζ(z), ℜ(z) may be used to write any meromorphic doubly periodic function F(z) as a sum of terms having each only one pole; for if in the expansion of F(z) near a pole z = a the terms with negative powers of z − a be
then the difference
| 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) |
and a corresponding formula for v in terms of V. If O be the centre of the circle, Q be the interior point z, P the point aE(iθ) of the circumference, and ω the angle which QP makes with OQ produced, this integral is at once found to be the same as
| u = | 1 | ∫ Udω − | 1 | ∫ Udθ |
| π | 2π |
of which the second part does not depend upon the position of z, and the equivalence of the integrals holds for every arc of integration.
Conversely, let U be any continuous real function on the circumference, U0 being the value of it at a point P0 of the circumference, and describe a small circle with centre at P0 cutting the given circle in A and B, so that for all points P of the arc AP0B we have |U − U0| < ε, where ε is a given small real quantity. Describe a further circle, centre P0 within the former, cutting the given circle in A′ and B′, and let Q be restricted to lie in the small space bounded by the arc A′P0B′ and this second circle; then for all positions of P upon the greater arc AB of the original circle QP2 is greater than a definite finite quantity which is not zero, say QP2 > D2. Consider now the integral
| u′ = | 1 | ∫ U | (a2 − r2) | dθ = | 1 | ∫ Udω − | 1 | ∫ Udθ, |
| 2π | a2 + r2 − 2ar cos (θ − φ) | π | 2π |
which we evaluate as the sum of two, respectively along the small arc AP0B and the greater arc AB. It is easy to verify that, for the whole circumference,
| U0 = | 1 | ∫ U0 | a2 − r2 | dθ = | 1 | ∫ U0 dω − | 1 | ∫ U0 dθ. |
| 2π | a2 + r2 − 2ar cos (θ − φ) | π | 2π |
Hence we can write
| u′ − U0 = | 1 | ∫ AP0B (U − U0)dω − | 1 | ∫ AP0B (U − U0)dθ + | 1 | ∫ AB (U − U0) | (a2 − r2) | dθ. |
| 2π | 2π | 2π | QP2 |
If the finite angle between QA and QB be called Φ and the finite angle AOB be called Θ, the sum of the first two components is numerically less than
| ε | (Φ + Θ). |
| 2π |
If the greatest value of |(U − U0)| on the greater arc AB be called H, the last component is numerically less than
| H | (a2 − r2) |
| D2 |
of which, when the circle, of centre P0, passing through A′B′ is sufficiently small, the factor a2 − r2 is arbitrarily small. Thus it appears that u′ is a function of the position of Q whose limit, when Q, interior to the original circle, approaches indefinitely near to P0, is U0. From the form
| u′ = | 1 | ∫ Udω − | 1 | ∫ Udθ, |
| π | 2π |
since the inclination of QP to a fixed direction is, when Q varies, P remaining fixed, a solution of the differential equation
| ∂2ψ | + | ∂2 | = 0, |
| ∂x2 | ∂y2 |
where z, = x + iy, is the point Q, we infer that u′ is a differentiable
