Before passing to this it may be convenient to make here a few remarks as to the periodicity of (single valued) monogenic functions. To say that ƒ(z) is periodic is to say that there exists a constant ω such that for every point z of the interior of the region of existence of ƒ(z) we have ƒ(z + ω) = ƒ(z). This involves, considering all existing periods ω = ρ + iσ, that there exists a lower limit of ρ² + σ² other than zero; for otherwise all the differential coefficients of ƒ(z) would be zero, and ƒ(z) a constant; we can then suppose that not both ρ and σ are numerically less than ε, where ε > σ. Hence, if g be any real quantity, since the range (−g, ... g) contains only a finite number of intervals of length ε, and there cannot be two periods ω = ρ + iσ such that με ⋜ ρ < (μ + 1)ε, νε ⋜ σ < (ν + 1)ε, where μ, ν are integers, it follows that there is only a finite number of periods for which both ρ and σ are in the interval (−g ... g). Considering then all the periods of the function which are real multiples of one period ω, and in particular those periods λω wherein 0 < λ ⋜ 1, there is a lower limit for λ, greater than zero, and therefore, since there is only a finite number of such periods for which the real and imaginary parts both lie between −g and g, a least value of λ, say λ0. If Ω = λ0ω and λ = Mλ0+ λ′, where M is an integer and 0 ⋜ λ′ < λ0, any period λω is of the form MΩ + λ′ω; since, however, Ω, MΩ and λω are periods, so also is λ′ω, and hence, by the construction of λ0, we have λ′ = 0; thus all periods which are real multiples of ω are expressible in the form MΩ where M is an integer, and Ω a period.If beside ω the functions have a period ω′ which is not a real multiple of ω, consider all existing periods of the form μω + νω′ wherein μ, ν are real, and of these those for which 0 ⋜ μ ⋜ 1, 0 < ν ⋜ 1;as before there is a least value for ν, actually occurring in one or more periods, say in the period Ω′ = μ0ω + ν0ω′; now take, if μω + νω′ be a period, ν = N′ν0+ ν′, where N′ is an integer, and 0 ⋜ ν′ < ν0; thence μω + νω′ = μω + N′(Ω′ − μ0ω) + ν′ω′; take then μ − Nμ0= Nλ0+ λ′, where N is an integer and λ0is as above, and 0 ⋜ λ′ < λ0; we thus have a period NΩ + N′Ω′ + λ′ω + ν′ω′, and hence a period λ′ω + ν′ω′, wherein λ′ < λ0, ν′ < ν0; hence ν′ = 0 and λ′ = 0. All periods of the form μω + νω′ are thus expressible in the form NΩ + N′Ω′, where Ω, Ω′ are periods and N, N′ are integers. But in fact any complex quantity, P + iQ, and in particular any other possible period of the function, is expressible, with μ, ν real, in the form μω + νω′; for if ω = ρ + iσ, ω′ = ρ′ + iσ′, this requires only P = μρ + νρ′, Q = μσ + νσ′, equations which, since ω′/ω is not real, always give finite values for μ and ν.It thus appears that if a single valued monogenic function of z be periodic, either all its periods are real multiples of one of them, and then all are of the form MΩ, where Ω is a period and M is an integer, or else, if the function have two periods whose ratio is not real, then all its periods are expressible in the form NΩ + N′Ω′, where Ω, Ω′ are periods, and N, N′ are integers. In the former case, putting ζ = 2πiz/Ω, and the function ƒ(z) = φ(ζ), the function φ(ζ) has, like exp (ζ), the period 2πi, and if we take t = exp (ζ) or ζ = λ(t) the function is a single valued function of t. If then in particular ƒ(z) is an integral function, regarded as a function of t, it has singularities only for t = 0 and t = ∞, and may be expanded in the formΣ∞−∞antn.Taking the case when the single valued monogenic function has two periods ω, ω′ whose ratio is not real, we can form a network of parallelograms covering the plane of z whose angular points are the points c + mω + m′ω′, wherein c is some constant and m, m′ are all possible positive and negative integers; choosing arbitrarily one of these parallelograms, and calling it the primary parallelogram, all the values of which the function is at all capable occur for points of this primary parallelogram, any point, z′, of the plane being, as it is called,congruentto a definite point, z, of the primary parallelogram, z′ − z being of the form mω + m′ω′, where m, m′ are integers. Such a function cannot be an integral function, since then, if, in the primary parallelogram |ƒ(z)| < M, it would also be the case, on a circle of centre the origin and radius R, that |ƒ(z)| < M, and therefore, if Σanznbe the expansion of the function, which is valid for an integral function for all finite values of z, we should have |an| < MR−n, which can be made arbitrarily small by taking R large enough. The function must then have singularities for finite values of z.We consider only functions for which these are poles. Of these there cannot be an infinite number in the primary parallelogram, since then those of these poles which are sufficiently near to one of the necessarily existing limiting points of the poles would be arbitrarily near to one another, contrary to the character of a pole. Supposing the constant c used in naming the corners of the parallelograms so chosen that no pole falls on the perimeter of a parallelogram, it is clear that the integral 1/(2πi)∫ƒ(z) dz round the perimeter of the primary parallelogram vanishes; for the elements of the integral corresponding to two such opposite perimeter points as z, z + ω (or as z, z + ω′) are mutually destructive. This integral is, however, equal to the sum of the residues of ƒ(z) at the poles interior to the parallelogram. Which sum is therefore zero. There cannot therefore be such a function having only one pole of the first order in any parallelogram; we shall see that there can be such a function with two poles only in any parallelogram, each of the first order, with residues whose sum is zero, and that there can be such a function with one pole of the second order, having an expansion near this pole of the form (z-a)−2+ (power series in z − a).Considering next the function φ(z) = [ƒ(z)]−1dƒ(z)/dz, it is easily seen that an ordinary point of ƒ(z) is an ordinary point of φ(z), that a zero of order m for ƒ(z) in the neighbourhood of which ƒ(z) has a form, (z − a)mmultiplied by a power series, is a pole of φ(z) of residue m, and that a pole of ƒ(z) of order n is a pole of φ(z) of residue −n; manifestly φ(z) has the two periods of ƒ(z). We thus infer, since the sum of the residues of φ(z) is zero, that for the function ƒ(z), the sum of the orders of its vanishing at points belonging to one parallelogram, Σm, is equal to the sum of the orders of its poles, Σn; which is briefly expressed by saying that the number of its zeros is equal to the number of its poles. Applying this theorem to the function ƒ(z) − A, where A is an arbitrary constant, we have the result, that the function ƒ(z) assumes the value A in one of the parallelograms as many times as it becomes infinite. Thus, by what is proved above, every conceivable complex value does arise as a value for the doubly periodic function ƒ(z) in any one of its parallelograms, and in fact at least twice. The number of times it arises is called theorderof the function; the result suggests a property of rational functions.Consider further the integral∫z [ƒ′(z)/ƒ(z)] dz, where ƒ′(z) = dƒ(z)/dz taken round the perimeter of the primary parallelogram; the contribution to this arising from two opposite perimeter points such as z and z + ω is of the form −ω∫z [ƒ′(z)/ƒ(z)] dz, which, as z increases from z0to z0+ ω′, gives, if λ denote the generalized logarithm, − ω {λ [ƒ(z0+ ω′)] − λ[ƒ(z0)]}, that is, since ƒ(z0+ ω′) = ƒ(z0), gives 2πiNω, where N is an integer; similarly the result of the integration along the other two opposite sides is of the form 2πiN′ω′, where N′ is an integer. The integral, however, is equal to 2πi times the sum of the residues of zƒ′(z) / ƒ(z) at the poles interior to the parallelogram. For a zero, of order m, of ƒ(z) at z = a, the contribution to this sum is 2πima, for a pole of order n at z = b the contribution is −2πinb; we thus infer that Σma − Σnb = Nω + N′ω′; this we express in words by saying that the sum of the values of z where ƒ(z) = 0 within any parallelogram is equal to the sum of the values of z where ƒ(z) = ∞ save for integral multiples of the periods. By considering similarly the function ƒ(z) − A where A is an arbitrary constant, we prove that each of these sums is equal to the sum of the values of z where the function takes the value A in the parallelogram.
Before passing to this it may be convenient to make here a few remarks as to the periodicity of (single valued) monogenic functions. To say that ƒ(z) is periodic is to say that there exists a constant ω such that for every point z of the interior of the region of existence of ƒ(z) we have ƒ(z + ω) = ƒ(z). This involves, considering all existing periods ω = ρ + iσ, that there exists a lower limit of ρ² + σ² other than zero; for otherwise all the differential coefficients of ƒ(z) would be zero, and ƒ(z) a constant; we can then suppose that not both ρ and σ are numerically less than ε, where ε > σ. Hence, if g be any real quantity, since the range (−g, ... g) contains only a finite number of intervals of length ε, and there cannot be two periods ω = ρ + iσ such that με ⋜ ρ < (μ + 1)ε, νε ⋜ σ < (ν + 1)ε, where μ, ν are integers, it follows that there is only a finite number of periods for which both ρ and σ are in the interval (−g ... g). Considering then all the periods of the function which are real multiples of one period ω, and in particular those periods λω wherein 0 < λ ⋜ 1, there is a lower limit for λ, greater than zero, and therefore, since there is only a finite number of such periods for which the real and imaginary parts both lie between −g and g, a least value of λ, say λ0. If Ω = λ0ω and λ = Mλ0+ λ′, where M is an integer and 0 ⋜ λ′ < λ0, any period λω is of the form MΩ + λ′ω; since, however, Ω, MΩ and λω are periods, so also is λ′ω, and hence, by the construction of λ0, we have λ′ = 0; thus all periods which are real multiples of ω are expressible in the form MΩ where M is an integer, and Ω a period.
If beside ω the functions have a period ω′ which is not a real multiple of ω, consider all existing periods of the form μω + νω′ wherein μ, ν are real, and of these those for which 0 ⋜ μ ⋜ 1, 0 < ν ⋜ 1;as before there is a least value for ν, actually occurring in one or more periods, say in the period Ω′ = μ0ω + ν0ω′; now take, if μω + νω′ be a period, ν = N′ν0+ ν′, where N′ is an integer, and 0 ⋜ ν′ < ν0; thence μω + νω′ = μω + N′(Ω′ − μ0ω) + ν′ω′; take then μ − Nμ0= Nλ0+ λ′, where N is an integer and λ0is as above, and 0 ⋜ λ′ < λ0; we thus have a period NΩ + N′Ω′ + λ′ω + ν′ω′, and hence a period λ′ω + ν′ω′, wherein λ′ < λ0, ν′ < ν0; hence ν′ = 0 and λ′ = 0. All periods of the form μω + νω′ are thus expressible in the form NΩ + N′Ω′, where Ω, Ω′ are periods and N, N′ are integers. But in fact any complex quantity, P + iQ, and in particular any other possible period of the function, is expressible, with μ, ν real, in the form μω + νω′; for if ω = ρ + iσ, ω′ = ρ′ + iσ′, this requires only P = μρ + νρ′, Q = μσ + νσ′, equations which, since ω′/ω is not real, always give finite values for μ and ν.
It thus appears that if a single valued monogenic function of z be periodic, either all its periods are real multiples of one of them, and then all are of the form MΩ, where Ω is a period and M is an integer, or else, if the function have two periods whose ratio is not real, then all its periods are expressible in the form NΩ + N′Ω′, where Ω, Ω′ are periods, and N, N′ are integers. In the former case, putting ζ = 2πiz/Ω, and the function ƒ(z) = φ(ζ), the function φ(ζ) has, like exp (ζ), the period 2πi, and if we take t = exp (ζ) or ζ = λ(t) the function is a single valued function of t. If then in particular ƒ(z) is an integral function, regarded as a function of t, it has singularities only for t = 0 and t = ∞, and may be expanded in the formΣ∞−∞antn.
Taking the case when the single valued monogenic function has two periods ω, ω′ whose ratio is not real, we can form a network of parallelograms covering the plane of z whose angular points are the points c + mω + m′ω′, wherein c is some constant and m, m′ are all possible positive and negative integers; choosing arbitrarily one of these parallelograms, and calling it the primary parallelogram, all the values of which the function is at all capable occur for points of this primary parallelogram, any point, z′, of the plane being, as it is called,congruentto a definite point, z, of the primary parallelogram, z′ − z being of the form mω + m′ω′, where m, m′ are integers. Such a function cannot be an integral function, since then, if, in the primary parallelogram |ƒ(z)| < M, it would also be the case, on a circle of centre the origin and radius R, that |ƒ(z)| < M, and therefore, if Σanznbe the expansion of the function, which is valid for an integral function for all finite values of z, we should have |an| < MR−n, which can be made arbitrarily small by taking R large enough. The function must then have singularities for finite values of z.
We consider only functions for which these are poles. Of these there cannot be an infinite number in the primary parallelogram, since then those of these poles which are sufficiently near to one of the necessarily existing limiting points of the poles would be arbitrarily near to one another, contrary to the character of a pole. Supposing the constant c used in naming the corners of the parallelograms so chosen that no pole falls on the perimeter of a parallelogram, it is clear that the integral 1/(2πi)∫ƒ(z) dz round the perimeter of the primary parallelogram vanishes; for the elements of the integral corresponding to two such opposite perimeter points as z, z + ω (or as z, z + ω′) are mutually destructive. This integral is, however, equal to the sum of the residues of ƒ(z) at the poles interior to the parallelogram. Which sum is therefore zero. There cannot therefore be such a function having only one pole of the first order in any parallelogram; we shall see that there can be such a function with two poles only in any parallelogram, each of the first order, with residues whose sum is zero, and that there can be such a function with one pole of the second order, having an expansion near this pole of the form (z-a)−2+ (power series in z − a).
Considering next the function φ(z) = [ƒ(z)]−1dƒ(z)/dz, it is easily seen that an ordinary point of ƒ(z) is an ordinary point of φ(z), that a zero of order m for ƒ(z) in the neighbourhood of which ƒ(z) has a form, (z − a)mmultiplied by a power series, is a pole of φ(z) of residue m, and that a pole of ƒ(z) of order n is a pole of φ(z) of residue −n; manifestly φ(z) has the two periods of ƒ(z). We thus infer, since the sum of the residues of φ(z) is zero, that for the function ƒ(z), the sum of the orders of its vanishing at points belonging to one parallelogram, Σm, is equal to the sum of the orders of its poles, Σn; which is briefly expressed by saying that the number of its zeros is equal to the number of its poles. Applying this theorem to the function ƒ(z) − A, where A is an arbitrary constant, we have the result, that the function ƒ(z) assumes the value A in one of the parallelograms as many times as it becomes infinite. Thus, by what is proved above, every conceivable complex value does arise as a value for the doubly periodic function ƒ(z) in any one of its parallelograms, and in fact at least twice. The number of times it arises is called theorderof the function; the result suggests a property of rational functions.
Consider further the integral∫z [ƒ′(z)/ƒ(z)] dz, where ƒ′(z) = dƒ(z)/dz taken round the perimeter of the primary parallelogram; the contribution to this arising from two opposite perimeter points such as z and z + ω is of the form −ω∫z [ƒ′(z)/ƒ(z)] dz, which, as z increases from z0to z0+ ω′, gives, if λ denote the generalized logarithm, − ω {λ [ƒ(z0+ ω′)] − λ[ƒ(z0)]}, that is, since ƒ(z0+ ω′) = ƒ(z0), gives 2πiNω, where N is an integer; similarly the result of the integration along the other two opposite sides is of the form 2πiN′ω′, where N′ is an integer. The integral, however, is equal to 2πi times the sum of the residues of zƒ′(z) / ƒ(z) at the poles interior to the parallelogram. For a zero, of order m, of ƒ(z) at z = a, the contribution to this sum is 2πima, for a pole of order n at z = b the contribution is −2πinb; we thus infer that Σma − Σnb = Nω + N′ω′; this we express in words by saying that the sum of the values of z where ƒ(z) = 0 within any parallelogram is equal to the sum of the values of z where ƒ(z) = ∞ save for integral multiples of the periods. By considering similarly the function ƒ(z) − A where A is an arbitrary constant, we prove that each of these sums is equal to the sum of the values of z where the function takes the value A in the parallelogram.
We pass now to the construction of a function having two arbitrary periods ω, ω′ of unreal ratio, which has a single pole of the second order in any one of its parallelograms.
For this consider first the network of parallelograms whose corners are the points Ω = mω + m′ω′, where m, m′ take all positive and negative integer values; putting a small circle about each corner of this network, let P be a point outside all these circles; this will be interior to a parallelogram whose corners in order may be denoted by z0, z0+ ω, z0+ ω + ω′, z0+ ω′; we shall denote z0, z0+ ω by A0, B0; this parallelogram Π0is surrounded by eight other parallelograms, forming with Π0a larger parallelogram Π1, of which one side, for instance, contains the points z0− ω − ω′, z0− ω′, z0− ω′ + ω, z0− ω′ + 2ω, which we shall denote by A1, B1, C1, D1. This parallelogram Π1is surrounded by sixteen of the original parallelograms, forming with Π1a still larger parallelogram Π2of which one side, for instance, contains the points z0− 2ω − 2ω′, z0− ω − 2ω′, z0− 2ω′, z0+ ω − 2ω′, z0+ 2ω − 2ω′, z0+ 3ω − 2ω′, which we shall denote by A2, B2, C2, D2, E2, F2. And so on. Now consider the sum of the inverse cubes of the distances of the point P from the corners of all the original parallelograms. The sum will contain the termsS0=1+(1+1+1)+(1+1+ ... +1)+ ...PA03PA13PB13PC13PA23PB23PE23and three other sets of terms, each infinite in number, formed in a similar way. If the perpendiculars from P to the sides A0B0, A1B1C1, A2B2C2D2E2, and so on, be p, p + q, p + 2q and so on, the sum S0is at most equal to1+3+5+ ... +2n + 1+ ...p3(p + q)3(p + 2q)3(p + nq)3of which the general term is ultimately, when n is large, in a ratio of equality with 2q−3n−2, so that the series S0is convergent, as we know the sum Σn−2to be; this assumes that p ≠ 0; if P be on A0B0the proof for the convergence of S0− 1/PA03, is the same. Taking the three other sums analogous to S0we thus reach the result that the seriesφ(z) = −2Σ (z − Ω)−3,where Ω is mω + m′ω′, and m, m′ are to take all positive and negative integer values, and z is any point outside small circles described with the points Ω as centres, isabsolutely convergent. Its sum is therefore independent of the order of its terms. By the nature of the proof, which holds for all positions of z outside the small circles spoken of, the series is also clearlyuniformly convergentoutside these circles. Each term of the series being a monogenic function of z, the series may therefore be differentiated and integrated outside these circles, and represents a monogenic function. It is clearly periodic with the periods ω, ω′; for φ(z + ω) is the same sum as φ(z) with the terms in a slightly different order. Thus φ(z + ω) = φ(z) and φ(z + ω′) = φ(z).Consider now the functionƒ(z) =1+∫z0{φ(z) +2}dz,z2z3where, for the subject of integration, the area of uniform convergence clearly includes the point z = 0; this givesdƒ(z)= φ(z)dzandƒ(z) =1+Σ′{1−1},z2(z − Ω)2Ω2wherein Σ′ is a sum excluding the term for which m = 0 and m′ = 0. Hence ƒ(z + ω) − ƒ(z) and ƒ(z + ω′) − ƒ(z) are both independent of z. Noticing, however, that, by its form, ƒ(z) is an even function of z, and putting z = −½ω, z = −½ω′ respectively, we infer that also ƒ(z) has the two periods ω and ω′. In the primary parallelogram Π0, however, ƒ(z) is only infinite at z = 0 in the neighbourhood of which its expansion is of the form z−2+ (power series in z). Thus ƒ(z) is such a doubly periodic function as was to be constructed, having in any parallelogram of periods only one pole, of the second order.
For this consider first the network of parallelograms whose corners are the points Ω = mω + m′ω′, where m, m′ take all positive and negative integer values; putting a small circle about each corner of this network, let P be a point outside all these circles; this will be interior to a parallelogram whose corners in order may be denoted by z0, z0+ ω, z0+ ω + ω′, z0+ ω′; we shall denote z0, z0+ ω by A0, B0; this parallelogram Π0is surrounded by eight other parallelograms, forming with Π0a larger parallelogram Π1, of which one side, for instance, contains the points z0− ω − ω′, z0− ω′, z0− ω′ + ω, z0− ω′ + 2ω, which we shall denote by A1, B1, C1, D1. This parallelogram Π1is surrounded by sixteen of the original parallelograms, forming with Π1a still larger parallelogram Π2of which one side, for instance, contains the points z0− 2ω − 2ω′, z0− ω − 2ω′, z0− 2ω′, z0+ ω − 2ω′, z0+ 2ω − 2ω′, z0+ 3ω − 2ω′, which we shall denote by A2, B2, C2, D2, E2, F2. And so on. Now consider the sum of the inverse cubes of the distances of the point P from the corners of all the original parallelograms. The sum will contain the terms
and three other sets of terms, each infinite in number, formed in a similar way. If the perpendiculars from P to the sides A0B0, A1B1C1, A2B2C2D2E2, and so on, be p, p + q, p + 2q and so on, the sum S0is at most equal to
of which the general term is ultimately, when n is large, in a ratio of equality with 2q−3n−2, so that the series S0is convergent, as we know the sum Σn−2to be; this assumes that p ≠ 0; if P be on A0B0the proof for the convergence of S0− 1/PA03, is the same. Taking the three other sums analogous to S0we thus reach the result that the series
φ(z) = −2Σ (z − Ω)−3,
where Ω is mω + m′ω′, and m, m′ are to take all positive and negative integer values, and z is any point outside small circles described with the points Ω as centres, isabsolutely convergent. Its sum is therefore independent of the order of its terms. By the nature of the proof, which holds for all positions of z outside the small circles spoken of, the series is also clearlyuniformly convergentoutside these circles. Each term of the series being a monogenic function of z, the series may therefore be differentiated and integrated outside these circles, and represents a monogenic function. It is clearly periodic with the periods ω, ω′; for φ(z + ω) is the same sum as φ(z) with the terms in a slightly different order. Thus φ(z + ω) = φ(z) and φ(z + ω′) = φ(z).
Consider now the function
where, for the subject of integration, the area of uniform convergence clearly includes the point z = 0; this gives
and
wherein Σ′ is a sum excluding the term for which m = 0 and m′ = 0. Hence ƒ(z + ω) − ƒ(z) and ƒ(z + ω′) − ƒ(z) are both independent of z. Noticing, however, that, by its form, ƒ(z) is an even function of z, and putting z = −½ω, z = −½ω′ respectively, we infer that also ƒ(z) has the two periods ω and ω′. In the primary parallelogram Π0, however, ƒ(z) is only infinite at z = 0 in the neighbourhood of which its expansion is of the form z−2+ (power series in z). Thus ƒ(z) is such a doubly periodic function as was to be constructed, having in any parallelogram of periods only one pole, of the second order.
It can be shown that any single valued meromorphic function of z with ω and ω′ as periods can be expressed rationally in terms of ƒ(z) and φ(z), and that [φ(z)]2is of the form 4[ƒ(z)]3+ Aƒ(z) + B, where A, B are constants.
To prove the last of these results, we write, for |z| < |Ω|,1−1=2z+3z²+ ...,(z − Ω)²Ω²Ω³Ω4and hence, if Σ′Ω−2n= σn, since Σ′Ω−(2n−1)= 0, we have, for sufficiently small z greater than zero,ƒ(z) = z−2+ 3σ2·z2+ 5σ3·z4+ ...andφ(z) = −2z−3+ 6σ2·z + 20σ3·z3+ ...;using these series we find that the functionF(z) = [φ(z)]² − 4[ƒ(z)]³ + 60σ2ƒ(z) + 140σ3contains no negative powers of z, being equal to a power series in z² beginning with a term in z². The function F(z) is, however, doubly periodic, with periods ω, ω′, and can only be infinite when either ƒ(z) or φ(z) is infinite; this follows from its form in ƒ(z) and φ(z); thus in one parallelogram of periods it can be infinite only when z = 0; we have proved, however, that it is not infinite, but, on the contrary, vanishes, when z = 0. Being, therefore, never infinite for finite values of z it is a constant, and therefore necessarily always zero. Putting therefore ƒ(z) = ζ and φ(z) = dζ/dz we see thatdz= (4ζ³ − 60σ2ζ − 140σ3)−1/2.dζHistorically it was in the discussion of integrals such as∫ dζ (4ζ³ − 60σ2·ζ − 140σ3)−1/2,regarded as a branch of Integral Calculus, that the doubly periodic functions arose. As in the familiar casez =∫ζ0(1 − ζ²)−1/2dζ,where ζ = sin z, it has proved finally to be simpler to regard ζ as a function of z. We shall come to the other point of view below, under § 20,Elliptic Integrals.
To prove the last of these results, we write, for |z| < |Ω|,
and hence, if Σ′Ω−2n= σn, since Σ′Ω−(2n−1)= 0, we have, for sufficiently small z greater than zero,
ƒ(z) = z−2+ 3σ2·z2+ 5σ3·z4+ ...
and
φ(z) = −2z−3+ 6σ2·z + 20σ3·z3+ ...;
using these series we find that the function
F(z) = [φ(z)]² − 4[ƒ(z)]³ + 60σ2ƒ(z) + 140σ3
contains no negative powers of z, being equal to a power series in z² beginning with a term in z². The function F(z) is, however, doubly periodic, with periods ω, ω′, and can only be infinite when either ƒ(z) or φ(z) is infinite; this follows from its form in ƒ(z) and φ(z); thus in one parallelogram of periods it can be infinite only when z = 0; we have proved, however, that it is not infinite, but, on the contrary, vanishes, when z = 0. Being, therefore, never infinite for finite values of z it is a constant, and therefore necessarily always zero. Putting therefore ƒ(z) = ζ and φ(z) = dζ/dz we see that
Historically it was in the discussion of integrals such as
∫ dζ (4ζ³ − 60σ2·ζ − 140σ3)−1/2,
regarded as a branch of Integral Calculus, that the doubly periodic functions arose. As in the familiar case
z =∫ζ0(1 − ζ²)−1/2dζ,
where ζ = sin z, it has proved finally to be simpler to regard ζ as a function of z. We shall come to the other point of view below, under § 20,Elliptic Integrals.
To prove that any doubly periodic function F(z) with periods ω, ω′, having poles at the points z = a1, ... z = amof a parallelogram, these being, for simplicity of explanation, supposed to be all of the first order, is rationally expressible in terms of φ(z) and ƒ(z), and we proceed as follows:—
Consider the expressionΦ(z) =(ζ, 1)m+ η(ζ, 1)m−2(ζ − A1) (ζ − A2)...(ζ − Am)where As= ƒ(as), ζ is an abbreviation for ƒ(z) and η for φ(z), and (ζ, 1)m, (ζ, 1)m−2, denote integral polynomials in ζ, of respective orders m and m − 2, so that there are 2m unspecified, homogeneously entering, constants in the numerator. It is supposed that no one of the points a1, ... amis one of the points mω + m′ω′ where f(z) = ∞. The function Φ(z) is a monogenic function of z with the periods ω, ω′, becoming infinite (and having singularities) only when (1) ζ = ∞ or (2) one of the factors ζ-Asis zero. In a period parallelogram including z = 0 the first arises only for z = 0; since for ζ = ∞, η is in a finite ratio to ζ3/2; the function Φ(z) for ζ = ∞ is not infinite provided the coefficient of ζmin (ζ, 1)mis not zero; thus Φ(z) is regular about z = 0. When ζ − As= 0, that is ƒ(z) = f(as), we have z = ±as+ mω + m′ω′, and no other values of z, m and m′ being integers; suppose the unspecified coefficients in the numerator so taken that the numerator vanished to the first order in each of the m points −a1, −a2, ... −am; that is, if φ(as) = Bs, and therefore φ(−as) = −Bs, so that we have the m relations(As, 1)m− Bs(As, 1)m−2= 0;then the function Φ(z) will only have the m poles a1, ... am. Denoting further the m zeros of F(z) by a1′, ... am′, putting ƒ(as′) = As′, φ(as′) = Bs′, suppose the coefficients of the numerator of Φ(z) to satisfy the further m − 1 conditions(As′, 1)m+ Bs′ (As′, 1)m−2= 0for s = 1, 2, ... (m − 1). The ratios of the 2m coefficients in the numerator of Φ(z) can always be chosen so that the m + (m − 1) linear conditions are all satisfied. Consider then the ratioF(z) / Φ(z);it is a doubly periodic function with no singularity other than the one pole am′. It is therefore a constant, the numerator of Φ(z) vanishing spontaneously in am′. We haveF(z) = AΦ(z),where A is a constant; by which F(z) is expressed rationally in terms of ƒ(z) and φ(z), as was desired.When z = 0 is a pole of F(z), say of order r, the other poles, each of the first order, being a1, ... am, similar reasoning can be applied to a function(ζ, 1)h+ η(ζ, 1)k,(ζ − A1) ... (ζ − Am)where h, k are such that the greater of 2h − 2m, 2k + 3 − 2m is equal to r; the case where some of the poles a1, ... amare multiple is to be met by introducing corresponding multiple factors in the denominator and taking a corresponding numerator. We give a solution of the general problem below, of a different form.One important application of the result is the theorem that the functions ƒ(z + t), φ(z + t), which are such doubly periodic function of z as have been discussed, can each be expressed, so far as they depend on z, rationally in terms of ƒ(z) and φ(z), and therefore, so far as they depend on z and t, rationally in terms of ƒ(z), ƒ(t), φ(z) and φ(t). It can in fact be shown, by reasoning analogous to that given above, thatƒ(z + t) + ƒ(z) + ƒ(t) = ¼[φ(z) − φ(t)]2.ƒ(z) − ƒ(t)This shows that if F(z) be any single valued monogenic function which is doubly periodic and of meromorphic character, then F(z + t) is an algebraic function of F(z) and F(t). Conversely any single valued monogenic function of meromorphic character, F(z), which is such that F(z + t) is an algebraic function of F(z) and F(t), can be shown to be a doubly periodic function, or a function obtained from such by degeneration (in virtue of special relations connecting the fundamental constants).The functions ƒ(z), φ(z) above are usually denoted by ℜ(z), ℜ′(z); further the fundamental differential equation is usually written(ℜ′z)² = 4(ℜz)³ − g2ℜz − g3,and the roots of the cubic on the right are denoted by e1, e2, e3; for the odd function, ℜ′z, we have, for the congruent arguments −½ωand ½ω, ℜ′ (½ω) = −ℜ′ (−½ω) = −ℜ′ (½ω), and hence ℜ′ (½ω) = 0; hence we can take e1= ℜ (½ω), e2= ℜ (½ω + ½ω′), e3= ℜ (½ω). It can then be proved that [ℜ(z) − e1] [ℜ (z + ½ω) − e1] = (e1− e2) (e1− e3), with similar equations for the other half periods. Consider more particularly the function ℜ(z) − e1; like ℜ(z) it has a pole of the second order at z = 0, its expansion in its neighbourhood being of the form z−2(1 − e1z2+ Az4+ ...); having no other pole, it has therefore either two zeros, or a double zero in a period parallelogram (ω, ω′). In fact near its zero ½ω its expansion is (x − ½ω) ℜ′ (½ω) + ½(z − ½ω)² ℜ″ (½ω) + ...; we have seen that ℜ′ (½ω) = 0; thus it has a zero of the second order wherever it vanishes. Thus it appears that the square root [ℜ(z) − e1]1/2, if we attach a definite sign to it for some particular value of z, is a single valued function of z; for it can at most have two values, and the only small circuits in the plane which could lead to an interchange of these values are those about either a pole or a zero, neither of which, as we have seen, has this effect; the function is therefore single valued for any circuit. Denoting the function, for a moment, by ƒ1(z), we have ƒ1(z + ω) = ±ƒ1(z), ƒ1(z + ω′) = ±ƒ1(z); it can be seen by considerations of continuity that the right sign in either of these equations does not vary with z; not both these signs can be positive, since the function has only one pole, of the first order, in a parallelogram (ω, ω′); from the expansion of ƒ1(z) about z = 0, namely z− 1(1 − ½e1z² + ...), it follows that ƒ1(z) is an odd function, and hence ƒ1(−½ω′) = −ƒ1(½ω′), which is not zero since [ƒ1(½ω′)]² = e3− e1, so that we have ƒ1(z + ω′) = −ƒ1(z); an equation f1(z + ω) = −ƒ1(z) would then give ƒ1(z + ω + ω′) = ƒ1(z), and hence ƒ1(½ω + ½ω′) = ƒ1(−½ω − ½ω′), of which the latter is −ƒ1(½ω + ½ω′); this would give ƒ1(½ω + ½ω′) = 0, while [ƒ1(½ω + ½ω′)]² = e2− e1. We thus infer that ƒ1(z + ω) = ƒ1(z), ƒ1(z + ω′) = −ƒ1(z), ƒ1(z + ω + ω′) = −ƒ1(z). The function ƒ1(z) is thus doubly periodic with the periods ω and 2ω′; in a parallelogram of which two sides are ω and 2ω′ it has poles at z = 0, z = ω′ each of the first order, and zeros of the first order at z = ½ω, z = ½ω + ω′; it is thus a doubly periodic function of the second order with two different poles of the first order in its parallelogram (ω, 2ω′). We may similarly consider the functions ƒ2(z) = [ℜ(z) − e2]1/2, ƒ3(z) = [ℜ(z) − e3]1/2; they giveƒ2(z + ω + ω′) = ƒ2(z), ƒ2(z + ω) = −ƒ2(z), ƒ2(z + ω′) = −ƒ2(z),ƒ3(z + ω′) = ƒ3z, ƒ3(z + ω) = −ƒ3(z), ƒ3(z + ω + ω′) = −ƒ3(z).Taking u = z (e1− e3)1/2, with a definite determination of the constant (e1− e3)1/2, it is usual, taking the preliminary signs so that for z = 0 each of zƒ1(z), zƒ2(z), zƒ3(z) is equal to +1, to putsn(u) =(e1− e3)1/2, cn(u) =ƒ1(z), dn(u) =f2(z),ƒ3(z)ƒ3(z)ƒ3(z)k² = (e2− e3) / (e1− e3), K = ½ω (e1− e3)1/2, iK′ = ½ω′ (e1− e3)1/2;thus sn(u) is an odd doubly periodic function of the second order with the periods 4K, 2iK, having poles of the first order at u = iK′, u = 2K + iK′, and zeros of the first order at u = 0, u = 2K; similarly cn(u), dn(u) are even doubly periodic functions whose periods can be written down, and sn²(u) + cn²(u) = 1, k²sn²(u) + dn²(u) = 1; if x = sn(u) we at once find, from the relations given here, thatdu= [(1 − x²) (1 − k²x²)]−1/2;dxif we put x = sinφ we havedu= [1 − k²sin²φ]−1/2,dφand if we call φ the amplitude of u, we may write φ = am(u), x = sin·am(u), which explains the origin of the notation sn(u). Similarly cn(u) is an abbreviation of cos·am(u), and dn(u) of Δam(u), where Δ(φ) meant (1 − k²sin²φ)1/2. The addition equation for each of the functions ƒ1(z), ƒ2(z), ƒ3(z) is very simple, beingƒ(z + t) = ½(∂+∂)logƒ(z) + ƒ(t)=ƒ(z)ƒ′(t) − ƒ(t)ƒ′(z),∂z∂iƒ(z) − ƒ(t)ƒ²(z) − ƒ²(t)where f1′(z) means dƒ1(z)/dz, which is equal to −ƒ2(z)·ƒ3(z), and ƒ²(z)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[ƒ1′(z)]2= [ƒ12(z) + e1− e2] [ƒ12(z) + e1− e3],and in fact we find(∂2−∂2)log [ƒ(z) + ƒ(t)] = ƒ2(z) − ƒ2(t) =(∂2−∂2)log [ƒ(z) − ƒ(t)];∂z2dt2∂z2dt2hence 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, d2denote respectively sn(u1), cn(u1), dn(u1), sn(u2), cn(u2), dn(u2), they can be put into the formssn(u1+ u2) = (s1c2d2+ s2c1d1) / D,cn(u1+ u2) = (c1c2− s1s2d1d2) / D,dn(u1+ u2) = (d1d2− k2s1s2c1c2) / D,whereD = 1 − k2s12s22.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 factƒ1(z) = ½ℜ′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 byℜ(z) = ℜ1(z) + ℜ1(z + ω′) − ℜ1(ω′),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 − ½(a′ + b′)] − ℜ [a′ − ½(a′ + b′)]} / {ℜ [z − ½(a + b)] − ℜ [a − ½(a + b)]};if the expansion of this function near z = a beλ(z − a)−1+ μ +Σn=1μn(z − a)n,the expansion near z = b is−λ (z − b)− 1+ μ +Σn=1(−1)nμn(z − b)n,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 byζ(z) −1=∫π0[1− ℜ(z)]dz =Σ′(1+1+z),zz2z − ΩΩΩ2for which as before we have equationsζ(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∫ ζ(z)dzround 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 equationσ(z)= exp{∫z0[ζ(x) −1]dz}= Π′[ (1 −z)exp(z+z2) ],zzΩΩ2Ω2we 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 equationsσ(z + ω)= −exp [2πiη(z + ½ω)],σ(z + ω′)= −exp [2πiη′ (z + ½ω′)].σ(z)σ(z)By means of these equations, if a1+ a2+ ... + am= a′1+ a′2+ ... + a′m, it is readily shown thatσ(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, ... amas its simple poles, and a′1, ... a′mas 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 beA1(z − a)−1+ A2(z − a)−2+ ... + Am+1(z − a)−(m+1),then the differenceF(z) − A1ζ (z − a) − A2ℜ (z − a) − ... +Am+1(−1)mℜm−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).
Consider the expression
where As= ƒ(as), ζ is an abbreviation for ƒ(z) and η for φ(z), and (ζ, 1)m, (ζ, 1)m−2, denote integral polynomials in ζ, of respective orders m and m − 2, so that there are 2m unspecified, homogeneously entering, constants in the numerator. It is supposed that no one of the points a1, ... amis one of the points mω + m′ω′ where f(z) = ∞. The function Φ(z) is a monogenic function of z with the periods ω, ω′, becoming infinite (and having singularities) only when (1) ζ = ∞ or (2) one of the factors ζ-Asis zero. In a period parallelogram including z = 0 the first arises only for z = 0; since for ζ = ∞, η is in a finite ratio to ζ3/2; the function Φ(z) for ζ = ∞ is not infinite provided the coefficient of ζmin (ζ, 1)mis not zero; thus Φ(z) is regular about z = 0. When ζ − As= 0, that is ƒ(z) = f(as), we have z = ±as+ mω + m′ω′, and no other values of z, m and m′ being integers; suppose the unspecified coefficients in the numerator so taken that the numerator vanished to the first order in each of the m points −a1, −a2, ... −am; that is, if φ(as) = Bs, and therefore φ(−as) = −Bs, so that we have the m relations
(As, 1)m− Bs(As, 1)m−2= 0;
then the function Φ(z) will only have the m poles a1, ... am. Denoting further the m zeros of F(z) by a1′, ... am′, putting ƒ(as′) = As′, φ(as′) = Bs′, suppose the coefficients of the numerator of Φ(z) to satisfy the further m − 1 conditions
(As′, 1)m+ Bs′ (As′, 1)m−2= 0
for s = 1, 2, ... (m − 1). The ratios of the 2m coefficients in the numerator of Φ(z) can always be chosen so that the m + (m − 1) linear conditions are all satisfied. Consider then the ratio
F(z) / Φ(z);
it is a doubly periodic function with no singularity other than the one pole am′. It is therefore a constant, the numerator of Φ(z) vanishing spontaneously in am′. We have
F(z) = AΦ(z),
where A is a constant; by which F(z) is expressed rationally in terms of ƒ(z) and φ(z), as was desired.
When z = 0 is a pole of F(z), say of order r, the other poles, each of the first order, being a1, ... am, similar reasoning can be applied to a function
where h, k are such that the greater of 2h − 2m, 2k + 3 − 2m is equal to r; the case where some of the poles a1, ... amare multiple is to be met by introducing corresponding multiple factors in the denominator and taking a corresponding numerator. We give a solution of the general problem below, of a different form.
One important application of the result is the theorem that the functions ƒ(z + t), φ(z + t), which are such doubly periodic function of z as have been discussed, can each be expressed, so far as they depend on z, rationally in terms of ƒ(z) and φ(z), and therefore, so far as they depend on z and t, rationally in terms of ƒ(z), ƒ(t), φ(z) and φ(t). It can in fact be shown, by reasoning analogous to that given above, that
This shows that if F(z) be any single valued monogenic function which is doubly periodic and of meromorphic character, then F(z + t) is an algebraic function of F(z) and F(t). Conversely any single valued monogenic function of meromorphic character, F(z), which is such that F(z + t) is an algebraic function of F(z) and F(t), can be shown to be a doubly periodic function, or a function obtained from such by degeneration (in virtue of special relations connecting the fundamental constants).
The functions ƒ(z), φ(z) above are usually denoted by ℜ(z), ℜ′(z); further the fundamental differential equation is usually written
(ℜ′z)² = 4(ℜz)³ − g2ℜz − g3,
and the roots of the cubic on the right are denoted by e1, e2, e3; for the odd function, ℜ′z, we have, for the congruent arguments −½ωand ½ω, ℜ′ (½ω) = −ℜ′ (−½ω) = −ℜ′ (½ω), and hence ℜ′ (½ω) = 0; hence we can take e1= ℜ (½ω), e2= ℜ (½ω + ½ω′), e3= ℜ (½ω). It can then be proved that [ℜ(z) − e1] [ℜ (z + ½ω) − e1] = (e1− e2) (e1− e3), with similar equations for the other half periods. Consider more particularly the function ℜ(z) − e1; like ℜ(z) it has a pole of the second order at z = 0, its expansion in its neighbourhood being of the form z−2(1 − e1z2+ Az4+ ...); having no other pole, it has therefore either two zeros, or a double zero in a period parallelogram (ω, ω′). In fact near its zero ½ω its expansion is (x − ½ω) ℜ′ (½ω) + ½(z − ½ω)² ℜ″ (½ω) + ...; we have seen that ℜ′ (½ω) = 0; thus it has a zero of the second order wherever it vanishes. Thus it appears that the square root [ℜ(z) − e1]1/2, if we attach a definite sign to it for some particular value of z, is a single valued function of z; for it can at most have two values, and the only small circuits in the plane which could lead to an interchange of these values are those about either a pole or a zero, neither of which, as we have seen, has this effect; the function is therefore single valued for any circuit. Denoting the function, for a moment, by ƒ1(z), we have ƒ1(z + ω) = ±ƒ1(z), ƒ1(z + ω′) = ±ƒ1(z); it can be seen by considerations of continuity that the right sign in either of these equations does not vary with z; not both these signs can be positive, since the function has only one pole, of the first order, in a parallelogram (ω, ω′); from the expansion of ƒ1(z) about z = 0, namely z− 1(1 − ½e1z² + ...), it follows that ƒ1(z) is an odd function, and hence ƒ1(−½ω′) = −ƒ1(½ω′), which is not zero since [ƒ1(½ω′)]² = e3− e1, so that we have ƒ1(z + ω′) = −ƒ1(z); an equation f1(z + ω) = −ƒ1(z) would then give ƒ1(z + ω + ω′) = ƒ1(z), and hence ƒ1(½ω + ½ω′) = ƒ1(−½ω − ½ω′), of which the latter is −ƒ1(½ω + ½ω′); this would give ƒ1(½ω + ½ω′) = 0, while [ƒ1(½ω + ½ω′)]² = e2− e1. We thus infer that ƒ1(z + ω) = ƒ1(z), ƒ1(z + ω′) = −ƒ1(z), ƒ1(z + ω + ω′) = −ƒ1(z). The function ƒ1(z) is thus doubly periodic with the periods ω and 2ω′; in a parallelogram of which two sides are ω and 2ω′ it has poles at z = 0, z = ω′ each of the first order, and zeros of the first order at z = ½ω, z = ½ω + ω′; it is thus a doubly periodic function of the second order with two different poles of the first order in its parallelogram (ω, 2ω′). We may similarly consider the functions ƒ2(z) = [ℜ(z) − e2]1/2, ƒ3(z) = [ℜ(z) − e3]1/2; they give
Taking u = z (e1− e3)1/2, with a definite determination of the constant (e1− e3)1/2, it is usual, taking the preliminary signs so that for z = 0 each of zƒ1(z), zƒ2(z), zƒ3(z) is equal to +1, to put
thus sn(u) is an odd doubly periodic function of the second order with the periods 4K, 2iK, having poles of the first order at u = iK′, u = 2K + iK′, and zeros of the first order at u = 0, u = 2K; similarly cn(u), dn(u) are even doubly periodic functions whose periods can be written down, and sn²(u) + cn²(u) = 1, k²sn²(u) + dn²(u) = 1; if x = sn(u) we at once find, from the relations given here, that
if we put x = sinφ we have
and if we call φ the amplitude of u, we may write φ = am(u), x = sin·am(u), which explains the origin of the notation sn(u). Similarly cn(u) is an abbreviation of cos·am(u), and dn(u) of Δam(u), where Δ(φ) meant (1 − k²sin²φ)1/2. The addition equation for each of the functions ƒ1(z), ƒ2(z), ƒ3(z) is very simple, being
where f1′(z) means dƒ1(z)/dz, which is equal to −ƒ2(z)·ƒ3(z), and ƒ²(z)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
[ƒ1′(z)]2= [ƒ12(z) + e1− e2] [ƒ12(z) + e1− e3],
and in fact we find
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, d2denote respectively sn(u1), cn(u1), dn(u1), sn(u2), cn(u2), dn(u2), they can be put into the forms
where
D = 1 − k2s12s22.
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
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
ℜ(z) = ℜ1(z) + ℜ1(z + ω′) − ℜ1(ω′),
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
if the expansion of this function near z = a be
λ(z − a)−1+ μ +Σn=1μn(z − a)n,
the expansion near z = b is
−λ (z − b)− 1+ μ +Σn=1(−1)nμn(z − b)n,
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
for which as before we have equations
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
∫ ζ(z)dz
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
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
By means of these equations, if a1+ a2+ ... + am= a′1+ a′2+ ... + a′m, it is readily shown that
is a doubly periodic function having a1, ... amas its simple poles, and a′1, ... a′mas 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
A1(z − a)−1+ A2(z − a)−2+ ... + Am+1(z − a)−(m+1),
then the difference
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
where U + iV is the value of the function at a point of the circumference and t = aE(iθ); this is the same as
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
hence by subtraction we have
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
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, U0being the value of it at a point P0of the circumference, and describe a small circle with centre at P0cutting 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 P0within 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 integralu′ =1∫U(a² − r²)dθ =1∫Udω −1∫Udθ,2πa² + r² − 2ar cos (θ − φ)π2πwhich 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∫U0a² − r²dθ =1∫U0dω −1∫U0dθ.2πa² + r² − 2ar cos (θ − φ)π2πHence we can writeu′ − U0=1∫AP0B(U − U0)dω −1∫AP0B(U − U0)dθ +1∫AB(U − U0)(a² − r²)dθ.2π2π2πQP²If 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 thanε(Φ + Θ).2πIf the greatest value of |(U − U0)| on the greater arc AB be called H, the last component is numerically less thanH(a² − r²)D²of 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 formu′ =1∫Udω −1∫Udθ,π2πsince 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∂y²where z, = x + iy, is the point Q, we infer that u′ is a differentiablefunction satisfying this equation; indeed, when r < a, we can write1∫U(a² − r²)dθ =1∫U[1 + 2rcos (θ − φ) + 2r²cos 2(θ − φ) + ...]dθ2πa² + r² − 2ar cos (θ − φ)2πaa²= a0+ a1x + b1y + a2(x² − y²) + 2b2xy + ...,wherea0=1∫Udθ, a1=1∫U cosθdθ, b1=1∫U sinθdθ,2ππaπaa2=1∫U cos 2θdθ, b2=1∫U sin 2θdθ.πa²πa²In 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, y0of 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 R0interior 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 R0the function satisfies the equation∂²P+∂P²= 0,∂x²∂y²we then call it a monogenic potential function. From this, save for an additive constant, there is defined another potential function by means of the equationQ =∫(x, y)(∂Pdy −∂Pdx).∂x∂yThe 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 R0is such that every closed path drawn in it is capable of being deformed continuously to a point lying within R0, that is, issimply 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 byζ = (z − c) exp (−P − iQ)will be developable by a power series in (z − z0) about every point z0interior 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 integralz = ∫ (t − a)α−1(t − b)β−1... dt,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)]1/2, 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 ½π 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)1/2[2ih/(1 − h4)]1/2+ ..., 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 functionz = [(ζ² − h²) / (1 − h²ζ²)]1/2being 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 (2⁄3πi), we have e1= 1, e2= ε², e3= ε, the half periods may be taken to be½ω =∫∞1dt, ½ω′ =∫∞e3dt= ½εω;2(t³ − 1)1/22(t³ − 1)1/2drawing 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 argument1⁄3(2ω + ω′), and D, of argument1⁄3(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 variablet = ½ iℜ′(u)takes each real value once as u passes along the perimeter of the triangle ODE, being as can be shown respectively ∞, 1, 0, − 1 at O, D, H, E, and takes every complex value of imaginary part positive once in the interior of this triangle. This leads tou =1⁄3i∫∞t(t2− 1)−2/3dtin accordance with the general theory.It can be deduced that τ = t2represents the triangle ODH on the upper half plane of τ, and ζ = (i − τ−1)1/2represents similarly the triangle OBD.
Conversely, let U be any continuous real function on the circumference, U0being the value of it at a point P0of the circumference, and describe a small circle with centre at P0cutting 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 P0within 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
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,
Hence we can write
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
If the greatest value of |(U − U0)| on the greater arc AB be called H, the last component is numerically less than
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
since the inclination of QP to a fixed direction is, when Q varies, P remaining fixed, a solution of the differential equation
where z, = x + iy, is the point Q, we infer that u′ is a differentiablefunction satisfying this equation; indeed, when r < a, we can write
= a0+ a1x + b1y + a2(x² − y²) + 2b2xy + ...,
where
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, y0of 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 R0interior 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 R0the function satisfies the equation
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
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 R0is such that every closed path drawn in it is capable of being deformed continuously to a point lying within R0, that is, issimply 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
ζ = (z − c) exp (−P − iQ)
will be developable by a power series in (z − z0) about every point z0interior 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
z = ∫ (t − a)α−1(t − b)β−1... dt,
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)]1/2, 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 ½π 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)1/2[2ih/(1 − h4)]1/2+ ..., 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
z = [(ζ² − h²) / (1 − h²ζ²)]1/2
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 (2⁄3πi), we have e1= 1, e2= ε², e3= ε, the half periods may be taken to be
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 argument1⁄3(2ω + ω′), and D, of argument1⁄3(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
t = ½ iℜ′(u)
takes each real value once as u passes along the perimeter of the triangle ODE, being as can be shown respectively ∞, 1, 0, − 1 at O, D, H, E, and takes every complex value of imaginary part positive once in the interior of this triangle. This leads to
u =1⁄3i∫∞t(t2− 1)−2/3dt
in accordance with the general theory.
It can be deduced that τ = t2represents the triangle ODH on the upper half plane of τ, and ζ = (i − τ−1)1/2represents similarly the triangle OBD.
§ 16.Multiple valued Functions. Algebraic Functions.—The explanations and definitions of a monogenic function hitherto given have been framed for the most part with a view to single valued functions. But starting from a power series, say in z − c, which represents a single value at all points of its circle of convergence, suppose that, by means of a derived series in z − c′, where c′ is interior to the circle of convergence, we can continue the function beyond this, and then by means of a series derived from the first derived series we can make a further continuation, and so on; it may well be that when, after a closed circuit, we again consider points in the first circle of convergence, the value represented may not agree with the original value. One example is the case z1/2, for which two values exist for any value of z; another is the generalized logarithm λ(z), for which there is an infinite number of values. In such cases, as before, the region of existence of the function consists of all points which can be reached by such continuations with power series, and the singular points, which are the limiting points of the point-aggregate constituting the region of existence, are those points in whose neighbourhood the radii of convergence of derived series have zero for limit. In this description the point z = ∞ does not occupy an exceptional position, a power series in z − c being transformed to a series in 1/z when z is near enough to c by means of z − c = c(1 − cz−1) [1 − (1 − cz−1)]−1, and a series in 1/z to a series in z − c, when z is near enough to c, by means of 1/z = 1/c [1 + (z − c / c)]−1.