The infinite series of most importance are those of which the general term is anzn, wherein anis a constant, and z is regarded as variable, n = 0, 1, 2, 3, ... Such a series is called a power series, if a real and positive number M exists such that for z = z0and every n, |anz0n| < M, a condition which is satisfied, for instance, if the series converges for z = z0, then it is at once proved that the series converges absolutely for every z for which |z| < |z0|, and converges uniformly over every range |z| < r′ for which r′ < |z0|. To every power series there belongs then a circle of convergence within which it converges absolutely and uniformly; the function of z represented by it is thus continuous within the circle (this being the result of a general property of uniformly convergent series of continuous functions); the sum for an interior point z is, however, continuous with the sum for a point z0on the circumference, as z approaches to z0provided the series converges for z = z0, as can be shown without much difficulty. Within a common circle of convergence two power series Σ anzn, Σ bnzncan be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series. If r1be less than the radius of convergence of a series Σ anznand for |z| = r1, the sum of the series be in absolute value less than a real positive quantity M, it can be shown that for |z| = r1every term is also less than M in absolute value, namely, |an| < Mr1−n. If in every arbitrarily small neighbourhood of z=0 there be a point for which two converging power series Σanzn, Σbnznagree in value, then the series are identical, or an= bn; thus also if Σanznvanish at z = 0 there is a circle of finite radius about z = 0 as centre within which no other points are found for which the sum of the series is zero. Considering a power series ƒ(z) = Σanznof radius of convergence R, if |z0| < R and we put z = z0+ t with |t| < R-|z0|, the resulting series Σan(z0+ t)nmay be regarded as a double series in z0and t, which, since |z0| + t < R, is absolutely convergent; it may then be arranged according to powers of t. Thus we may write ƒ(z) = Σ Antn; hence A0= ƒ(z0), and we have [ƒ(z0+ t) − ƒ(z0)]/t = Σn=1Antn−l, wherein the continuous series on the right reduces to A1for t = 0; thus the ratio on the left has a definite limit when t = 0, equal namely to A1or Σnanz0n − 1. In other words, the original series may legitimately be differentiated at any interior point z0of its circle of convergence. Repeating this process we find ƒ(z0+ t) = Σtnƒ(n)(z0)/n!, where ƒ(n)(z0) is the nth differential coefficient. Repeating for this power series, in t, the argument applied about z = 0 for Σanzn, we infer that for the series ƒ(z) every point which reduces it to zero is an isolated point, and of such points only a finite number lie within a circle which is within the circle of convergence of ƒ(z).Perhaps the simplest possible power series is ez= exp(z) = 1 + z2/2! + z3/3! + ... of which the radius of convergence is infinite. By multiplication we have exp(z)·exp(z1) = exp(z + z1). In particular when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy). Now the functionsU0= sin y, V0= 1 − cos y, U1= y − sin y,V1=1⁄2y² − 1 + cos y, U2=1⁄6y³ − y + sin y, V2=1⁄24y4−1⁄2y2+ 1 − cos y, ...all vanish for y = 0, and the differential coefficient of any one after the first is the preceding one; as a function (of a real variable) is increasing when its differential coefficient is positive, we infer, for y positive, that each of these functions is positive; proceeding to a limit we hence infer thatcos y = 1 −1⁄2y² +1⁄24y4− ..., sin y = y −1⁄6y³ +1⁄120y5− ...,for positive, and hence, for all values of y. We thus have exp(iy) = cos y + i sin y, and exp (z) = exp (x)·(cos y + i sin y). In other words, the modulus of exp (z) is exp (x) and the phase is y. Hence alsoexp(z + 2πi) = exp(x) [cos (y + 2π) + i sin(y + 2π)],which we express by saying that exp (z) has the period 2πi, and hence also the period 2kπi, where k is an arbitrary integer. From the fact that the constantly increasing function exp (x) can vanish only for x = 0, we at once prove that exp (z) has no other periods.Taking in the plane of z an infinite strip lying between the lines y = 0, y = 2π and plotting the function ζ = exp (z) upon a new plane, it follows at once from what has been said that every complex value of ζ arises when z takes in turn all positions in this strip, and that no value arises twice over. The equation ζ = exp(z) thus defines z, regarded as depending upon ζ, with only an additive ambiguity 2kπi, where k is an integer. We write z = λ(ζ); when ζ is real this becomes the logarithm of ζ; in general λ(ζ) = log |ζ| + i ph (ζ) + 2kπi, where k is an integer; and when ζ describes a closed circuit surrounding the origin the phase of ζ increases by 2π, or k increases by unity. Differentiating the series for ζ we have dζ/dz = ζ, so that z, regarded as depending upon ζ, is also differentiable, with dz/dζ = ζ− 1. On the other hand, consider the series ζ − 1 − ½(ζ− 1)2+1⁄3(ζ − 1)3− ...; it converges when ζ = 2 and hence converges for |ζ − 1| < 1; its differential coefficient is, however, 1 − (ζ − 1) + (ζ − 1)2− ..., that is, (1 + ζ − 1)− 1. Wherefore if φ(ζ) denote this series, for |ζ − 1| < 1, the difference λ(ζ) − φ(ζ), regarded as a function of ξ and η, has vanishing differential coefficients; if we take the value of λ(ζ) which vanishes when ζ = 1 we infer thence that for |ζ − 1| < 1, λ(ζ) = Σn=1[(−1)(n−1)/n (ζ − 1)n. It is to be remarked that it is impossible for ζ while subject to |ζ − 1| < 1 to make a circuit about the origin. For values of ζ for which |ζ − 1| ≮ 1, we can also calculate λ(ζ) with the help of infinite series, utilizing the fact that λ(ζζ′) = λ(ζ) + λ(ζ′).The function λ(ζ) is required to define ζawhen ζ and a are complex numbers; this is defined as exp [aλ(ζ)], that is as Σn=0an[λ(ζ)]n/n!. When a is a real integer the ambiguity of λ(ζ) is immaterial here, since exp [aλ(ζ) + 2kaπi] = exp [aλ(ζ)]; when a is of the form 1/q, where q is a positive integer, there are q values possible for ζ1/q, of the form exp [1/q λ(ζ)] exp (2kπi/q), with k = 0, 1, ... q − 1, all other values of k leading to one of these; the qth power of any one of these values is ζ; when a = p/q, where p, q are integers without common factor, q being positive, we have ζp/q= (ζ1/q)p. The definition of the symbol ζais thus a generalization of the ordinary definition of a power, when the numbers are real. As an example, let it be required to find the meaning of ii; the number i is of modulus unity and phase ½π; thus λ(i) = i (½π + 2kπ); thusii= exp (−½π − 2kπ) = exp (−½π) exp (−2kπ),is always real, but has an infinite number of values.The function exp (z) is used also to define a generalized form of the cosine and sine functions when z is complex; we write, namely, cos z = ½[exp (iz) + exp (−iz)] and sin z = −½i [exp (iz) − exp(−iz)]. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, cos i = 1 + 1/2! + 1/4! + ... is obviously greater than unity.
The infinite series of most importance are those of which the general term is anzn, wherein anis a constant, and z is regarded as variable, n = 0, 1, 2, 3, ... Such a series is called a power series, if a real and positive number M exists such that for z = z0and every n, |anz0n| < M, a condition which is satisfied, for instance, if the series converges for z = z0, then it is at once proved that the series converges absolutely for every z for which |z| < |z0|, and converges uniformly over every range |z| < r′ for which r′ < |z0|. To every power series there belongs then a circle of convergence within which it converges absolutely and uniformly; the function of z represented by it is thus continuous within the circle (this being the result of a general property of uniformly convergent series of continuous functions); the sum for an interior point z is, however, continuous with the sum for a point z0on the circumference, as z approaches to z0provided the series converges for z = z0, as can be shown without much difficulty. Within a common circle of convergence two power series Σ anzn, Σ bnzncan be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series. If r1be less than the radius of convergence of a series Σ anznand for |z| = r1, the sum of the series be in absolute value less than a real positive quantity M, it can be shown that for |z| = r1every term is also less than M in absolute value, namely, |an| < Mr1−n. If in every arbitrarily small neighbourhood of z=0 there be a point for which two converging power series Σanzn, Σbnznagree in value, then the series are identical, or an= bn; thus also if Σanznvanish at z = 0 there is a circle of finite radius about z = 0 as centre within which no other points are found for which the sum of the series is zero. Considering a power series ƒ(z) = Σanznof radius of convergence R, if |z0| < R and we put z = z0+ t with |t| < R-|z0|, the resulting series Σan(z0+ t)nmay be regarded as a double series in z0and t, which, since |z0| + t < R, is absolutely convergent; it may then be arranged according to powers of t. Thus we may write ƒ(z) = Σ Antn; hence A0= ƒ(z0), and we have [ƒ(z0+ t) − ƒ(z0)]/t = Σn=1Antn−l, wherein the continuous series on the right reduces to A1for t = 0; thus the ratio on the left has a definite limit when t = 0, equal namely to A1or Σnanz0n − 1. In other words, the original series may legitimately be differentiated at any interior point z0of its circle of convergence. Repeating this process we find ƒ(z0+ t) = Σtnƒ(n)(z0)/n!, where ƒ(n)(z0) is the nth differential coefficient. Repeating for this power series, in t, the argument applied about z = 0 for Σanzn, we infer that for the series ƒ(z) every point which reduces it to zero is an isolated point, and of such points only a finite number lie within a circle which is within the circle of convergence of ƒ(z).
Perhaps the simplest possible power series is ez= exp(z) = 1 + z2/2! + z3/3! + ... of which the radius of convergence is infinite. By multiplication we have exp(z)·exp(z1) = exp(z + z1). In particular when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy). Now the functions
U0= sin y, V0= 1 − cos y, U1= y − sin y,
V1=1⁄2y² − 1 + cos y, U2=1⁄6y³ − y + sin y, V2=1⁄24y4−1⁄2y2+ 1 − cos y, ...
all vanish for y = 0, and the differential coefficient of any one after the first is the preceding one; as a function (of a real variable) is increasing when its differential coefficient is positive, we infer, for y positive, that each of these functions is positive; proceeding to a limit we hence infer that
cos y = 1 −1⁄2y² +1⁄24y4− ..., sin y = y −1⁄6y³ +1⁄120y5− ...,
for positive, and hence, for all values of y. We thus have exp(iy) = cos y + i sin y, and exp (z) = exp (x)·(cos y + i sin y). In other words, the modulus of exp (z) is exp (x) and the phase is y. Hence also
exp(z + 2πi) = exp(x) [cos (y + 2π) + i sin(y + 2π)],
which we express by saying that exp (z) has the period 2πi, and hence also the period 2kπi, where k is an arbitrary integer. From the fact that the constantly increasing function exp (x) can vanish only for x = 0, we at once prove that exp (z) has no other periods.
Taking in the plane of z an infinite strip lying between the lines y = 0, y = 2π and plotting the function ζ = exp (z) upon a new plane, it follows at once from what has been said that every complex value of ζ arises when z takes in turn all positions in this strip, and that no value arises twice over. The equation ζ = exp(z) thus defines z, regarded as depending upon ζ, with only an additive ambiguity 2kπi, where k is an integer. We write z = λ(ζ); when ζ is real this becomes the logarithm of ζ; in general λ(ζ) = log |ζ| + i ph (ζ) + 2kπi, where k is an integer; and when ζ describes a closed circuit surrounding the origin the phase of ζ increases by 2π, or k increases by unity. Differentiating the series for ζ we have dζ/dz = ζ, so that z, regarded as depending upon ζ, is also differentiable, with dz/dζ = ζ− 1. On the other hand, consider the series ζ − 1 − ½(ζ− 1)2+1⁄3(ζ − 1)3− ...; it converges when ζ = 2 and hence converges for |ζ − 1| < 1; its differential coefficient is, however, 1 − (ζ − 1) + (ζ − 1)2− ..., that is, (1 + ζ − 1)− 1. Wherefore if φ(ζ) denote this series, for |ζ − 1| < 1, the difference λ(ζ) − φ(ζ), regarded as a function of ξ and η, has vanishing differential coefficients; if we take the value of λ(ζ) which vanishes when ζ = 1 we infer thence that for |ζ − 1| < 1, λ(ζ) = Σn=1[(−1)(n−1)/n (ζ − 1)n. It is to be remarked that it is impossible for ζ while subject to |ζ − 1| < 1 to make a circuit about the origin. For values of ζ for which |ζ − 1| ≮ 1, we can also calculate λ(ζ) with the help of infinite series, utilizing the fact that λ(ζζ′) = λ(ζ) + λ(ζ′).
The function λ(ζ) is required to define ζawhen ζ and a are complex numbers; this is defined as exp [aλ(ζ)], that is as Σn=0an[λ(ζ)]n/n!. When a is a real integer the ambiguity of λ(ζ) is immaterial here, since exp [aλ(ζ) + 2kaπi] = exp [aλ(ζ)]; when a is of the form 1/q, where q is a positive integer, there are q values possible for ζ1/q, of the form exp [1/q λ(ζ)] exp (2kπi/q), with k = 0, 1, ... q − 1, all other values of k leading to one of these; the qth power of any one of these values is ζ; when a = p/q, where p, q are integers without common factor, q being positive, we have ζp/q= (ζ1/q)p. The definition of the symbol ζais thus a generalization of the ordinary definition of a power, when the numbers are real. As an example, let it be required to find the meaning of ii; the number i is of modulus unity and phase ½π; thus λ(i) = i (½π + 2kπ); thus
ii= exp (−½π − 2kπ) = exp (−½π) exp (−2kπ),
is always real, but has an infinite number of values.
The function exp (z) is used also to define a generalized form of the cosine and sine functions when z is complex; we write, namely, cos z = ½[exp (iz) + exp (−iz)] and sin z = −½i [exp (iz) − exp(−iz)]. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, cos i = 1 + 1/2! + 1/4! + ... is obviously greater than unity.
§4.Of Functions of a Complex Variable in General.—We have in what precedes shown how to generalize the ordinary rational, algebraic and logarithmic functions, and considered more general cases, of functions expressible by power series in z. With the suggestions furnished by these cases we can frame a general definition. So far our use of the plane upon which z is represented has been only illustrative, the results being capable of analytical statement. In what follows this representation is vital to the mode of expression we adopt; as then the properties of numbers cannot be ultimately based upon spatial intuitions, it is necessary to indicate what are the geometrical ideas requiring elucidation.
Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be counter-clockwise; another square, also of side a, may be added to this, so that there is a side common; this common side being erased we have a composite region with a definite direction of perimeter; to this a third square of the same size may be attached, so that there is a side common to it and one of the former squares, and this common side may be erased. If this process be continued any number of times we obtain a region of the plane bounded by one or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of description, which is such that the region is on the left of the describing point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, it being understood that there is assigned an upper limit for the greatest side of a triangle, and a lower limit for the smallest angle. In the former method, each square may be divided into four others by lines through its centre parallel to its sides; in the latter method each triangle may be divided into four others by lines joining the middle points of its sides; this halves the sides and preserves the angles. When we speak of aregionof the plane in general, unless the contrary is stated, we shall suppose it capable of being generated in this latter way by means of a finite number of triangles, there being an upper limit to the length of a side of the triangle and a lower limit to the size of an angle of the triangle. We shall also require to speak of apathin the plane; this is to be understood as capable of arising as a limit of a polygonal path of finite length, there being a definite direction or sense of description at every point of the path, which therefore never meets itself. From this the meaning of a closed path is clear. The boundary points of a region form one or more closed paths, but, in general, it is only in a limiting sense that the interior points of a closed path are a region.There is a logical principle also which must be referred to. We frequently have cases where, about every interior or boundary, point z0of a certain region a circle can be put, say of radius r0, such that for all points z of the region which are interior to this circle, for which, that is, |z − z0| < r0, a certain property holds. Assuming that to r0is given the value which is the upper limit for z0, of the possible values, we may call the points |z − z0| < r0, the neighbourhood belonging to orproperto z0, and may speak of the property as the property (z, z0). The value of r0will in general vary with z0; what is in most cases of importance is the question whether the lower limit of r0for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z, z0) is of a kind which we may call extensive; such, namely, that if it holds, for some position of z0and all positions of z, within a certain region, then the property (z, z1) holds within a circle of radius R about any interior point z1of this region for all points z for which the circle |z − z1| = R is within the region. Also in this case r0varies continuously with z0. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (1) forsomepoint z0within or upon the boundary of the sub-region, (2) foreverypoint z within or upon the boundary of the sub-region.We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (z, z0) holds for all points z and some point z0of the region, be calledsuitable: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the preceding, which converge to a point, say ζ; after a certain stage all these will be interior to the proper region of ζ; this, however, is contrary to the supposition that they are all unsuitable.We now make some applications of this result (B). Suppose a definite finite real value attached to every interior or boundary point of the region, say ƒ(x, y). It may have a finite upper limit H for the region, so that no point (x, y) exists for which ƒ(x, y) > H, but points (x, y) exist for which ƒ(x, y) > H − ε, however small ε may be; if not we say that its upper limit is infinite. There is then at least one point of the region such that, for points of the region within a circle about this point, the upper limit of ƒ(x, y) is H, however small the radius of the circle be taken; for if not we can put about every point of the region a circle within which the upper limit of ƒ(x, y) is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function ƒ(x, y) is the radius r0of the neighbourhood proper to any point z0, spoken of above. We can hence prove the statement (A) above.Suppose the property (z, z0) extensive, and, if possible, that the lower limit of r0is zero. Let then ζ be a point such that the lower limit of r0is zero for points z0within a circle about ζ however small; let r be the radius of the neighbourhood proper to ζ; take z0so that |z0-ζ| < ½r; the property (z, z0), being extensive, holds within a circle, centre z0, of radius r − |z0− ζ|, which is greater than |z0− ζ|, and increases to r as |z0− ζ| diminishes; this being true for all points z0near ζ, the lower limit of r0is not zero for the neighbourhood of ζ, contrary to what was supposed. This proves (A). Also, as is here shown that r0⋝ r − |z0− ζ|, may similarly be shown that r ⋝ r0− |z0− ζ|. Thus r0differs arbitrarily little from r when |z0− ζ| is sufficiently small; that is, r0varies continuously with z0. Next suppose the function ƒ(x, y), which has a definite finite value at every point of the region considered, to be continuous but not necessarily real, so that about every point z0, within or upon the boundary of the region, η being an arbitrary real positive quantity assigned beforehand, a circle is possible, so that for all points z of the region interior to this circle, we have |ƒ(x, y) −ƒ(x0, y0)| < ½η, and therefore (x′, y′) being any other point interior to this circle, |ƒ(x′, y′) − ƒ(x, y)| < η. We can then apply the result (A) obtained above, taking for the neighbourhood proper to any point z0the circular area within which, for any two points (x, y), (x′, y′), we have |ƒ(x′, x′) − ƒ(x, y)| < η. This is clearly an extensive property. Thus, a number r is assignable, greater than zero, such that, for any two points (x, y), (x′, y′) within a circle |z − z0| = r about any point z0, we have |ƒ(x′, y′) − ƒ(x, y)| < η, and, in particular, |ƒ(x, y) −ƒ(x0, y0)| < η, where η is an arbitrary real positive quantity agreed upon beforehand.Take now any path in the region, whose extreme points are z0, z, and let z1, ... zn−1be intermediate points of the path, in order; denote the continuous function ƒ(x, y) by ƒ(z), and let ƒrdenote any quantity such that |ƒr− ƒ(zr)| ⋜ |ƒ(zr+1) − ƒ(zr)|; consider the sum(z1− z0)ƒ0+ (z2− z1)ƒ1+ ... + (z − zn−1)ƒn−1.By the definition of a path we can suppose, n being large enough, that the intermediate points z1, ... zn − 1are so taken that if zi, zi + 1be any two points intermediate, in order, to zrand zr + 1, we have |zi + i-zi| < |zr+1− zr|; we can thus suppose |z1− z0|, |z2− z1|, ... |z − zn−1|all to converge constantly to zero. This being so, we can show that the sum above has a definite limit. For this it is sufficient, as in the case of an integral of a function of one real variable, to prove this to be so when the convergence is obtained by taking new points of division intermediate to the former ones. If, however, zr, 1, zr, 2, ... zr, m−1be intermediate in order to zrand zr+1, and |ƒr, i− ƒ(zr, i)| < |ƒ(zr, i+1) − ƒ(zr, i)|, the difference between Σ(zr+1− zr)ƒrandΣ{ (zr, 1-zr)ƒr, 0+ (zr, 2− zr, 1)ƒr, 1+ ... + (zr+1− zr, m−1)ƒr, m−1},which is equal toΣrΣi(zr, i+1− zr, i) (ƒr, i− ƒr),is, when |zr+1− zr| is small enough, to ensure |ƒ(zr+1) − ƒ(zr)| < η, less in absolute value thanΣ2ηΣ|zr, i+1− zr, i|,which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 2ηS, and is therefore arbitrarily small.The limit in question is called∫zz0ƒ(z)dz. In particular when ƒ(z) = 1, it is obvious from the definition that its value is z − z0; when ƒ(z) = z, by taking ƒr= ½(zr+1− zr), it is equally clear that its value is ½(z² − z0²); these results will be applied immediately.Suppose now that to every interior and boundary point z0of a certain region there belong two definite finite numbers ƒ(z0), F(z0), such that, whatever real positive quantity η may be, a real positive number ε exists for which the condition|ƒ(z) − ƒ(z0)− F(z0)|< η,z − z0which we describe as the condition (z, z0), is satisfied for every point z, within or upon the boundary of the region, satisfying the limitation |z − z0| < ε. Then ƒ(z0) is called a differentiable function of the complex variable z0over this region, its differential coefficient being F(z0). The function ƒ(z0) is thus a continuous function of the realvariables x0, y0, where z0= x0+ iy0, over the region; it will appear that F(z0) is also continuous and in fact also a differentiable function of z0.Supposing η to be retained the same for all points z0of the region, and σ0to be the upper limit of the possible values of ε for the point z0, it is to be presumed that σ0will vary with z0, and it is not obvious as yet that the lower limit of the values of σ0as z0varies over the region may not be zero. We can, however, show that the region can be divided into a finite number of sub-regions for each of which the condition (z, z0), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position of z0, within or upon the boundary of this sub-region. This is proved above as result (B).Hence it can be proved that, for a differentiable function ƒ(z), the integral∫zz1ƒ(z)dz has the same value by whatever path within the region we pass from z1to z. This we prove by showing that when taken round a closed path in the region the integral ∫ƒ(z)dz vanishes. Consider first a triangle over which the condition (z, z0) holds, for some position of z0and every position of z, within or upon the boundary of the triangle. Then asƒ(z) = ƒ(z0) + (z − z0) F(z0) + ηθ(z − z0), where |θ| < 1,we have∫ƒ(z)dz = [ƒ(z0) − z0F(z0)] ∫dz + F(z0) ∫zdz + η∫θ(z − z0)dz,which, as the path is closed, is η ∫θ(z − z0)dz. Now, from the theorem that the absolute value of a sum is less than the sum of the absolute values of the terms, this last is less, in absolute value, than ηap, where a is the greatest side of the triangle and p is its perimeter; if Δ be the area of the triangle, we have Δ = ½ab sin C > (α/π) ba, where α is the least angle of the triangle, and hence a(a + b + c) < 2a(b + c) < 4πΔ/α; the integral ∫ƒ(z)dz round the perimeter of the triangle is thus < 4πηΔ/α. Now consider any region made up of triangles, as before explained, in each of which the condition (z, z0) holds, as in the triangle just taken. The integral ∫ƒ(z)dz round the boundary of the region is equal to the sum of the values of the integral round the component triangles, and thus less in absolute value than 4πηK/α, where K is the whole area of the region, and α is the smallest angle of the component triangles. However small η be taken, such a division of the region into a finite number of component triangles has been shown possible; the integral round the perimeter of the region is thus arbitrarily small. Thus it is actually zero, which it was desired to prove. Two remarks should be added: (1) The theorem is proved only on condition that the closed path of integration belongs to the region at every point of which the conditions are satisfied. (2) The theorem, though proved only when the region consists of triangles, holds also when the boundary points of the region consist of one or more closed paths, no two of which meet.Hence we can deduce the remarkable result that the value of ƒ(z) at any interior point of a region is expressible in terms of the value of ƒ(z) at the boundary points. For consider in the original region the function ƒ(z)/(z − z0), where z0is an interior point: this satisfies the same conditions as ƒ(z) except in the immediate neighbourhood of z0. Taking out then from the original region a small regular polygonal region with z0as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon becomes a circle, it appears that the integral∫dzƒ(z)/(z − z0) round the boundary of the original region is equal to the same integral taken counter-clockwise round a small circle having z0as centre; on this circle, however, if z − z0= rE(iθ), dz/(z − z0) = idθ, and ƒ(z) differs arbitrarily little from f(z0) if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to 2πiƒ(z0). Hence ƒ(z0) = 1/2πi∫(dtƒ(t)/(t − z0), where this integral is round the boundary of the original region. From this it appears thatF(z0) = lim.ƒ(z) − ƒ(z0)=1∫dtƒ(t)z − z02πi(t − z0)²also round the boundary of the original region. This form shows, however, that F(z0) is a continuous, finite, differentiable function of z0over the whole interior of the original region.
Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be counter-clockwise; another square, also of side a, may be added to this, so that there is a side common; this common side being erased we have a composite region with a definite direction of perimeter; to this a third square of the same size may be attached, so that there is a side common to it and one of the former squares, and this common side may be erased. If this process be continued any number of times we obtain a region of the plane bounded by one or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of description, which is such that the region is on the left of the describing point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, it being understood that there is assigned an upper limit for the greatest side of a triangle, and a lower limit for the smallest angle. In the former method, each square may be divided into four others by lines through its centre parallel to its sides; in the latter method each triangle may be divided into four others by lines joining the middle points of its sides; this halves the sides and preserves the angles. When we speak of aregionof the plane in general, unless the contrary is stated, we shall suppose it capable of being generated in this latter way by means of a finite number of triangles, there being an upper limit to the length of a side of the triangle and a lower limit to the size of an angle of the triangle. We shall also require to speak of apathin the plane; this is to be understood as capable of arising as a limit of a polygonal path of finite length, there being a definite direction or sense of description at every point of the path, which therefore never meets itself. From this the meaning of a closed path is clear. The boundary points of a region form one or more closed paths, but, in general, it is only in a limiting sense that the interior points of a closed path are a region.
There is a logical principle also which must be referred to. We frequently have cases where, about every interior or boundary, point z0of a certain region a circle can be put, say of radius r0, such that for all points z of the region which are interior to this circle, for which, that is, |z − z0| < r0, a certain property holds. Assuming that to r0is given the value which is the upper limit for z0, of the possible values, we may call the points |z − z0| < r0, the neighbourhood belonging to orproperto z0, and may speak of the property as the property (z, z0). The value of r0will in general vary with z0; what is in most cases of importance is the question whether the lower limit of r0for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z, z0) is of a kind which we may call extensive; such, namely, that if it holds, for some position of z0and all positions of z, within a certain region, then the property (z, z1) holds within a circle of radius R about any interior point z1of this region for all points z for which the circle |z − z1| = R is within the region. Also in this case r0varies continuously with z0. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (1) forsomepoint z0within or upon the boundary of the sub-region, (2) foreverypoint z within or upon the boundary of the sub-region.
We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (z, z0) holds for all points z and some point z0of the region, be calledsuitable: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the preceding, which converge to a point, say ζ; after a certain stage all these will be interior to the proper region of ζ; this, however, is contrary to the supposition that they are all unsuitable.
We now make some applications of this result (B). Suppose a definite finite real value attached to every interior or boundary point of the region, say ƒ(x, y). It may have a finite upper limit H for the region, so that no point (x, y) exists for which ƒ(x, y) > H, but points (x, y) exist for which ƒ(x, y) > H − ε, however small ε may be; if not we say that its upper limit is infinite. There is then at least one point of the region such that, for points of the region within a circle about this point, the upper limit of ƒ(x, y) is H, however small the radius of the circle be taken; for if not we can put about every point of the region a circle within which the upper limit of ƒ(x, y) is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function ƒ(x, y) is the radius r0of the neighbourhood proper to any point z0, spoken of above. We can hence prove the statement (A) above.
Suppose the property (z, z0) extensive, and, if possible, that the lower limit of r0is zero. Let then ζ be a point such that the lower limit of r0is zero for points z0within a circle about ζ however small; let r be the radius of the neighbourhood proper to ζ; take z0so that |z0-ζ| < ½r; the property (z, z0), being extensive, holds within a circle, centre z0, of radius r − |z0− ζ|, which is greater than |z0− ζ|, and increases to r as |z0− ζ| diminishes; this being true for all points z0near ζ, the lower limit of r0is not zero for the neighbourhood of ζ, contrary to what was supposed. This proves (A). Also, as is here shown that r0⋝ r − |z0− ζ|, may similarly be shown that r ⋝ r0− |z0− ζ|. Thus r0differs arbitrarily little from r when |z0− ζ| is sufficiently small; that is, r0varies continuously with z0. Next suppose the function ƒ(x, y), which has a definite finite value at every point of the region considered, to be continuous but not necessarily real, so that about every point z0, within or upon the boundary of the region, η being an arbitrary real positive quantity assigned beforehand, a circle is possible, so that for all points z of the region interior to this circle, we have |ƒ(x, y) −ƒ(x0, y0)| < ½η, and therefore (x′, y′) being any other point interior to this circle, |ƒ(x′, y′) − ƒ(x, y)| < η. We can then apply the result (A) obtained above, taking for the neighbourhood proper to any point z0the circular area within which, for any two points (x, y), (x′, y′), we have |ƒ(x′, x′) − ƒ(x, y)| < η. This is clearly an extensive property. Thus, a number r is assignable, greater than zero, such that, for any two points (x, y), (x′, y′) within a circle |z − z0| = r about any point z0, we have |ƒ(x′, y′) − ƒ(x, y)| < η, and, in particular, |ƒ(x, y) −ƒ(x0, y0)| < η, where η is an arbitrary real positive quantity agreed upon beforehand.
Take now any path in the region, whose extreme points are z0, z, and let z1, ... zn−1be intermediate points of the path, in order; denote the continuous function ƒ(x, y) by ƒ(z), and let ƒrdenote any quantity such that |ƒr− ƒ(zr)| ⋜ |ƒ(zr+1) − ƒ(zr)|; consider the sum
(z1− z0)ƒ0+ (z2− z1)ƒ1+ ... + (z − zn−1)ƒn−1.
By the definition of a path we can suppose, n being large enough, that the intermediate points z1, ... zn − 1are so taken that if zi, zi + 1be any two points intermediate, in order, to zrand zr + 1, we have |zi + i-zi| < |zr+1− zr|; we can thus suppose |z1− z0|, |z2− z1|, ... |z − zn−1|all to converge constantly to zero. This being so, we can show that the sum above has a definite limit. For this it is sufficient, as in the case of an integral of a function of one real variable, to prove this to be so when the convergence is obtained by taking new points of division intermediate to the former ones. If, however, zr, 1, zr, 2, ... zr, m−1be intermediate in order to zrand zr+1, and |ƒr, i− ƒ(zr, i)| < |ƒ(zr, i+1) − ƒ(zr, i)|, the difference between Σ(zr+1− zr)ƒrand
Σ{ (zr, 1-zr)ƒr, 0+ (zr, 2− zr, 1)ƒr, 1+ ... + (zr+1− zr, m−1)ƒr, m−1},
which is equal to
ΣrΣi(zr, i+1− zr, i) (ƒr, i− ƒr),
is, when |zr+1− zr| is small enough, to ensure |ƒ(zr+1) − ƒ(zr)| < η, less in absolute value than
Σ2ηΣ|zr, i+1− zr, i|,
which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 2ηS, and is therefore arbitrarily small.
The limit in question is called∫zz0ƒ(z)dz. In particular when ƒ(z) = 1, it is obvious from the definition that its value is z − z0; when ƒ(z) = z, by taking ƒr= ½(zr+1− zr), it is equally clear that its value is ½(z² − z0²); these results will be applied immediately.
Suppose now that to every interior and boundary point z0of a certain region there belong two definite finite numbers ƒ(z0), F(z0), such that, whatever real positive quantity η may be, a real positive number ε exists for which the condition
which we describe as the condition (z, z0), is satisfied for every point z, within or upon the boundary of the region, satisfying the limitation |z − z0| < ε. Then ƒ(z0) is called a differentiable function of the complex variable z0over this region, its differential coefficient being F(z0). The function ƒ(z0) is thus a continuous function of the realvariables x0, y0, where z0= x0+ iy0, over the region; it will appear that F(z0) is also continuous and in fact also a differentiable function of z0.
Supposing η to be retained the same for all points z0of the region, and σ0to be the upper limit of the possible values of ε for the point z0, it is to be presumed that σ0will vary with z0, and it is not obvious as yet that the lower limit of the values of σ0as z0varies over the region may not be zero. We can, however, show that the region can be divided into a finite number of sub-regions for each of which the condition (z, z0), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position of z0, within or upon the boundary of this sub-region. This is proved above as result (B).
Hence it can be proved that, for a differentiable function ƒ(z), the integral∫zz1ƒ(z)dz has the same value by whatever path within the region we pass from z1to z. This we prove by showing that when taken round a closed path in the region the integral ∫ƒ(z)dz vanishes. Consider first a triangle over which the condition (z, z0) holds, for some position of z0and every position of z, within or upon the boundary of the triangle. Then as
ƒ(z) = ƒ(z0) + (z − z0) F(z0) + ηθ(z − z0), where |θ| < 1,
we have
∫ƒ(z)dz = [ƒ(z0) − z0F(z0)] ∫dz + F(z0) ∫zdz + η∫θ(z − z0)dz,
which, as the path is closed, is η ∫θ(z − z0)dz. Now, from the theorem that the absolute value of a sum is less than the sum of the absolute values of the terms, this last is less, in absolute value, than ηap, where a is the greatest side of the triangle and p is its perimeter; if Δ be the area of the triangle, we have Δ = ½ab sin C > (α/π) ba, where α is the least angle of the triangle, and hence a(a + b + c) < 2a(b + c) < 4πΔ/α; the integral ∫ƒ(z)dz round the perimeter of the triangle is thus < 4πηΔ/α. Now consider any region made up of triangles, as before explained, in each of which the condition (z, z0) holds, as in the triangle just taken. The integral ∫ƒ(z)dz round the boundary of the region is equal to the sum of the values of the integral round the component triangles, and thus less in absolute value than 4πηK/α, where K is the whole area of the region, and α is the smallest angle of the component triangles. However small η be taken, such a division of the region into a finite number of component triangles has been shown possible; the integral round the perimeter of the region is thus arbitrarily small. Thus it is actually zero, which it was desired to prove. Two remarks should be added: (1) The theorem is proved only on condition that the closed path of integration belongs to the region at every point of which the conditions are satisfied. (2) The theorem, though proved only when the region consists of triangles, holds also when the boundary points of the region consist of one or more closed paths, no two of which meet.
Hence we can deduce the remarkable result that the value of ƒ(z) at any interior point of a region is expressible in terms of the value of ƒ(z) at the boundary points. For consider in the original region the function ƒ(z)/(z − z0), where z0is an interior point: this satisfies the same conditions as ƒ(z) except in the immediate neighbourhood of z0. Taking out then from the original region a small regular polygonal region with z0as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon becomes a circle, it appears that the integral∫dzƒ(z)/(z − z0) round the boundary of the original region is equal to the same integral taken counter-clockwise round a small circle having z0as centre; on this circle, however, if z − z0= rE(iθ), dz/(z − z0) = idθ, and ƒ(z) differs arbitrarily little from f(z0) if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to 2πiƒ(z0). Hence ƒ(z0) = 1/2πi∫(dtƒ(t)/(t − z0), where this integral is round the boundary of the original region. From this it appears that
also round the boundary of the original region. This form shows, however, that F(z0) is a continuous, finite, differentiable function of z0over the whole interior of the original region.
§ 5.Applications.—The previous results have manifold applications.
(1) If an infinite series of differentiable functions of z be uniformly convergent along a certain path lying with the region of definition of the functions, so that S(2) = u0(z) + u1(z) + ... + un−1(z) + Rn(z), where |Rn(z)| < ε for all points of the path, we have∫zz0S(z)dz =∫zz0u0(z)dz +∫zz0u1(z)dz + ... +∫zz0un−1(z)dz +∫zz0Rn(z)dz,wherein, in absolute value,∫zz0Rn(z)dz < εL, if L be the length of the path. Thus the series may be integrated, and the resulting series is also uniformly convergent.(2) If ƒ(x, y) be definite, finite and continuous at every point of a region, and over any closed path in the region ∫ƒ(x, y)dz = 0, then ψ(z) =∫zz0ƒ(x, y)dz, for interior points z0, z, is a differentiable function of z, having for its differential coefficient the function ƒ(x, y), which is therefore also a differentiable function of z at interior points.(3) Hence if the series u0(z) + u1(z) + ... to ∞ be uniformly convergent over a region, its terms being differentiable functions of z, then its sum S(z) is a differentiable function of z, whose differential coefficient, given by (1/2πi) ∫ 2πi/(t − z)², is obtainable by differentiating the series. This theorem, unlike (1), does not hold for functions of a real variable.(4) If the region of definition of a differentiable function ƒ(z) include the region bounded by two concentric circles of radii r, R, with centre at the origin, and z0be an interior point of this region,ƒ(z0) =1∫ƒ(t)dt−1∫ƒ(t)dt,2πiRt− z02πirt− z0where the integrals are both counter-clockwise round the two circumferences respectively; putting in the first (t − z0)−1=Σn=0z0n/tn+1, and in the second (t − z0)−1= −Σn=0tn/z0n+1, we find ƒ(z0) =Σ∞−∞Anz0n, wherein An= (1/2πi)∫[ƒ(t)/tn+1] dt, taken round any circle, centre the origin, of radius intermediate between r and R. Particular cases are: (α) when the region of definition of the function includes the whole interior of the outer circle; then we may take r = 0, the coefficients Anfor which n < 0 all vanish, and the function ƒ(z0) is expressed for the whole interior |z0| < R by a power seriesΣ∞0Anz0n. In other words,about every interior point c of the region of definition a differentiable function of z is expressible by a power series in z − c; a very important result.(β) If the region of definition, though not including the origin, extends to within arbitrary nearness of this on all sides, and at the same time the product zmƒ(z) has a finite limit when |z| diminishes to zero, all the coefficients Anfor which n < −m vanish, and we havef(z0) = A−mz0−m+ A−m+1z0−m+1+ ... + A−1z0−1+ A0+ A1z0... to ∞.Such a case occurs, for instance, when ƒ(z) = cosec z, the number m being unity.
(1) If an infinite series of differentiable functions of z be uniformly convergent along a certain path lying with the region of definition of the functions, so that S(2) = u0(z) + u1(z) + ... + un−1(z) + Rn(z), where |Rn(z)| < ε for all points of the path, we have
wherein, in absolute value,∫zz0Rn(z)dz < εL, if L be the length of the path. Thus the series may be integrated, and the resulting series is also uniformly convergent.
(2) If ƒ(x, y) be definite, finite and continuous at every point of a region, and over any closed path in the region ∫ƒ(x, y)dz = 0, then ψ(z) =∫zz0ƒ(x, y)dz, for interior points z0, z, is a differentiable function of z, having for its differential coefficient the function ƒ(x, y), which is therefore also a differentiable function of z at interior points.
(3) Hence if the series u0(z) + u1(z) + ... to ∞ be uniformly convergent over a region, its terms being differentiable functions of z, then its sum S(z) is a differentiable function of z, whose differential coefficient, given by (1/2πi) ∫ 2πi/(t − z)², is obtainable by differentiating the series. This theorem, unlike (1), does not hold for functions of a real variable.
(4) If the region of definition of a differentiable function ƒ(z) include the region bounded by two concentric circles of radii r, R, with centre at the origin, and z0be an interior point of this region,
where the integrals are both counter-clockwise round the two circumferences respectively; putting in the first (t − z0)−1=Σn=0z0n/tn+1, and in the second (t − z0)−1= −Σn=0tn/z0n+1, we find ƒ(z0) =Σ∞−∞Anz0n, wherein An= (1/2πi)∫[ƒ(t)/tn+1] dt, taken round any circle, centre the origin, of radius intermediate between r and R. Particular cases are: (α) when the region of definition of the function includes the whole interior of the outer circle; then we may take r = 0, the coefficients Anfor which n < 0 all vanish, and the function ƒ(z0) is expressed for the whole interior |z0| < R by a power seriesΣ∞0Anz0n. In other words,about every interior point c of the region of definition a differentiable function of z is expressible by a power series in z − c; a very important result.
(β) If the region of definition, though not including the origin, extends to within arbitrary nearness of this on all sides, and at the same time the product zmƒ(z) has a finite limit when |z| diminishes to zero, all the coefficients Anfor which n < −m vanish, and we have
f(z0) = A−mz0−m+ A−m+1z0−m+1+ ... + A−1z0−1+ A0+ A1z0... to ∞.
Such a case occurs, for instance, when ƒ(z) = cosec z, the number m being unity.
§ 6.Singular Points.—Theregion of existenceof a differentiable function of z is an unclosed aggregate of points, each of which is an interior point of a neighbourhood consisting wholly of points of the aggregate, at every point of which the function is definite and finite and possesses a unique finite differential coefficient. Every point of the plane, not belonging to the aggregate, which is a limiting point of points of the aggregate, such, that is, that points of the aggregate lie in every neighbourhood of this, is called asingular pointof the function.
About every interior point z0of the region of existence the function may be represented by a power series in z − z0, and the series converges and represents the function over any circle centre at z0which contains no singular point in its interior. This has been proved above. And it can be similarly proved, putting z = 1/ζ, that if the region of existence of the function contains all points of the plane for which |z| > R, then the function is representable for all such points by a power series in z− 1or ζ; in such case we say that the region of existence of the function contains the point z = ∞. A series in z− 1has a finite limit when |z| = ∞; a series in z cannot remain finite for all points z for which |z| > R; for if, for |z| = R, the sum of a power series Σanznin z is in absolute value less than M, we have |an| < Mr−n, and therefore, if M remains finite for all values of r however great, an= 0. Thus the region of existence of a function if it contains all finite points of the plane cannot contain the point z = ∞; such is, for instance, the case of the function exp (z) = Σzn/n!. This may be regarded as a particular case of a well-known result (§ 7), that the circumference of convergence of any power series representing the function contains at least one singular point. As an extreme case functions exist whose region of existence is circular, there being a singular point in every arc of the circumference, however small; for instance, this is the case for the functions represented for |z| < 1 by the seriesΣn=0zm, where m = n², the seriesΣn=0zmwhere m = n!, and the seriesΣn=1zm/(m + 1)(m + 2) where m = an, a being a positive integer, although in the last case the series actually converges for every point of the circle of convergence |z| = 1. If z be a point interior to the circle of convergence of a series representing the function, the series may be rearranged in powers of z − z0; as z0approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in z − z0diminish to zero; when, however, a circle can be put about z0, not containing any singular point of the function, but containing points outside the circle of convergence of the original series, then the series in z − z0gives the value of the function for these external points. If the function be supposed to be given only for the interior of the original circle, by the original power series, the series in z − z0converging beyond the original circle gives what is known as ananalytical continuationof the function. It appears from what hasbeen proved that the value of the function at all points of its region of existence can be obtained from its value, supposed given by a series in one original circle, by a succession of such processes of analytical continuation.
About every interior point z0of the region of existence the function may be represented by a power series in z − z0, and the series converges and represents the function over any circle centre at z0which contains no singular point in its interior. This has been proved above. And it can be similarly proved, putting z = 1/ζ, that if the region of existence of the function contains all points of the plane for which |z| > R, then the function is representable for all such points by a power series in z− 1or ζ; in such case we say that the region of existence of the function contains the point z = ∞. A series in z− 1has a finite limit when |z| = ∞; a series in z cannot remain finite for all points z for which |z| > R; for if, for |z| = R, the sum of a power series Σanznin z is in absolute value less than M, we have |an| < Mr−n, and therefore, if M remains finite for all values of r however great, an= 0. Thus the region of existence of a function if it contains all finite points of the plane cannot contain the point z = ∞; such is, for instance, the case of the function exp (z) = Σzn/n!. This may be regarded as a particular case of a well-known result (§ 7), that the circumference of convergence of any power series representing the function contains at least one singular point. As an extreme case functions exist whose region of existence is circular, there being a singular point in every arc of the circumference, however small; for instance, this is the case for the functions represented for |z| < 1 by the seriesΣn=0zm, where m = n², the seriesΣn=0zmwhere m = n!, and the seriesΣn=1zm/(m + 1)(m + 2) where m = an, a being a positive integer, although in the last case the series actually converges for every point of the circle of convergence |z| = 1. If z be a point interior to the circle of convergence of a series representing the function, the series may be rearranged in powers of z − z0; as z0approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in z − z0diminish to zero; when, however, a circle can be put about z0, not containing any singular point of the function, but containing points outside the circle of convergence of the original series, then the series in z − z0gives the value of the function for these external points. If the function be supposed to be given only for the interior of the original circle, by the original power series, the series in z − z0converging beyond the original circle gives what is known as ananalytical continuationof the function. It appears from what hasbeen proved that the value of the function at all points of its region of existence can be obtained from its value, supposed given by a series in one original circle, by a succession of such processes of analytical continuation.
§ 7.Monogenic Functions.—This suggests an entirely different way of formulating the fundamental parts of the theory of functions of a complex variable, which appears to be preferable to that so far followed here.
Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z − z0, about any point z0interior to its circle of convergence, and the new series converges certainly for |z − z0| < r − |z0|, if r be the original radius of convergence. If for every position of z0this is the greatest radius of convergence of the derived series, then the original series represents a function existing only within its circle of convergence. If for some position of z0the derived series converges for |z − z0| < r − |z0| + D, then it can be shown that for points z, interior to the original circle, lying in the annulus r − |z0| < |z − z0| < r − |z0| + D, the value represented by the derived series agrees with that represented by the original series. If for another point z1interior to the original circle the derived series converges for |z − z1| < r − |z1| + E, and the two circles |z − z0| = r − |z0| + D, |z − z1| = r − |z1| + E have interior points common, lying beyond |z| = r, then it can be shown that the values represented by these series at these common points agree. Either series then can be used to furnish an analytical continuation of the function as originally defined. Continuing this process of continuation as far as possible, we arrive at the conception of the function as defined by an aggregate of power series of which every one has points of convergence common with some one or more others; the whole aggregate of points of the plane which can be so reached constitutes the region of existence of the function; the limiting points of this region are the points in whose neighbourhood the derived series have radii of convergence diminishing indefinitely to zero; these are the singular points. The circle of convergence of any of the series has at least one such singular point upon its circumference. So regarded the function is called amonogenicfunction, the epithet having reference to the single origin, by one power series, of the expressions representing the function; it is also sometimes called amonogenic analyticalfunction, or simply ananalyticalfunction; all that is necessary to define it is the value of the function and of all its differential coefficients, at some one point of the plane; in the method previously followed here it was necessary to suppose the function differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy’s theorem, that ∫ƒ(z)dz = 0 over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior of its circle of convergence. There is another advantage belonging to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a monogenic function is by no means necessarily single valued—it may quite well happen that starting from a particular power series, converging over a certain circle, and applying the process of analytical continuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. The notion of basing the theory of functions on the theory of power series is, after Newton, largely due to Lagrange, who has some interesting remarks in this regard at the beginning of hisThéorie des fonctions analytiques. He applies the idea, however, primarily to functions of a real variable for which the expression by power series is only of very limited validity; for functions of a complex variable probably the systematization of the theory owes most to Weierstrass, whose use of the word monogenic is that adopted above. In what follows we generally suppose this point of view to be regarded as fundamental.
Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z − z0, about any point z0interior to its circle of convergence, and the new series converges certainly for |z − z0| < r − |z0|, if r be the original radius of convergence. If for every position of z0this is the greatest radius of convergence of the derived series, then the original series represents a function existing only within its circle of convergence. If for some position of z0the derived series converges for |z − z0| < r − |z0| + D, then it can be shown that for points z, interior to the original circle, lying in the annulus r − |z0| < |z − z0| < r − |z0| + D, the value represented by the derived series agrees with that represented by the original series. If for another point z1interior to the original circle the derived series converges for |z − z1| < r − |z1| + E, and the two circles |z − z0| = r − |z0| + D, |z − z1| = r − |z1| + E have interior points common, lying beyond |z| = r, then it can be shown that the values represented by these series at these common points agree. Either series then can be used to furnish an analytical continuation of the function as originally defined. Continuing this process of continuation as far as possible, we arrive at the conception of the function as defined by an aggregate of power series of which every one has points of convergence common with some one or more others; the whole aggregate of points of the plane which can be so reached constitutes the region of existence of the function; the limiting points of this region are the points in whose neighbourhood the derived series have radii of convergence diminishing indefinitely to zero; these are the singular points. The circle of convergence of any of the series has at least one such singular point upon its circumference. So regarded the function is called amonogenicfunction, the epithet having reference to the single origin, by one power series, of the expressions representing the function; it is also sometimes called amonogenic analyticalfunction, or simply ananalyticalfunction; all that is necessary to define it is the value of the function and of all its differential coefficients, at some one point of the plane; in the method previously followed here it was necessary to suppose the function differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy’s theorem, that ∫ƒ(z)dz = 0 over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior of its circle of convergence. There is another advantage belonging to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a monogenic function is by no means necessarily single valued—it may quite well happen that starting from a particular power series, converging over a certain circle, and applying the process of analytical continuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. The notion of basing the theory of functions on the theory of power series is, after Newton, largely due to Lagrange, who has some interesting remarks in this regard at the beginning of hisThéorie des fonctions analytiques. He applies the idea, however, primarily to functions of a real variable for which the expression by power series is only of very limited validity; for functions of a complex variable probably the systematization of the theory owes most to Weierstrass, whose use of the word monogenic is that adopted above. In what follows we generally suppose this point of view to be regarded as fundamental.
§ 8.Some Elementary Properties of Single Valued Functions.—Apoleis a singular point of the function ƒ(z) which is not a singularity of the function 1/ƒ(z); this latter function is therefore, by the definition, capable of representation about this point, z0, by a series [ƒ(z)]−1= Σan(z − z0)n. If herein a0is not zero we can hence derive a representation for ƒ(z) as a power series about z0, contrary to the hypothesis that z0is a singular point for this function. Hence a0= 0; suppose also a1= 0, a2= 0, ... am−1= 0, but am± 0. Then [ƒ(z)]−1= (z − z0)m[am+ am+1(z − z0) + ...], and hence (z − z0)mƒ(z) = am−1+ Σbn(z − z0)n, namely, the expression of ƒ(z) about z = z0contains a finite number of negative powers of z − z0and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity.
The integral ∫ƒ(z)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 2πiA1, where A1is the coefficient of (z − z0)−1in the expansion of ƒ(z) at the pole; this coefficient has therefore a certain uniqueness, and it is called theresidue of ƒ(z) at the pole. Considering a region in which there are no other singularities than poles, all these being interior points,the integral (1/2πi)∫ƒ(z)dz round the boundary of this region is equal to the sum of the residues at the included poles, a very important result. Any singular point of a function which is not a pole is called anessential singularity; if it be isolated the function is capable, in the neighbourhood of this point, of approaching arbitrarily near to any assigned value. For, the point being isolated, the function can be represented, in its neighbourhood, as we have proved, by a seriesΣ∞−∞an(z − z0)n; it thus cannot remain finite in the immediate neighbourhood of the point. The point is necessarily an isolated essential singularity also of the function {ƒ(z) − A}−1for if this were expressible by a power series about the point, so would also the function ƒ(z) be; as {ƒ(z) − A}− 1approaches infinity, so does ƒ(z) approach the arbitrary value A. Similar remarks apply to the point z = ∞, the function being regarded as a function of ζ = z−1. In the neighbourhood of an essential singularity, which is a limiting point also of poles, the function clearly becomes infinite. For an essential singularity which is not isolated the same result does not necessarily hold.
The integral ∫ƒ(z)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 2πiA1, where A1is the coefficient of (z − z0)−1in the expansion of ƒ(z) at the pole; this coefficient has therefore a certain uniqueness, and it is called theresidue of ƒ(z) at the pole. Considering a region in which there are no other singularities than poles, all these being interior points,the integral (1/2πi)∫ƒ(z)dz round the boundary of this region is equal to the sum of the residues at the included poles, a very important result. Any singular point of a function which is not a pole is called anessential singularity; if it be isolated the function is capable, in the neighbourhood of this point, of approaching arbitrarily near to any assigned value. For, the point being isolated, the function can be represented, in its neighbourhood, as we have proved, by a seriesΣ∞−∞an(z − z0)n; it thus cannot remain finite in the immediate neighbourhood of the point. The point is necessarily an isolated essential singularity also of the function {ƒ(z) − A}−1for if this were expressible by a power series about the point, so would also the function ƒ(z) be; as {ƒ(z) − A}− 1approaches infinity, so does ƒ(z) approach the arbitrary value A. Similar remarks apply to the point z = ∞, the function being regarded as a function of ζ = z−1. In the neighbourhood of an essential singularity, which is a limiting point also of poles, the function clearly becomes infinite. For an essential singularity which is not isolated the same result does not necessarily hold.
A single valued function is said to be anintegralfunction when it has no singular points except z = ∞. Such is, for instance, an integral polynomial, which has z = ∞ for a pole, and the functions exp (z) which has z = ∞ as an essential singularity. A function which has no singular points for finite values of z other than poles is called ameromorphicfunction. If it also have a pole at z = ∞ it is arationalfunction; for then, if a1, ... asbe its finite poles, of orders m1; m2, ... ms, the product (z − a1)m1... (z − as)msƒ(z) is an integral function with a pole at infinity, capable therefore, for large values of z, of an expression (z−1)−mΣr=0ar(z−1)r; thus (z − a1)m1... (z − as)msƒ(z) is capable of a formΣr=0brzr, but z−mΣr=0brzrremains finite for z = ∞. Therefore br+1= br+2= ... = 0, andƒ(z) is a rational function.