VOLUME XI SLICE III Frost to Fyzabad (continued 7)

In particular, to take the simplest case, if C0, C be simple closed polygons, and Γ be a path to which C approximates by taking the number of sides of C continually greater, we can find a rational function differing arbitrarily little from ƒ(z) for all points of R0 whose poles are at one finite point c external to Γ. By a transformation of the form t − c = r−1, with the appropriate change in the rational function, we can suppose this point c to be at infinity, in which case the rational function becomes a polynomial. Suppose ε1, ε2, ... to be an indefinitely continued sequence of real positive numbers, converging to zero, and Pr to be the polynomial such that, within C0, |Pr − ƒ(z)| < εr; then the infinite series of polynomials

P1(z) + {P2(z) − P1(z)} + {P3(z) − P2(z)} + ...,

whose sum to n terms is Pn(z), converges for all finite values of z and represents ƒ(z) within C0.

When C consists of a series of disconnected polygons, some of which may include others, and, by increasing indefinitely the number of sides of the polygons C, the points C become the boundary points Γ of a region, we can suppose the poles of the rational function, constructed to approximate to ƒ(z) within R0, to be at points of Γ. A series of rational functions of the form

H1(z) + {H2(z) − H1(z)} + {H3(z) − H2(z)} + ...

then, as before, represents ƒ(z) within R0. And R0 may be taken to coincide as nearly as desired with the interior of the region bounded by Γ.

§ 11. Expression of (1 − z)−1 by means of Polynomials. Applications.—We pursue the ideas just cursorily explained in some further detail.

Let c be an arbitrary real positive quantity; putting the complex variable ζ = ξ + iη, enclose the points ζ = l, ζ = 1 + c by means of (i.) the straight lines η = ±a, from ξ = l to ξ = 1 + c, (ii.) a semicircle convex to ζ = 0 of equation (ξ − 1)2 + η2 = a2, (iii.) a semicircle concave to ζ = 0 of equation (ξ − 1 − c)2 + η2 = a2. The quantities c and a are to remain fixed. Take a positive integer r so that 1/r (c/a) is less than unity, and put σ = 1/r (c/a). Now take

c1 = 1 + c/r, c2 = 1 + 2c/r, ... cr = 1 + c;

if n1, n2, ... nr, be positive integers, the rational function

1 { 1 − ( c1 − 1 ) n1 }

1 − ζ c1 − ζ

is finite at ζ = 1, and has a pole of order n1 at ζ = c1; the rational function

1 { 1 − ( c1 − 1 ) n1 } { 1 − ( c2 − c1 ) n2 } n1

1 − ζ c1 − ζ c2 − ζ

is thus finite except for ζ = c2, where it has a pole of order n1n2; finally, writing

xs = ( cs − cs−1 ) ns,

cs − ζ

the rational function

U = (1 − ζ)−1 (1 − x1) (1 − x2)n1 (1 − x3)n1n2 ... (1 − xr)n1n2 ... nr − 1

has a pole only at ζ = 1 + c, of order n1n2 ... nr.

The difference (1 − ζ)−1 − U is of the form (1 − ζ)−1P, where P, of the form

1 − (1 − ρ1) (1 − ρ2)...(1 − ρk),

in which there are equalities among ρ1, ρ2, ... ρk, is of the form

Σρ1 − Σρ1ρ2 + Σρ1ρ2ρ3 − ...;

therefore, if |ri| = |ρi|, we have

|P| < Σ r1 + Σ r1r2 + Σ r1r2r3 + ... < (1 + r1) (1 + r2)...(1 + rk) − 1;

now, so long as ζ is without the closed curve above described round ζ = 1, ζ = 1 + c, we have

| 1 | < 1 , | cm − cm−1 | < c/r < σ,

1 − ζ a cm − ζ a

and hence

|(1 − ζ)−1 − U| < a−1 {(1 + σn1) (1 + σn2)n1 (1 + σn3)n1n2 ... (1 + σnr)n1n2 ... nr−1 − 1}.

Take an arbitrary real positive ε, and μ, a positive number, so that εmu − 1 < εa, then a value of n1 such that σn1 < μ/(1 + μ) and therefore σn1/(1 − σn1 < μ, and values for n2, n3 ... such that σn2 < 1/n1 σ2n1, σn3 < 1/n1n2 σ3n1, ... σnr < 1/(n1 ... nr−1) σnr n1; then, as 1 + x < ex, we have

|(−ζ)−1 − U| < a−1 {exp (σn1 + n1σn2 + n1n2σn3 + ... + n1n2 ... nr−1σnr) − 1},

and therefore less than

a−1 {exp (σn1 + σ2n1 + ... + σnr n1) − 1},

which is less than

1 [ exp ( σn1 ) − 1 ]

a 1 − σn1

and therefore less than ε.

The rational function U, with a pole at ζ = 1 + c, differs therefore from (1 − ζ)−1, for all points outside the closed region put about ζ = 1, ζ = l + c, by a quantity numerically less than ε. So long as a remains the same, r and σ will remain the same, and a less value of ε will require at most an increase of the numbers n1, n2, ... nr; but if a be taken smaller it may be necessary to increase r, and with this the complexity of the function U.

Now put

z = cζ , ζ = (c + 1)z ;

c + 1 − ζ c + z

thereby the points ζ = 0, 1, 1 + c become the points z = 0, 1, ∞, the function (1 − z)−1 being given by (1 − z)−1 = c(c + 1)−1 (1 − ζ)−1 + (c + 1)−1; the function U becomes a rational function of z with a pole only at z = ∞, that is, it becomes a polynomial in z, say [(c + 1)/c] H − 1/c, where H is also a polynomial in z, and

1 − H = c [ 1 − U ];

1 − z c + 1 1 − ζ

the lines η = ±a become the two circles expressed, if z = x + iy, by

(x + c)² + y² = ± c(c + 1) y,

a

the points (η = 0, ξ = 1 − a), (η = 0, ξ = 1 + c + a) become respectively the points (y = 0, x = c(1 − a)/(c + a), (y = 0, x = −c(l + c + a)/a), whose limiting positions for a = 0 are respectively (y = 0, x = 1), (y = 0, x = −∞). The circle (x + c)² + y² = c(c + 1)y/a can be written

y = (x + c)² + (x + c)4 {μ + √[μ² − (x + c)²]}−2,

2μ 2μ

where μ = ½c(c + 1)/a; its ordinate y, for a given value of x, can therefore be supposed arbitrarily small by taking a sufficiently small.

We have thus proved the following result; taking in the plane of z any finite region of which every interior and boundary point is at a finite distance, however short, from the points of the real axis for which 1 ⋜ x ⋜ ∞, we can take a quantity a, and hence, with an arbitrary c, determine a number r; then corresponding to an arbitrary εs, we can determine a polynomial Ps, such that, for all points interior to the region, we have

|(1 − z−1) − Ps| < εs;

thus the series of polynomials

P1 + (P2 − P1) + (P3 − P2) + ...,

constructed with an arbitrary aggregate of real positive numbers ε1, ε2, ε3, ... with zero as their limit, converges uniformly and represents (1 − z)−1 for the whole region considered.

§ 12. Expansion of a Monogenic Function in Polynomials, over a Star Region.—Now consider any monogenic function ƒ(z) of which the origin is not a singular point; joining the origin to any singular point by a straight line, let the part of this straight line, produced beyond the singular point, lying between the singular point and z = ∞, be regarded as a barrier in the plane, the portion of this straight line from the origin to the singular point being erased. Consider next any finite region of the plane, whose boundary points constitute a path of integration, in a sense previously explained, of which every point is at a finite distance greater than zero from each of the barriers before explained; we suppose this region to be such that any line joining the origin to a boundary point, when produced, does not meet the boundary again. For every point x in this region R we can then write

2πiƒ(x) = ∫ ƒ(t) ƒ(t) ,

t 1 − xt−1

where ƒ(x) represents a monogenic branch of the function, in case it be not everywhere single valued, and t is on the boundary of the region. Describe now another region R0 lying entirely within R, and let x be restricted to be within R0 or upon its boundary; then for any point t on the boundary of R, the points z of the plane for which zt− 1 is real and positive and equal to or greater than 1, being points for which |z| = |t| or |z| > |t|, are without the region R0, and not infinitely near to its boundary points. Taking then an arbitrary real positive ε we can determine a polynomial in xt− 1, say P(xt−1), such that for all points x in R0 we have

|(1 − xt−1)−1 − P(xt−1)| < ε;

the form of this polynomial may be taken the same for all points t on the boundary of R, and hence, if E be a proper variable quantity of modulus not greater than ε,

| 2πiƒ(x) − ∫ dt ƒ(t)P(xt−1) | = | ∫ dt ƒ(t)E | ⋜ εLM,

t t

where L is the length of the path of integration, the boundary of R, and M is a real positive quantity such that upon this boundary |t−1 ƒ(t)| < M. If now

P (xt−1) = c0 + c1xt−1 + ... + cm xm t−m,

and

1 ∫ t−r−1 ƒ(t)dt = μr,

2πi

this gives

|ƒ(x) − {c0μ0 + c1μ1x + ... + cmμmxm}| ⋜ εLM/2π,

where the quantities μ0, μ1, μ2, ... are the coefficients in the expansion of ƒ(x) about the origin.

If then an arbitrary finite region be constructed of the kind explained, excluding the barriers joining the singular points of ƒ(x) to x = ∞, it is possible, corresponding to an arbitrary real positive number σ, to determine a number m, and a polynomial Q(x), of order m, such that for all interior points of this region

|ƒ(x) − Q(x)| < σ.

Hence as before, within this region ƒ(x) can be represented by a series of polynomials, converging uniformly; when ƒ(x) is not a single valued function the series represents one branch of the function.

The same result can be obtained without the use of Cauchy’s integral. We explain briefly the character of the proof. If a monogenic function of t, φ(t) be capable of expression as a power series in t − x about a point x, for |t − x| ⋜ ρ, and for all points of this circle |φ(t)| < g, we know that |φ(n)(x)| < gρ−n(n!). Hence, taking |z| < 1⁄3ρ, and, for any assigned positive integer μ, taking m so that for n > m we have (μ + n)μ < (3⁄2)n, we have

| φ(μ + n)(x)·zn | < φ(μ + n)(x) (μ + n)μ |z|n < g ( 3 ) n ( ρ ) n < g ,

n! (μ + n)! ρμ + n 2 3 ρμ 2n

and therefore

φμ (x + z) = Σ mn=0 φ(μ + n) (x) zn + εμ,

n!

where

|εμ| < g Σ ∞n=m+1 1 < g .

ρμ 2n ρμ 2m

Now draw barriers as before, directed from the origin, joining the singular point of φ(z) to z = ∞, take a finite region excluding all these barriers, let ρ be a quantity less than the radii of convergence of all the power series developments of φ(z) about interior points of this region, so chosen moreover that no circle of radius ρ with centre at an interior point of the region includes any singular point of φ(z), let g be such that |φ(z)| < g for all circles of radius ρ whose centres are interior points of the region, and, x being any interior point of the region, choose the positive integer n so that 1/n |x| < 1⁄3ρ; then take the points a1 = x/n, a2 = 2x/n, a3 = 3x/n, ... an = x; it is supposed that the region is so taken that, whatever x may be, all these are interior points of the region. Then by what has been said, replacing x, z respectively by 0 and x/n, we have

φ(μ) (a1) = Σ m1λ1=0 φ(μ + λ1) (0) ( x ) λ1 + αμ

λ1! n

with

αμ < g/ρμ 2m1,

provided (μ + m1 + 1)μ < (2⁄3)m1 + 1; in fact for μ ⋜ 2n2n−2 it is sufficient to take m1 = n2n; by another application of the same inequality, replacing x, z respectively by a1 and x/n, we have

φ(μ) (a2) = Σ m2λ2=0 φ(μ + λ2) (a1) ( x ) λ2 + β′μ ,

λ2! n

where

|β′μ| < g / ρμ 2m2

provided (μ + m2 + 1)μ < (3⁄2)m2 + 1; we take m2 = n2n − 2, supposing μ < 2n2n−4. So long as λ2 ⋜ m2 ⋜ n2n−2 and μ < 2n2n−4 we have μ + λ2 < 2n2n−2, and we can use the previous inequality to substitute here for φ(μ + λ2) (a1). When this is done we find

φ(μ) (a2) = Σ m2λ2=0 Σ m1λ1=0 φ(μ + λ1 + λ2) (0) ( x ) λ1 + λ2 + βμ ,

λ1! λ2! n

where |βμ| < 2g/ρμ 2m2, the numbers m1, m2 being respectively n2n and n2n−2.

Applying then the original inequality to φ(μ) (a3) = φ(μ) (a2 + x/n), and then using the series just obtained, we find a series for φ(μ) (a3). This process being continued, we finally obtain

φ(x) = Σ m1λ1=0 Σ m2λ2=0 ... Σ mnλn=0 φh (0) ( x ) h + ε ,

K n

where h = λ1 + λ2 + ... + λn, K = λ1! λ2! ... λn!, m1 = n2n, m2 = n2n−2, ..., mn= n², |ε| < 2g/2mn.

By this formula φ(x) is represented, with any required degree of accuracy, by a polynomial, within the region in question; and thence can be expressed as before by a series of polynomials converging uniformly (and absolutely) within this region.

§ 13. Application of Cauchy’s Theorem to the Determination of Definite Integrals.—Some reference must be made to a method whereby real definite integrals may frequently be evaluated by use of the theorem of the vanishing of the integral of a function of a complex variable round a contour within which the function is single valued and non singular.

We are to evaluate an integral ∫ ba ƒ(x)dx; we form a closed contour of which the portion of the real axis from x = a to x = b forms a part, and consider the integral ∫ƒ(z)dz round this contour, supposing that the value of this integral can be determined along the curve forming the completion of the contour. The contour being supposed such that, within it, ƒ(z) is a single valued and finite function of the complex variable z save at a finite number of isolated interior points, the contour integral is equal to the sum of the values of ∫ƒ(z)dz taken round these points. Two instances will suffice to explain the method. (1) The integral ∫ ∞0 [(tan x)/x] dx is convergent if it be understood to mean the limit when ε, ζ, σ, ... all vanish of the sum of the integrals

∫ 1/2π−ε0 tan x dx, ∫ 3/2π−ζ1/2π+ε tan x dx, ∫ 5/2π−σ3/2π+ζ tan x dx, ...

x x x

Now draw a contour consisting in part of the whole of the positive and negative real axis from x = −nπ to x = +nπ, where n is a positive integer, broken by semicircles of small radius whose centres are the points x = ±½π, x = ±¾π, ... , the contour containing also the lines x = nπ and x = −nπ for values of y between 0 and nπ tan α, where α is a small fixed angle, the contour being completed by the portion of a semicircle of radius nπ sec α which lies in the upper half of the plane and is terminated at the points x = ±nπ, y = nπ tan α. Round this contour the integral ∫ [(tan z / z)] dz has the value zero. The contributions to this contour integral arising from the semicircles of centres −½(2s − 1)π, + ½(2s − 1)π, supposed of the same radius, are at once seen to have a sum which ultimately vanishes when the radius of the semicircles diminishes to zero. The part of the contour lying on the real axis gives what is meant by 2 ∫ nπ0 [(tan x / x)] dx. The contribution to the contour integral from the two straight portions at x = ±nπ is

∫ nπ tan α0 idy ( tan iy − tan iy )

nπ + iy −nπ + iy

where i tan iy, = −[exp(y) − exp(−y)]/[exp(y) + exp(−y)], is a real quantity which is numerically less than unity, so that the contribution in question is numerically less than

∫ nπ tan α0 dy 2nπ , that is than 2α.

n²π² + y²

Finally, for the remaining part of the contour, for which, with R = nπ sec α, we have z = R(cos θ + i sin θ) = RE(iθ), we have

dz = idθ, i tan z = exp(−R sin θ) E(iR cos θ) − exp(R sin θ) E(−iR cos θ) ;

z exp(−R sin θ) E(iR cos θ) + exp(R sin θ) E(−iR cos θ)

when n and therefore R is very large, the limit of this contribution to the contour integral is thus

− ∫ π−αα dθ = − (π − 2α).

Making n very large the result obtained for the whole contour is

2 ∫ ∞0 tan x dx − (π − 2α) − 2αε = 0,

x

where ε is numerically less than unity. Now supposing α to diminish to zero we finally obtain

∫ ∞0 tan x dx = π .

x 2

(2) For another case, to illustrate a different point, we may take the integral

∫ za−1 dz,

1 + z

wherein a is real quantity such that 0 < a < 1, and the contour consists of a small circle, z = rE(iθ), terminated at the points x = r cos α, y = ± r sin α, where α is small, of the two lines y = ± r sin α for r cos α ⋜ x ⋜ R cos β, where R sin β = r sin α, and finally of a large circle z = RE(iφ), terminated at the points x = R cos β, y = ±R sin β. We suppose α and β both zero, and that the phase of z is zero for r cos a ⋜ x ⋜ R cos β, y = r sin α = R sin β. Then on r cos α ⋜ x ⋜ R cos β, y = −r sin α, the phase of z will be 2π, and zα − 1 will be equal to xα − 1 exp [2πi(a − 1)], where x is real and positive. The two straight portions of the contour will thus together give a contribution

[1 − exp(2πiα)] ∫ R cos βr cos α xa−1 dx.

1 + x

It can easily be shown that if the limit of zƒ(z) for z = 0 is zero, the integral ∫ƒ(z)dz taken round an arc, of given angle, of a small circle enclosing the origin is ultimately zero when the radius of the circle diminishes to zero, and if the limit of zƒ(z) for z = ∞ is zero, the same integral taken round an arc, of given angle, of a large circle whose centre is the origin is ultimately zero when the radius of the circle increases indefinitely; in our case with ƒ(z) = zα−1/(1 + z), we have zƒ(z) = za/(1 + z), which, for 0 < a < 1, diminishes to zero both for z = 0 and for z = ∞. Thus, finally the limit of the contour integral when r = 0, R = ∞ is

[1 − exp(2πiα)] ∫ ∞0 xα−1 dx.

1 + x

Within the contour ƒ(z) is single valued, and has a pole at z = 1; at this point the phase of z is π and za−1 is exp [iπ(a − 1)] or − exp(iπa); this is then the residue of ƒ(z) at z = −1; we thus have

[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.

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 z0 to 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 Π0 is surrounded by eight other parallelograms, forming with Π0 a 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 Π1 is surrounded by sixteen of the original parallelograms, forming with Π1 a still larger parallelogram Π2 of 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

S0 = 1 + ( 1 + 1 + 1 ) + ( 1 + 1 + ... + 1 ) + ...

PA03 PA13 PB13 PC13 PA23 PB23 PE23

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 S0 is at most equal to

1 + 3 + 5 + ... + 2n + 1 + ...

p3 (p + q)3 (p + 2q)3 (p + nq)3

of which the general term is ultimately, when n is large, in a ratio of equality with 2q−3 n−2, so that the series S0 is convergent, as we know the sum Σn−2 to be; this assumes that p ≠ 0; if P be on A0B0 the proof for the convergence of S0 − 1/PA03, is the same. Taking the three other sums analogous to S0 we 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, is absolutely 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 clearly uniformly convergent outside 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

ƒ(z) = 1 + ∫ z0 { φ(z) + 2 } dz,

z2 z3

where, for the subject of integration, the area of uniform convergence clearly includes the point z = 0; this gives

dƒ(z) = φ(z)

dz

and

ƒ(z) = 1 + Σ′ { 1 − 1 } ,

z2 (z − Ω)2 Ω2

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)]2 is 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)² Ω² Ω³ Ω4

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

dz = (4ζ³ − 60σ2ζ − 140σ3)−1/2.

dζ

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)−1/2 dζ,

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 = am of 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 expression

Φ(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, ... am is 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-As is 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 ζm in (ζ, 1)m is 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

(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, ... am are 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

ƒ(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 give

ƒ2(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 put

sn(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, that

du = [(1 − x²) (1 − k²x²)]−1/2;

dx

if we put x = sinφ we have

du = [1 − k²sin²φ]−1/2,

dφ

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

ƒ(z + t) = ½ ( ∂ + ∂ ) log ƒ(z) + ƒ(t) = ƒ(z)ƒ′(t) − ƒ(t)ƒ′(z) ,

∂z ∂i ƒ(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)];

∂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

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

ƒ1(z) = ½ 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

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

{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 be

λ(z − a)−1 + μ + Σ n=1 μn (z − a)n,

the expansion near z = b is

−λ (z − b)− 1 + μ + Σ n=1 (−1)n μn (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

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

∫ 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

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 + 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 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

A1(z − a)−1 + A2(z − a)−2 + ... + Am+1(z − a)−(m+1),

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 function satisfying this equation; indeed, when r < a, we can write

1 ∫ U (a² − r²) dθ = 1 ∫ U [ 1 + 2 r cos (θ − φ) + 2 r² cos 2(θ − φ) + ... ] dθ

2π a² + r² − 2ar cos (θ − φ) 2π a a²

= a0 + a1x + b1y + a2 (x² − y²) + 2b2xy + ...,

where

a0 = 1 ∫ Udθ, a1 = 1 ∫ U cosθ dθ, b1 = 1 ∫ U sinθ dθ,

2π π a π a

a2 = 1 ∫ U cos 2θ dθ, b2 = 1 ∫ U sin 2θ dθ.

π a² π a²

In this series the terms of order n are sums, with real coefficients, of the various integral polynomials of dimension n which satisfy the equation ∂²ψ/∂x² + ∂²ψ/∂y²; the series is thus the real part of a power series in z, and is capable of differentiation and integration within its region of convergence.

Conversely we may suppose a function, P, defined for the interior of a finite region R of the plane of the real variables x, y, capable of expression about any interior point x0, y0 of this region by a power series in x − x0, y − y0, with real coefficients, these various series being obtainable from one of them by continuation. For any region R0 interior to the region specified, the radii of convergence of these power series will then have a lower limit greater than zero, and hence a finite number of these power series suffice to specify the function for all points interior to R0. Each of these series, and therefore the function, will be differentiable; suppose that at all points of R0 the function satisfies the equation

∂²P + ∂P² = 0,

∂x² ∂y²

we then call it a monogenic potential function. From this, save for an additive constant, there is defined another potential function by means of the equation

Q = ∫ (x, y) ( ∂P dy − ∂P dx ).

∂x ∂y

The functions P, Q, being given by a finite number of power series, will be single valued in R0, and P + iQ will be a monogenic function of z within R0· In drawing this inference it is supposed that the region R0 is such that every closed path drawn in it is capable of being deformed continuously to a point lying within R0, that is, is simply connected.

Suppose in particular, c being any point interior to R0, that P approaches continuously, as z approaches to the boundary of R, to the value log r, where r is the distance of c to the points of the perimeter of R. Then the function of z expressed by

ζ = (z − c) exp (−P − iQ)

will be developable by a power series in (z − z0) about every point z0 interior to R0, and will vanish at z = c; while on the boundary of R it will be of constant modulus unity. Thus if it be plotted upon a plane of ζ the boundary of R will become a circle of radius unity with centre at ζ=0, this latter point corresponding to z=c. A closed path within R0, passing once round z=c, will lead to a closed path passing once about ζ = 0. Thus every point of the interior of R will give rise to one point of the interior of the circle. The converse is also true, but is more difficult to prove; in fact, the differential coefficient dζ/dz does not vanish for any point interior to R. This being assumed, we obtain a conformal representation of the interior of the region R upon the interior of a circle, in which the arbitrary interior point c of R corresponds to the centre of the circle, and, by utilizing the arbitrary constant arising in determining the function Q, an arbitrary point of the boundary of R corresponds to an arbitrary point of the circumference of the circle.

There thus arises the problem of the determination of a real monogenic potential function, single valued and finite within a given arbitrary region, with an assigned continuous value at all points of the boundary of the region. When the region is circular this problem is solved by the integral 1/π ∫ Udω − 1/π ∫ Udθ previously given. When the region is bounded by the outermost portions of the circumferences of two overlapping circles, it can hence be proved that the problem also has a solution; more generally, consider a finite simply connected region, whose boundary we suppose to consist of a single closed path in the sense previously explained, ABCD; joining A to C by two non-intersecting paths AEC, AFC lying within the region, so that the original region may be supposed to be generated by the overlapping regions AECD, CFAB, of which the common part is AECF; suppose now the problem of determining a single valued finite monogenic potential function for the region AECD with a given continuous boundary value can be solved, and also the same problem for the region CFAB; then it can be shown that the same problem can be solved for the original area. Taking indeed the values assigned for the original perimeter ABCD, assume arbitrarily values for the path AEC, continuous with one another and with the values at A and C; then determine the potential function for the interior of AECD; this will prescribe values for the path CFA which will be continuous at A and C with the values originally proposed for ABC; we can then determine a function for the interior of CFAB with the boundary values so prescribed. This in its turn will give values for the path AEC, so that we can determine a new function for the interior of AECD. With the values which this assumes along CFA we can then again determine a new function for the interior of CFAB. And so on. It can be shown that these functions, so alternately determined, have a limit representing such a potential function as is desired for the interior of the original region ABCD. There cannot be two functions with the given perimeter values, since their difference would be a monogenic potential function with boundary value zero, which can easily be shown to be everywhere zero. At least two other methods have been proposed for the solution of the same problem.

A particular case of the problem is that of the conformal representation of the interior of a closed polygon upon the upper half of the plane of a complex variable t. It can be shown without much difficulty that if a, b, c, ... be real values of t, and α, β, γ, ... be n real numbers, whose sum is n − 2, the integral

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

½ω = ∫ ∞1 dt , ½ω′ = ∫ ∞e3 dt = ½εω;

2(t³ − 1)1/2 2(t³ − 1)1/2

drawing the equilateral triangle whose vertices are O, of argument O, A of argument ω, and B of argument ω + ω′ = −ε²ω, and the equilateral triangle whose angular points are O, B and C, of argument ω′, let E, of argument 1⁄3(2ω + ω′), and D, of argument 1⁄3(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 = 1⁄3 i ∫ ∞t (t2 − 1)−2/3dt

in accordance with the general theory.

It can be deduced that τ = t2 represents the triangle ODH on the upper half plane of τ, and ζ = (i − τ−1)1/2 represents 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.

The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation ƒ(s, z) = p0sn + p1sn−1 + ... + pn = 0, wherein p0, p1, ... pn are integral polynomials in z. Assuming ƒ(s, z) incapable of being written as a product of polynomials rational in s and z, and excepting values of z for which the polynomial coefficient of sn vanishes, as also the values of z for which beside ƒ(s, z) = 0 we have also ∂f(s, z)/∂s = 0, and also in general the point z = ∞, the roots of this equation about any point z=c are given by n power series in z − c. About a finite point z = c for which the equation ∂f(s, z)/∂s = 0 is satisfied by one or more of the roots s of ƒ(s, z) = 0, the n roots break up into a certain number of cycles, the r roots of a cycle being given by a set of power series in a radical (z − c)1/r, these series of the cycle being obtainable from one another by replacing (z − c)1/r by ω(z − r)1/r, where ω, equal to exp (2πih/r), is one of the rth roots of unity. Putting then z − c = tr we may say that the r roots of a cycle are given by a single power series in t, an increase of 2π in the phase of t giving an increase of 2πr in the phase of z − c. This single series in t, giving the values of s belonging to one cycle in the neighbourhood of z = c when the phase of z − c varies through 2πr, is to be looked upon as defining a single place among the aggregate of values of z and s which satisfy ƒ(s, z) = 0; two such places may be at the same point (z = c, s = d) without coinciding, the corresponding power series for the neighbouring points being different. Thus for an ordinary value of z, z = c, there are n places for which the neighbouring values of s are given by n power series in z − c; for a value of z for which ∂f(s, z)/∂s = 0 there are less than n places. Similar remarks hold for the neighbourhood of z = ∞; there may be n places whose neighbourhood is given by n power series in z− 1 or fewer, one of these being associated with a series in t, where t = (z−1)1/r; the sum of the values of r which thus arise is always n. In general, then, we may say, with t of one of the forms (z − c), (z − c)1/r, z−1, (z−1)1/r. that the neighbourhood of any place (c, d) for which ƒ(c, d) = 0 is given by a pair of expressions z = c + P(t), s = d + Q(t), where P(t) is a (particular case of a) power series vanishing for t = 0, and Q(t) is a power series vanishing for t = 0, and t vanishes at (c, d), the expression z − c being replaced by z−1 when c is infinite, and similarly the expression s − d by s−1 when d is infinite. The last case arises when we consider the finite values of z for which the polynomial coefficient of sn vanishes. Of such a pair of expressions we may obtain a continuation by writing t = t0 + λ1τ + λ2τ² + ..., where τ is a new variable and λ1 is not zero; in particular for an ordinary finite place this equation simply becomes t = t0 + τ. It can be shown that all the pairs of power series z = c + P(t), s = d + Q(t) which are necessary to represent all pairs of values of z, s satisfying the equation ƒ(s, z) = 0 can be obtained from one of them by this process of continuation, a fact which we express by saying that the equation ƒ(s, z) = 0 defines a monogenic algebraic construct. With less accuracy we may say that an irreducible algebraic equation ƒ(s, z) = 0 determines a single monogenic function s of z.

Any rational function of z and s, where ƒ(s, z) = 0, may be considered in the neighbourhood of any place (c, d) by substituting therein z = c + P(t), s = d + Q(t); the result is necessarily of the form tmH(t), where H(t) is a power series in t not vanishing for t=0 and m is an integer. If this integer is positive, the function is said to vanish to order m at the place; if this integer is negative, = −μ, the function is infinite to order μ at the place. More generally, if A be an arbitrary constant, and, near (c, d), R(s, z) −A is of the form tmH(t), where m is positive, we say that R(s, z) becomes m times equal to A at the place; if R(s, z) is infinite of order μ at the place, so also is R(s, z) − A. It can be shown that the sum of the values of m at all the places, including the places z = ∞, where R(s, z) vanishes, which we call the number of zeros of R(s, z) on the algebraic construct, is finite, and equal to the sum of the values of μ where R(s, z) is infinite, and more generally equal to the sum of the values of m where R(s, z) = A; this we express by saying that a rational function R(s, z) takes any value (including ∞) the same number of times on the algebraic construct; this number is called the order of the rational function.

That the total number of zeros of R(s, z) is finite is at once obvious, these values being obtainable by rational elimination of s between ƒ(s, z) = 0, R(s, z) = 0. That the number is equal to the total number of infinities is best deduced by means of a theorem which is also of more general utility. Let R(s, z) be any rational function of s, z, which are connected by ƒ(s, z) = 0; about any place (c, d) for which z = c + P(t), s = d + Q(t), expand the product

R(s, z) dz

dt

in powers of t and pick out the coefficient of t−1. There is only a finite number of places of this kind. The theorem is that the sum of these coefficients of t−1 is zero. This we express by

[ R(s, z) dz ]t−1 = 0.

dt

The theorem holds for the case n=1, that is, for rational functions of one variable z; in that case, about any finite point we have z − c = t, and about z = ∞ we have z−1 = t, and therefore dz/dt = −t−2; in that case, then, the theorem is that in any rational function of z,

Σ ( A1 + A2 + ... + Am ) + Pzh + Qzh−1 + ... + R,

z − a (z − a)² (z − a)m

the sum ΣA1 of the sum of the residues at the finite poles is equal to the coefficient of 1/z in the expansion, in ascending powers of 1/z, about z = ∞; an obvious result. In general, if for a finite place of the algebraic construct associated with ƒ(s, z) = 0, whose neighbourhood is given by z = c + tr, s = d + Q(t), there be a coefficient of t−1 in R(s, z) dz/dt, this will be r times the coefficient of t−r in R(s, z) or R[d + Q(t), c + tr], namely will be the coefficient of t−r in the sum of the r series obtainable from R [d + Q(t), c + tr] by replacing t by ωt, where ω is an rth root of unity; thus the sum of the coefficients of t−1 in R(s, z) dz/dt for all the places which arise for z = c, and the corresponding values of s, is equal to the coefficient of (z − c)−1 in R(s1, z) + R(s2z) + ... + R(sn, z), where s1, ... sn are the n values of s for a value of z near to z = c; this latter sum Σ R(si, z) is, however, a rational function of z only. Similarly, near z = ∞, for a place given by z−1=tr, s = d + Q(t), or s−1 = Q(t), the coefficient of t−1 in R(s, z) dz/dt is equal to −r times the coefficient of tr in R[d + Q(t), t−r], that is equal to the negative coefficient of z−l in the sum of the r series R[d + Q(ωt), t−r], so that, as before, the sum of the coefficients of t−1 in R(s, z) dz/dt at the various places which arise for z = ∞ is equal to the negative coefficient of z− 1 in the same rational function of z, Σ R(si, z). Thus, from the corresponding theorem for rational functions of one variable, the general theorem now being proved is seen to follow.

Apply this theorem now to the rational function of s and z,

1 dR(s, z) ;

R(s, z) dz

at a zero of R(s, z) near which R(s, z) = tmH(t), we have

1 dR(s, z) dz = d {λ [R(s, z)] },

R(s, z) dz dt dt

where λ denotes the generalized logarithmic function, that is equal to

mt−1 + power series in t;

similarly at a place for which R(s, z) = t−μK(t); the theorem

[ 1 dR(s, z) dz ]t−1 = 0

R(s, z) dz dt

thus gives Σm = Σμ, or, in words, the total number of zeros of R(s, z) on the algebraic construct is equal to the total number of its poles. The same is therefore true of the function R(s, z) − A, where A is an arbitrary constant; thus the number in question, being equal to the number of poles of R(s, z) − A, is equal also to the number of times that R(s, z) = A on the algebraic construct.

We have seen above that all single valued doubly periodic meromorphic functions, with the same periods, are rational functions of two variables s, z connected by an equation of the form s² = 4z³ + Az + B. Taking account of the relation connecting these variables s, z with the argument of the doubly periodic functions (which was above denoted by z), it can then easily be seen that the theorem now proved is a generalization of the theorem proved previously establishing for a doubly periodic function a definite order. There exists a generalization of another theorem also proved above for doubly periodic functions, namely, that the sum of the values of the argument in one parallelogram of periods for which a doubly periodic function takes a given value is independent of that value; this generalization, known as Abel’s Theorem, is given § 17 below.

§ 17. Integrals of Algebraic Functions.—In treatises on Integral Calculus it is proved that if R(z) denote any rational function, an indefinite integral ∫R(z)dz can be evaluated in terms of rational and logarithmic functions, including the inverse trigonometrical functions. In generalization of this it was long ago discovered that if s² = az² + bz + c and R(s, z) be any rational function of s, z any integral ∫R(s, z)dz can be evaluated in terms of rational functions of s, z and logarithms of such functions; the simplest case is ∫s− 1dz or ∫(az² + bz + c) −1/2dz. More generally if f(s, z) = 0 be such a relation connecting s, z that when θ is an appropriate rational function of s and z both s and z are rationally expressible, in virtue of ƒ(s, z) = 0 in terms of θ, the integral ∫R(s, z)dz is reducible to a form ∫H(θ)dθ, where H(θ) is rational in θ, and can therefore also be evaluated by rational functions and logarithms of rational functions of s and z. It was natural to inquire whether a similar theorem holds for integrals ∫R(s, z)dz wherein s² is a cubic polynomial in z. The answer is in the negative. For instance, no one of the three integrals

∫ dz , ∫ zdz , ∫ dz

s s (z − c)s

can be expressed by rational and logarithms of rational functions of s and z; but it can be shown that every integral ∫R(s, z)dz can be expressed by means of integrals of these three types together with rational and logarithms of rational functions of s and z (see below under § 20, Elliptic Integrals). A similar theorem is true when s² = quartic polynomial in z; in fact when s² = A(z − a) (z − b) (z − c) (z − d), putting y = s(z − a)−2, x = (z − a)−1, we obtain y2 = cubic polynomial in x. Much less is the theorem true when the fundamental relation ƒ(s, z) = 0 is of more general type. There exists then, however, a very general theorem, known as Abel’s Theorem, which may be enunciated as follows: Beside the rational function R(s, z) occurring in the integral ∫R(s, z)dz, consider another rational function H(s, z); let (a1), ... (am) denote the places of the construct associated with the fundamental equation ƒ(s, z) = 0, for which H(s, z) is equal to one value A, each taken with its proper multiplicity, and let (b1), ... (bm) denote the places for which H(s, z) = B, where B is another value; then the sum of the m integrals ∫ (bi)(ai) R(s, z)dz is equal to the sum of the coefficients of t−1 in the expansions of the function

R(s, z) dz λ ( H(s, z) − B ),

dt H(s, z) − A

where λ denotes the generalized logarithmic function, at the various places where the expansion of R(s, z)dz/dt contains negative powers of t. This fact may be obtained at once from the equation

[ 1 R(s, z) dz ]t−1 = 0,

H(s, z) − μ dt

wherein μ is a constant. (For illustrations see below, under § 20, Elliptic Integrals.)

§ 18. Indeterminateness of Algebraic Integrals.—The theorem that the integral ∫ xa ƒ(z)dz is independent of the path from a to z, holds only on the hypothesis that any two such paths are equivalent, that is, taken together from the complete boundary of a region of the plane within which ƒ(z) is finite and single valued, besides being differentiable. Suppose that these conditions fail only at a finite number of isolated points in the finite part of the plane. Then any path from a to z is equivalent, in the sense explained, to any other path together with closed paths beginning and ending at the arbitrary point a each enclosing one or more of the exceptional points, these closed paths being chosen, when ƒ(z) is not a single valued function, so that the final value of ƒ(z) at a is equal to its initial value. It is necessary for the statement that this condition may be capable of being satisfied.

For instance, the integral ∫ z1 z−1dz is liable to an additive indeterminateness equal to the value obtained by a closed path about z = 0, which is equal to 2πi; if we put u = ∫ z1 z−1dz and consider z as a function of u, then we must regard this function as unaffected by the addition of 2πi to its argument u; we know in fact that z = exp (u) and is a single valued function of u, with the period 2πi. Or again the integral ∫ z0 (1 + z²)−1dz is liable to an additive indeterminateness equal to the value obtained by a closed path about either of the points z = ±i; thus if we put u = ∫ z0 (1 + z²)−1dz, the function z of u is periodic with period π, this being the function tan (u). Next we take the integral u = ∫ (z)(0) (1 − z²)−1/2dz, agreeing that the upper and lower limits refer not only to definite values of z, but to definite values of z each associated with a definite determination of the sign of the associated radical (1 − z²)−1/2. We suppose 1 + z, 1 − z each to have phase zero for z = 0; then a single closed circuit of z = −1 will lead back to z = 0 with (l − z²)1/2 = −1; the additive indeterminateness of the integral, obtained by a closed path which restores the initial value of the subject of integration, may be obtained by a closed circuit containing both the points ±1 in its interior; this gives, since the integral taken about a vanishing circle whose centre is either of the points z = ±1 has ultimately the value zero, the sum

∫ −10 dz + ∫ 0−1 dz + ∫ 10 dz + ∫ 01 dz ,

(1 − z²)1/2 −(1 − z²)1/2 −(1 − z²)1/2 (1 − z²)1/2

where, in each case, (1 − z²)1/2 is real and positive; that is, it gives

−4 ∫ 10 dz

(1 − z²)1/2

or 2π. Thus the additive indeterminateness of the integral is of the form 2kπ, where k is an integer, and the function z of u, which is sin (u), has 2π for period. Take now the case

u = ∫ (z)(z0) dz ,

√{ (z − a) (z − b) (z − c) (z − d) }

adopting a definite determination for the phase of each of the factors z − a, z − b, z − c, z − d at the arbitrary point z0, and supposing the upper limit to refer, not only to a definite value of z, but also to a definite determination of the radical under the sign of integration. From z0 describe a closed loop about the point z = a, consisting, suppose, of a straight path from z0 to a, followed by a vanishing circle whose centre is at a, completed by the straight path from a to z0. Let similar loops be imagined for each of the points b, c, d, no two of these having a point in common. Let A denote the value obtained by the positive circuit of the first loop; this will be in fact equal to twice the integral taken from z0 along the straight path to a; for the contribution due to the vanishing circle is ultimately zero, and the effect of the circuit of this circle is to change the sign of the subject of integration. After the circuit about a, we arrive back at z0 with the subject of integration changed in sign; let B, C, D denote the values of the integral taken by the loops enclosing respectively b, c and d when in each case the initial determination of the subject of integration is that adopted in calculating A. If then we take a circuit from z0 enclosing both a and b but not either c or d, the value obtained will be A − B, and on returning to z0 the subject of integration will have its initial value. It appears thus that the integral is subject to an additive indeterminateness equal to any one of the six differences such as A − B. Of these there are only two linearly independent; for clearly only A − B, A − C, A − D are linearly independent, and in fact, as we see by taking a closed circuit enclosing all of a, b, c, d, we have A − B + C − D = 0; for there is no other point in the plane beside a, b, c, d about which the subject of integration suffers a change of sign, and a circuit enclosing all of a, b, c, d may by putting z = 1/ζ be reduced to a circuit about ζ = 0 about which the value of the integral is zero. The general value of the integral for any position of z and the associated sign of the radical, when we start with a definite determination of the subject of integration, is thus seen to be of the form u0 + m(A − B) + n(A − C), where m and n are integers. The value of A − B is independent of the position of z0, being obtainable by a single closed positive circuit about a and b only; it is thus equal to twice the integral taken once from a to b, with a proper initial determination of the radical under the sign of integration. Similar remarks to the above apply to any integral ∫ H(z)dz, in which H(z) is an algebraic function of z; in any such case H(z) is a rational function of z and a quantity s connected therewith by an irreducible rational algebraic equation ƒ(s, z) = 0. Such an integral ƒK(z, s)dz is called an Abelian Integral.

§ 19. Reversion of an Algebraic Integral.—In a limited number of cases the equation u = ∫ [z0 to z] H(z)dz, in which H(z) is an algebraic function of z, defines z as a single valued function of u. Several cases of this have been mentioned in the previous section; from what was previously proved under § 14, Doubly Periodic Functions, it appears that it is necessary for this that the integral should have at most two linearly independent additive constants of indeterminateness; for instance, for an integral

u = ∫ zz0 [(z − a) (z − b) (z − c) (z − d) (z − e) (z − f) ]−1/2dz,