If for a single valued function F(z) every singular point in the finite part of the plane is isolated there can only be a finite number of these in any finite part of the plane, and they can be taken to be a1, a2, a3, ... with |a1| ⋜ |a2| ⋜ |a3| ... and limit |an| = ∞. About asthe function is expressible asΣ∞−∞An(z − as)n; let ƒs(z) =Σ1−∞An(z − as)nbe the sum of the negative powers in this expansion. Assuming z = 0 not to be a singular point, let ƒs(z) be expanded in powers of z, in the formΣn=0Cnzn, and μsbe chosen so that Fs(z) = ƒs(z) −Σμs−11Cnzn=Σ∞μsCnznis, for |z| < rs< |as|, less in absolute value than the general term εsof a fore-agreed convergent series of real positive terms. Then the series φ(z) =Σ∞s=1Fs(z) converges uniformly in any finite region of the plane, other than at the points as, and is expressible about any point by a power series, and near as, φ(z) − fs(z) is expressible by a power series in z − as. Thus F(z) − φ(z) is an integral function. In particular when all the finite singularities of F(z) are poles, F(z) is hereby expressed as the sum of an integral function and a series of rational functions. The condition |Fs(z)| < εsis imposed only to render the series ΣFs(z) uniformly convergent; this condition may in particular cases be satisfied by a seriesΣGs(z) where Gs(z) = ƒs(z) −Σνs−11Cnznand νs< μs. An example of the theorem is the function π cot πz − z− 1for which, taking at first only half the poles, ƒs(z) = 1/(z − s); in this case the seriesΣFs(z) where Fs(z) = (z − s)−1+ s−1is uniformly convergent; thus π cot πz − z−1−Σ∞−∞[(z − s)−1+ s−1], where s = 0 is excluded from the summation, is an integral function. It can be proved that this integral function vanishes.Considering an integral function ƒ(z), if there be no finite positions of z for which this function vanishes, the function λ[ƒ(z)] is at once seen to be an integral function, φ(z), or ƒ(z) = exp[φ(z)]; if however great R may be there be only a finite number of values of z for which ƒ(z) vanishes, say z = a1, ... am, then it is at once seen that ƒ(z) = exp [φ(z)]. (z − a1)h1...(z − am)hm, where φ(z) is an integral function, and h1, ... hmare positive integers. If, however, ƒ(z) vanish for z = a1, a2... where |a1| ⋜ |a2| ⋜ ... and limit |an| = ∞, and if for simplicity we assume that z − 0 is not a zero and all the zeros a1, a2, ... are of the first order, we find, by applying the preceding theorem to the function [1 / ƒ(z)] [dƒ(z) / dz], that ƒ(z) = exp [φ(z)]Π∞n=1{(1 − z/an) exp φn(z)}, where φ(z) is an integral function, and φn(z) is an integral polynomial of the form φn(z) = z/an+ z2/2an2+ ... + zs/sans. The number s may be the same for all values of n, or it may increase indefinitely with n; it is sufficient in any case to take s = n. In particular for the functionsinπx/πx, we havesin πx=Π∞−∞{ (1 −x)exp(x) },πxnnwhere n = 0 is excluded from the product. Or again we have1= xeCxΠ∞n=1{ (1 +x)exp(−x) },Γ(x)nnwhere C is a constant, and Γ(x) is a function expressible when x is real and positive by the integral∫∞0e−ttx−1dt.There exist interesting investigations as to the connexion of the value of s above, the law of increase of the modulus of the integral function ƒ(z), and the law of increase of the coefficients in the series ƒ(z) =Σanznas n increases (see the bibliography below underIntegral Functions). It can be shown, moreover, that an integral function actually assumes every finite complex value, save, in exceptional cases, one value at most. For instance, the function exp (z) assumes every finite value except zero (see below under § 21,Modular Functions).
If for a single valued function F(z) every singular point in the finite part of the plane is isolated there can only be a finite number of these in any finite part of the plane, and they can be taken to be a1, a2, a3, ... with |a1| ⋜ |a2| ⋜ |a3| ... and limit |an| = ∞. About asthe function is expressible asΣ∞−∞An(z − as)n; let ƒs(z) =Σ1−∞An(z − as)nbe the sum of the negative powers in this expansion. Assuming z = 0 not to be a singular point, let ƒs(z) be expanded in powers of z, in the formΣn=0Cnzn, and μsbe chosen so that Fs(z) = ƒs(z) −Σμs−11Cnzn=Σ∞μsCnznis, for |z| < rs< |as|, less in absolute value than the general term εsof a fore-agreed convergent series of real positive terms. Then the series φ(z) =Σ∞s=1Fs(z) converges uniformly in any finite region of the plane, other than at the points as, and is expressible about any point by a power series, and near as, φ(z) − fs(z) is expressible by a power series in z − as. Thus F(z) − φ(z) is an integral function. In particular when all the finite singularities of F(z) are poles, F(z) is hereby expressed as the sum of an integral function and a series of rational functions. The condition |Fs(z)| < εsis imposed only to render the series ΣFs(z) uniformly convergent; this condition may in particular cases be satisfied by a seriesΣGs(z) where Gs(z) = ƒs(z) −Σνs−11Cnznand νs< μs. An example of the theorem is the function π cot πz − z− 1for which, taking at first only half the poles, ƒs(z) = 1/(z − s); in this case the seriesΣFs(z) where Fs(z) = (z − s)−1+ s−1is uniformly convergent; thus π cot πz − z−1−Σ∞−∞[(z − s)−1+ s−1], where s = 0 is excluded from the summation, is an integral function. It can be proved that this integral function vanishes.
Considering an integral function ƒ(z), if there be no finite positions of z for which this function vanishes, the function λ[ƒ(z)] is at once seen to be an integral function, φ(z), or ƒ(z) = exp[φ(z)]; if however great R may be there be only a finite number of values of z for which ƒ(z) vanishes, say z = a1, ... am, then it is at once seen that ƒ(z) = exp [φ(z)]. (z − a1)h1...(z − am)hm, where φ(z) is an integral function, and h1, ... hmare positive integers. If, however, ƒ(z) vanish for z = a1, a2... where |a1| ⋜ |a2| ⋜ ... and limit |an| = ∞, and if for simplicity we assume that z − 0 is not a zero and all the zeros a1, a2, ... are of the first order, we find, by applying the preceding theorem to the function [1 / ƒ(z)] [dƒ(z) / dz], that ƒ(z) = exp [φ(z)]Π∞n=1{(1 − z/an) exp φn(z)}, where φ(z) is an integral function, and φn(z) is an integral polynomial of the form φn(z) = z/an+ z2/2an2+ ... + zs/sans. The number s may be the same for all values of n, or it may increase indefinitely with n; it is sufficient in any case to take s = n. In particular for the functionsinπx/πx, we have
where n = 0 is excluded from the product. Or again we have
where C is a constant, and Γ(x) is a function expressible when x is real and positive by the integral∫∞0e−ttx−1dt.
There exist interesting investigations as to the connexion of the value of s above, the law of increase of the modulus of the integral function ƒ(z), and the law of increase of the coefficients in the series ƒ(z) =Σanznas n increases (see the bibliography below underIntegral Functions). It can be shown, moreover, that an integral function actually assumes every finite complex value, save, in exceptional cases, one value at most. For instance, the function exp (z) assumes every finite value except zero (see below under § 21,Modular Functions).
The two theorems given above, the one, known as Mittag-Leffler’s theorem, relating to the expression as a sum of simpler functions of a function whose singular points have the point z = ∞ as their only limiting point, the other, Weierstrass’s factor theorem, giving the expression of an integral function as a product of factors each with only one zero in the finite part of the plane, may be respectively generalized as follows:—
I. If a1, a2, a3, ... be an infinite series of isolated points having the points of the aggregate (c) as their limiting points, so that in any neighbourhood of a point of (c) there exists an infinite number of the points a1, a2, ..., and with every point aithere be associated a polynomial in (z − ai)−1, say gi; then there exists a single valued function whose region of existence excludes only the points (a) and the points (c), having in a point aia pole whereat the expansion consists of the terms gi, together with a power series in z − ai; the function is expressible as an infinite series of terms gi− γi, where γiis also a rational function.II. With a similar aggregate (a), with limiting points (c), suppose with every point aithere is associated a positive integer ri. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order riat the point ai, but not elsewhere, expressible in the formΠ∞n=1(1 −an− cn)rnexp (gn),z − cnwhere with every point anis associated a proper point cnof (c), andgn= rnΣμns=11(an− cn)s,sz − cnμnbeing a properly chosen positive integer.If it should happen that the points (c) determine a path dividing the plane into separated regions, as, for instance, if an= R(1 − n−1) exp (iπ √2·n), when (c) consists of the points of the circle |z| = R, the product expression above denotes different monogenic functions in the different regions, not continuable into one another.
I. If a1, a2, a3, ... be an infinite series of isolated points having the points of the aggregate (c) as their limiting points, so that in any neighbourhood of a point of (c) there exists an infinite number of the points a1, a2, ..., and with every point aithere be associated a polynomial in (z − ai)−1, say gi; then there exists a single valued function whose region of existence excludes only the points (a) and the points (c), having in a point aia pole whereat the expansion consists of the terms gi, together with a power series in z − ai; the function is expressible as an infinite series of terms gi− γi, where γiis also a rational function.
II. With a similar aggregate (a), with limiting points (c), suppose with every point aithere is associated a positive integer ri. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order riat the point ai, but not elsewhere, expressible in the form
where with every point anis associated a proper point cnof (c), and
μnbeing a properly chosen positive integer.
If it should happen that the points (c) determine a path dividing the plane into separated regions, as, for instance, if an= R(1 − n−1) exp (iπ √2·n), when (c) consists of the points of the circle |z| = R, the product expression above denotes different monogenic functions in the different regions, not continuable into one another.
§ 9.Construction of a Monogenic Function with a given Region of Existence.—A series of isolated points interior to a given region can be constructed in infinitely many ways whose limiting points are the boundary points of the region, or are boundary points of the region of such denseness that one of them is found in the neighbourhood of every point of the boundary, however small. Then the application of the last enunciated theorem gives rise to a function having no singularities in the interior of the region, but having a singularity in a boundary point in every small neighbourhood of every boundary point; this function has the given region as region of existence.
§ 10.Expression of a Monogenic Function by means of Rational Functions in a given Region.—Suppose that we have a region R0of the plane, as previously explained, for all the interior or boundary points of which z is finite, and let its boundary points, consisting of one or more closed polygonal paths, no two of which have a point in common, be called C0. Further suppose that all the points of this region, including the boundary points, are interior points of another region R, whose boundary is denoted by C. Let z be restricted to be within or upon the boundary of C0; let a, b, ... be finite points upon C or outside R. Then when b is near enough to a, the fraction (a − b)/(z − b) is arbitrarily small for all positions of z; say
the rational function of the complex variable t,
in which n is a positive integer, is not infinite at t = a, but has a pole at t = b. By taking n large enough, the value of this function, for all positions z of t belonging to R0, differs as little as may be desired from (t − a)−1. By taking a sum of terms such as
we can thus build a rational function differing, in value, in R0, as little as may be desired from a given rational function
ƒ =ΣAp(t − a)−p,
and differing, outside R or upon the boundary of R, from ƒ, in the fact that while ƒ is infinite at t = a, F is infinite only at t = b. By a succession of steps of this kind we thus have the theorem that, given a rational function of t whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in R0arbitrarily little from the former, whose poles are all at the point c.
Now any monogenic function ƒ(t) whose region of definition includes C and the interior of R can be represented at all points z in R0byƒ(z) =1∫ƒ(t)dt,2πit − zwhere the path of integration is C. This integral is the limit of a sumS =1Σƒ(ti) (ti+1− ti),2πiti− zwhere the points tiare upon C; and the proof we have given of the existence of the limit shows that the sum S converges to ƒ(z) uniformly in regard to z, when z is in R0, so that we can suppose, when the subdivision of C into intervals ti+1− ti, has been carried sufficiently far, that|S − ƒ(z)| < ε,for all points z of R0, where ε is arbitrary and agreed upon beforehand. The function S is, however, a rational function of z with poles upon C, that is external to R0. We can thus find a rational function differing arbitrarily little from S, and therefore arbitrarily little from ƒ(z), for all points z of R0, with poles at arbitrary positions outside R0which can be reached by finite continuous curves lying outside R from the points of C.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 R0whose 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 Prto be the polynomial such that, within C0, |Pr− ƒ(z)| < εr; then the infinite series of polynomialsP1(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 formH1(z) + {H2(z) − H1(z)} + {H3(z) − H2(z)} + ...then, as before, represents ƒ(z) within R0. And R0may be taken to coincide as nearly as desired with the interior of the region bounded by Γ.
Now any monogenic function ƒ(t) whose region of definition includes C and the interior of R can be represented at all points z in R0by
where the path of integration is C. This integral is the limit of a sum
where the points tiare upon C; and the proof we have given of the existence of the limit shows that the sum S converges to ƒ(z) uniformly in regard to z, when z is in R0, so that we can suppose, when the subdivision of C into intervals ti+1− ti, has been carried sufficiently far, that
|S − ƒ(z)| < ε,
for all points z of R0, where ε is arbitrary and agreed upon beforehand. The function S is, however, a rational function of z with poles upon C, that is external to R0. We can thus find a rational function differing arbitrarily little from S, and therefore arbitrarily little from ƒ(z), for all points z of R0, with poles at arbitrary positions outside R0which can be reached by finite continuous curves lying outside R from the points of C.
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 R0whose 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 Prto 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 R0may be taken to coincide as nearly as desired with the interior of the region bounded by Γ.
§ 11.Expression of(1 − z)−1by 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 takec1= 1 + c/r, c2= 1 + 2c/r, ... cr= 1 + c;if n1, n2, ... nr, be positive integers, the rational function1{1 −(c1− 1)n1}1 − ζc1− ζis finite at ζ = 1, and has a pole of order n1at ζ = c1; the rational
function1{1 −(c1− 1)n1} {1 −(c2− c1)n2}n11 − ζc1− ζc2− ζis thus finite except for ζ = c2, where it has a pole of order n1n2;
finally, writingxs=(cs− cs−1)ns,cs− ζthe rational functionU = (1 − ζ)−1(1 − x1) (1 − x2)n1(1 − x3)n1n2... (1 − xr)n1n2... nr − 1has 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 form1 − (1 − ρ1) (1 − ρ2)...(1 − ρk),in which there are equalities among ρ1, ρ2, ... ρk, is of the formΣρ1− 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| 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 is finite at ζ = 1, and has a pole of order n1at ζ = c1; the rational
function is thus finite except for ζ = c2, where it has a pole of order n1n2;
finally, writing the rational function 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 now, so long as ζ is without the closed curve above described round
ζ = 1, ζ = 1 + c, we have and hence Take an arbitrary real positive ε, and μ, a positive number, so that
εmu− 1 < εa, then a value of n1such that σn1< μ/(1 + μ) and therefore
σn1/(1 − σn1< μ, and values for n2, n3... such that σn2< 1/n1σ2n1,
σn3< 1/n1n2σ3n1, ... σnr< 1/(n1... nr−1) σnrn1; then, as 1 + x < ex, we have and therefore less than a−1{exp (σn1+ σ2n1+ ... + σnrn1) − 1}, which is less than 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 thereby the points ζ = 0, 1, 1 + c become the points z = 0, 1, ∞, the
function (1 − z)−1being 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 the lines η = ±a become the two circles expressed, if z = x + iy, by 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 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)−1for 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 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 R0lying entirely within R,
and let x be restricted to be within R0or upon its boundary; then
for any point t on the boundary of R, the points z of the plane for
which zt− 1is 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 R0we 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, 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−1ƒ(t)| < M. If now P (xt−1) = c0+ c1xt−1+ ... + cmxmt−m, and 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 and therefore where 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 with αμ< g/ρμ2m1, provided (μ + m1+ 1)μ< (2⁄3)m1 + 1; in fact for μ ⋜ 2n2n−2it is sufficient
to take m1= n2n; by another application of the same inequality,
replacing x, z respectively by a1and x/n, we have where |β′μ| < g / ρμ2m2 provided (μ + m2+ 1)μ< (3⁄2)m2+ 1; we take m2= n2n − 2, supposing
μ < 2n2n−4. So long as λ2⋜ m2⋜ n2n−2and μ < 2n2n−4we 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 where |βμ| < 2g/ρμ2m2, the numbers m1, m2being respectively n2nand 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 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∫baƒ(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−ε0tan xdx,∫3/2π−ζ1/2π+εtan xdx,∫5/2π−σ3/2π+ζtan xdx, ...xxxNow 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 α0idy(tan iy−tan iy)nπ + iy−nπ + iywhere 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 α0dy2nπ, that is than 2α.n²π² + y²Finally, 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 havedz= idθ, i tan z =exp(−R sin θ) E(iR cos θ) − exp(R sin θ) E(−iR cos θ);zexp(−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−ααdθ = − (π − 2α).Making n very large the result obtained for the whole contour is2∫∞0tan xdx − (π − 2α) − 2αε = 0,xwhere ε is numerically less than unity. Now supposing α to diminish
to zero we finally obtain∫∞0tan xdx =π.x2(2) For another case, to illustrate a different point, we may take the
integral∫za−1dz,1 + zwherein 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 − 1will be equal to
xα − 1exp [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 αxa−1dx.1 + xIt 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)]∫∞0xα−1dx.1 + xWithin 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−1is exp [iπ(a − 1)] or − exp(iπa);
this is then the residue of ƒ(z) at z = −1; we thus have[1 − exp (2πia)]∫∞0xa−1dx = −2πi exp(iπa),1 + xthat is∫∞0xa−1dx = π cosec (aπ).1 + x We are to evaluate an integral∫baƒ(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 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 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 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 when n and therefore R is very large, the limit of this contribution
to the contour integral is thus −∫π−ααdθ = − (π − 2α). Making n very large the result obtained for the whole contour is where ε is numerically less than unity. Now supposing α to diminish
to zero we finally obtain (2) For another case, to illustrate a different point, we may take the
integral 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 − 1will be equal to
xα − 1exp [2πi(a − 1)], where x is real and positive. The two straight
portions of the contour will thus together give a contribution 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 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−1is exp [iπ(a − 1)] or − exp(iπa);
this is then the residue of ƒ(z) at z = −1; we thus have that is § 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.
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