| [1 − exp (2πia)] ∫ ∞0 | xa−1 | dx = −2πi exp(iπa), |
| 1 + x |
that is
| ∫ ∞0 | xa−1 | dx = π cosec (aπ). |
| 1 + x |
§ 14. Doubly Periodic Functions.—An excellent illustration of the preceding principles is furnished by the theory of single valued functions having in the finite part of the plane no singularities but poles, which have two periods.
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 λ0 is 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 form Σ ∞−∞ an tn.
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, congruent to 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 Σan zn be 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)]−1 dƒ(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)m multiplied 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 the order of the function; the result suggests a property of rational functions.