Part 27
whose sum to n terms is P_n(z), converges for all finite values of z and represents [f](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 [Gamma] of a region, we can suppose the poles of the rational function, constructed to approximate to [f](z) within R0, to be at points of [Gamma]. A series of rational functions of the form
H1(z) + {H2(z) - H1(z)} + {H3(z) - H2(z)} + ...
then, as before, represents [f](z) within R0. And R0 may be taken to coincide as nearly as desired with the interior of the region bounded by [Gamma].
S 11. _Expression of (1 - z)^(-1) by means of Polynomials. Applications._--We pursue the ideas just cursorily explained in some further detail.
Let c be an arbitrary real positive quantity; putting the complex variable [zeta] = [xi] + i[eta], enclose the points [zeta] = l, [zeta] = 1 + c by means of (i.) the straight lines [eta] = [+-]a, from [xi] = l to [xi] = 1 + c, (ii.) a semicircle convex to [zeta] = 0 of equation ([xi] - 1)^2 + [eta]^2 = a^2, (iii.) a semicircle concave to [zeta] = 0 of equation ([xi] - 1 - c)^2 + [eta]^2 = a^2. 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 [sigma] = 1/r (c/a). Now take
c1 = 1 + c/r, c2 = 1 + 2c/r, ... c_r = 1 + c;
if n1, n2, ... n_r, be positive integers, the rational function _ _ 1 | / c1 - 1 \^n1 | ---------- |1 - ( ----------- ) | 1 - [zeta] |_ \ c1 - [zeta] / _|
is finite at [zeta] = 1, and has a pole of order n1 at [zeta] = c1; the rational function _ _ _ _ 1 | / c1 - 1 \^n1 | | / c2 - c1 \^n2 |^n1 ---------- |1 - ( ------------ ) | |1 - ( ----------- ) | 1 - [zeta] |_ \ c1 - [zeta] / _| |_ \ c2 - [zeta]/ _|
is thus finite except for [zeta] = c2, where it has a pole of order n1n2; finally, writing
/ c_s - c_(s-1) \^n_s x_s = ( ------------- ), \ c_s - [zeta] /
the rational function
U = (1 - [zeta])^(-1) (1 - x1)(1 - x2)^n1 (1 - x3)^n1n2 ... (1 - x_r)^(n1n2 ... n_(r - 1))
has a pole only at [zeta] = 1 + c, of order n1n2 ... n_r.
The difference (1 - [zeta])^(-1) - U is of the form (1 - [zeta])^(-1)P, where P, of the form
1 - (1 - [rho]1)(1 - [rho]2)...(1 - [rho]_k),
in which there are equalities among [rho]1, [rho]2, ... [rho]_k, is of the form
[Sigma][rho]1 - [Sigma][rho]1[rho]2 + [Sigma][rho]1[rho]2[rho]3 - ...;
therefore, if |r_i| = |[rho]_i|, we have
|P| < [Sigma]r1 + [Sigma]r1r2 + [Sigma]r1r2r3 + ... < (1 + r1)(1 + r2)...(1 + r_k) - 1;
now, so long as [zeta] is without the closed curve above described round [zeta] = 1, [zeta] = 1 + c, we have
| 1 | 1 |c_m - c_(m-1)| c/r |----------| < ---, |-------------| < --- < [sigma], |1 - [zeta]| a |c_m - [zeta] | a
and hence
|(1 - [zeta])^(-1) - U| < a^(-1) {(1 + [sigma]^n1) (1 + [sigma]^n2)^n1 (1 + [sigma]^n3)^n1n2 ... (1 + [sigma]^n_r)^(n1n2 ... n_(r-1)) - 1}.
Take an arbitrary real positive [epsilon], and [mu], a positive number, so that [epsilon]^[mu] - 1 < [epsilon]a, then a value of n1 such that [sigma]^n1 < [mu]/(1 + [mu]) and therefore [sigma]^n1 / (1 - [sigma]^n1 < [mu], and values for n2, n3 ... such that[sigma]^n2 < 1/n1 [sigma]^2n1, [sigma]^n3 < 1/n1n2 [sigma]^{3n1, ... [sigma]^n_r} < 1/(n1...n_(r - 1)} [sigma]^n_rn1; then, as 1 + x < e^x, we have |(-[zeta])^(-1) - U| < a^-1 {exp([sigma]^n1 + n1[sigma]^n2 + n1n2[sigma]^n3 + ... + n1n2...n_(r - 1)[sigma]^n_r) - 1}, and therefore less than
a^(-1) {exp([sigma]^n1 + [sigma]^2n1 + ... + [sigma]^n_rn1) - 1},
which is less than _ _ 1 | / [sigma]^n1 \ | -- |exp ( -------------- ) - 1 | a |_ \1 - [sigma]^n1/ _|
and therefore less than [epsilon].
The rational function U, with a pole at [zeta] = 1 + c, differs therefore from (1 - [zeta])^(-1), for all points outside the closed region put about [zeta] = 1, [zeta] = l + c, by a quantity numerically less than [epsilon]. So long as a remains the same, r and [sigma] will remain the same, and a less value of [epsilon] will require at most an increase of the numbers n1, n2, ... n_r; but if a be taken smaller it may be necessary to increase r, and with this the complexity of the function U.
Now put
c[zeta] (c + 1)z z = --------------, [zeta] = --------; c + 1 - [zeta] c + z
thereby the points [zeta] = 0, 1, 1 + c become the points z = 0, 1, [oo], the function (1 - z)^(-1) being given by (1-z)^(-1) = c(c + 1)^(-1)(1 - [zeta])^(-1) + (c + 1)^(-1); the function U becomes a rational function of z with a pole only at z = [oo], that is, it becomes a polynomial in z, say [(c + 1)/c]H - 1/c, where H is also a polynomial in z, and _ _ 1 c | 1 | ----- - H = ----- | ---------- - U |; 1 - z c + 1 |_1 + [zeta] _|
the lines [eta] = [+-]a become the two circles expressed, if z = x + iy, by
c(c + 1) (x + c)^2 + y^2 = [+-] -------- y, a
the points ([eta] = 0, [xi] = 1 - a), ([eta] = 0, [xi] = 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 = -[oo]). The circle (x + c)^2 + y^2 = c(c + 1)y/a can be written
(x + c)^2 (x + c)^4 y = --------- + --------- {[mu] + [root][[mu]^2 - (x + c)^2]}^(-2), 2[mu] 2[mu]
where [mu] = 1/2c(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 =< [oo], we can take a quantity a, and hence, with an arbitrary c, determine a number r; then corresponding to an arbitrary [epsilon]_s, we can determine a polynomial P_s, such that, for all points interior to the region, we have
|(1 - z^(-1)) - P_s| < [epsilon]_s;
thus the series of polynomials
P1 + (P2 - P1) + (P3 - P2) + ...,
constructed with an arbitrary aggregate of real positive numbers [epsilon]1, [epsilon]2, [epsilon]3, ... with zero as their limit, converges uniformly and represents (1-z)^(-1) for the whole region considered.
S 12. _Expansion of a Monogenic Function in Polynomials, over a Star Region._--Now consider any monogenic function [f](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 = [oo], 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 _ / dt [f](t) 2[pi]i[f](x) = | -- -----------, _/ t 1 - xt^(-1)
where [f](x) represents a monogenic branch of the function, in case it be not everywhere single valued, and t is on the boundary of the region. Describe now another region R0 lying entirely within R, and let x be restricted to be within R0 or upon its boundary; then for any point t on the boundary of R, the points z of the plane for which zt^(-1) is real and positive and equal to or greater than 1, being points for which |z| = |t| or |z| > |t|, are without the region R0, and not infinitely near to its boundary points. Taking then an arbitrary real positive [epsilon] we can determine a polynomial in xt^(-1), say P(xt^(-1)), such that for all points x in R0 we have
|[1 - xt^(-1)]^(-1) - P[xt^(-1)]| < [epsilon];
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 [epsilon], _ _ | / dt | | / dt | |2[pi]i[f](x) - | --[f](t)P(xt^(-1))| = | | --[f](t)E| <= [epsilon]LM, | _/ t | | _/ t |
where L is the length of the path of integration, the boundary of R, and M is a real positive quantity such that upon this boundary |t^(-1)[f](t)| < M. If now
P(xt^(-1)) = c0 + c1xt^(-1) + ... + c_mx^mt^(-m),
and _ 1 / ------ | t^(-r-1)[f](t)dt = [mu]_r, 2[pi]i _/
this gives
|[f](x) - {c0[mu]0 + c1[mu]1x + ... + c_m[mu]_mx^m}| =< [epsilon]LM/2[pi],
where the quantities [mu]0, [mu]1, [mu]2, ... are the coefficients in the expansion of [f](x) about the origin.
If then an arbitrary finite region be constructed of the kind explained, excluding the barriers joining the singular points of [f](x) to x = [oo], it is possible, corresponding to an arbitrary real positive number [sigma], to determine a number m, and a polynomial Q(x), of order m, such that for all interior points of this region
|[f](x) - Q(x)| < [sigma].
Hence as before, within this region [f](x) can be represented by a series of polynomials, converging uniformly; when [f](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, [phi](t) be capable of expression as a power series in t-x about a point x, for |t - x| =< [rho], and for all points of this circle |[phi](t)| < g, we know that |[phi]^(n)(x)| < g[rho]^(-n)(n!). Hence, taking |z| < 1/3[rho], and, for any assigned positive integer [mu], taking m so that for n > m we have ([mu] + n)^[mu] < (3/2)^n, we have
|[phi]^(([mu] + n))(x)z^n| [phi]^([mu] + n)(x) |------------------------| < -------------------([mu] + n)^([mu])|z|^n | n! | ([mu] + n)!
g /3 \ /[rho]\ g < ---------------- ( -- )^n ( ----- )^n < --------------, [rho]^([mu] + n) \2 / \ 3 / [rho]^[mu] 2^n
and therefore
_m \ [phi]^([mu] + n)(x) [phi]^([mu])(x + z) = /_ ------------------- z^n + [epsilon]_[mu], n=0 n!
where
g _[oo] 1 g |[epsilon]_([mu])| < ---------- \ --- < -------------- [rho]^[mu] /_ 2^n [rho]^[mu] 2^m n=m+1
Now draw barriers as before, directed from the origin, joining the singular point of [phi](z) to z = [oo], take a finite region excluding all these barriers, let [rho] be a quantity less than the radii of convergence of all the power series developments of [phi](z) about interior points of this region, so chosen moreover that no circle of radius [rho] with centre at an interior point of the region includes any singular point of [phi](z), let g be such that |[phi](z)| < g for all circles of radius [rho] 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| 1/3 - [rho]; then take the points a1 = x/n, a2 = 2x/n, a3 = 3x/n, ... a_n = 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
_m1 [phi]^([mu] + [lambda]1)(0) /x \^[lambda]1 [phi]^([mu]) (a{1}) = \ --------------------------- ( -- ) + [alpha]_[mu] /_ [lambda]{1}! \n / [lambda]1=0
with
|[alpha]{[mu]}| < g/[rho]^[mu] 2^m1,
provided ([mu] + m1 + 1)^[mu] < (3/2)^(m1+1); in fact for [mu] =< 2n^(2n-2) it is sufficient to take m1 = n^2n; by another application of the same inequality, replacing x, z respectively by a1 and x/n, we have
_ m2 [phi]^([mu]+[lambda]2)(a1) /x \^[lambda]2 [phi]^([mu])(a2) = \ -------------------------- ( -- ) + [beta]'_[mu], /_ [lambda]{2}! \n / [lambda]2=0
where
|[beta]'[mu]| < g/[rho]^[mu] 2^m2
provided ([mu] + m2 + 1)^[mu] < {3/2}^(m2 + 1); we take m2 = n^(2n - 2), supposing [mu] < 2^(2n - 4). So long as [lambda]2 =< = m} =< n^(2n - 2) and [mu] < 2n^(2n - 4) we have [mu] + [lambda]{2} < 2n^(2n - 2), and we can use the previous inequality to substitute here for [phi]^([mu] + [lambda]2) (a1). When this is done we find
_ m2 _ m1 [phi]^([mu] + [lambda]1 + [lambda]2)(0) [phi]^([mu])(a2) = \ \ --------------------------------------- /_ /_ [lambda]1! [lambda]2! [lambda]2=0 [lambda]1=0
/x \ ^[lambda]1 + [lambda]2 ( -- ) + [beta]_[mu], \n /
where |[beta]_[mu]| < 2g/[rho]^[mu] 2^(m2), the numbers m1, m2 being respectively n^2n and n^(2n - 2).
Applying then the original inequality to [phi]^([mu]) (a3) = [phi]^([mu]) (a2 + x/n), and then using the series just obtained, we find a series for [phi]^([mu]) (a3). This process being continued, we finally obtain
_ m1 _ m2 _ m_n [phi](x) = \ \ ... \ [phi]^(h)(0) /x \^h /_ /_ /_ ----------- ( -- ) + [epsilon], [lambda]1=0 [lambda]2=0 [lambda]_n=0 K \n /
where h = [lambda]1 + [lambda]2 + ... + [lambda]_n , K = [lambda]1! [lambda]2! ... [lambda]_n!, m1 = n^(2n), m1 = n^(2n - 2), ... , m1 = n^2, |[epsilon]| < 2g/2^(m_n).
By this formula [phi](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.
S 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 [int][a to b] [f](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 [int][f](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, [f](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 [int][f](z)dz taken round these points. Two instances will suffice to explain the method. (1) The integral [int][0 to [oo]] (tan x)/x dx is convergent if it be understood to mean the limit when [epsilon], [zeta], [sigma], ... all vanish of the sum of the integrals
_1/2[pi]-[epsilon] _(3/2)[pi]-[zeta] _(5/2)[pi]-[sigma] / tan x / tan x / tan x | ----- dx, | ----- dx, | ----- dx, ... _/ 0 x _/1/2[pi]+[epsilon] x _/(3/2)[pi]+[zeta] x
Now draw a contour consisting in part of the whole of the positive and negative real axis from x = -n[pi] to x = + n[pi], where n is a positive integer, broken by semicircles of small radius whose centres are the points x = [+-]1/2[pi], x = [+-]3/4[pi], ... , the contour containing also the lines x = n[pi] and x = -n[pi] for values of y between 0 and n[pi] tan [alpha], where [alpha] is a small fixed angle, the contour being completed by the portion of a semicircle of radius n[pi] sec [alpha] which lies in the upper half of the plane and is terminated at the points x = [+-]n[pi], y = n[pi] tan [alpha]. Round this contour the integral [int](tan z /z) dz has the value zero. The contributions to this contour integral arising from the semicircles of centres -1/2(2s - 1)[pi], + 1/2(2s - 1)[pi], 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 [int][0 to n[pi]](tan x / x) dx. The contribution to the contour integral from the two straight portions at x = [+-]n[pi] is
_n[pi] tan [alpha] / / tan iy tan iy \ | idy ( ---------- - ----------- ) _/ 0 \n[pi] + iy -n[pi] + iy /
where i tan iy, = -[exp(y) - exp(-y)]/[exp(y) + exp(-y)], is a real quantity which is numerically less than unity, so that the contribution in question is numerically less than
_n[pi] tan [alpha] / 2n[pi] | dy ---------------, that is than 2[alpha]. _/ 0 n^2[pi]^2 + y^2
Finally, for the remaining part of the contour, for which, with R = n[pi] sec [alpha], we have z = R(cos [theta] + i sin [theta]) = RE(i[theta]), we have
dz -- = id[theta], i tan z = z
exp(-R sin [theta]) E(iR cos [theta]) - exp(R sin [theta]) E(-iR cos [theta]) -----------------------------------------------------------------------------; exp(-R sin [theta]) E(iR cos [theta]) + exp(R sin [theta]) E(-iR cos [theta])
when n and therefore R is very large, the limit of this contribution to the contour integral is thus _ / [pi]-[alpha] - | d[theta] = -([pi] - 2[alpha]). _/ [alpha]
Making n very large the result obtained for the whole contour is _ / [oo] tan x 2 | ----- dx - ([pi] - 2[alpha]) - 2[alpha][epsilon] = 0; _/ 0 x
where [epsilon] is numerically less than unity. Now supposing [alpha] to diminish to zero we finally obtain _ / [oo] tan x [pi] | ----- dx = ---- _/ 0 x 2
(2) For another case, to illustrate a different point, we may take the integral _ / z^(a-1) | ------- dz, _/ 1 + z
wherein a is real quantity such that 0 < a < 1, and the contour consists of a small circle, z = rE(i[theta]), terminated at the points x = r cos [alpha], y = [+-] r sin [alpha], where [alpha] is small, of the two lines y = [+-] r sin [alpha] for r cos [alpha] =< x =< R cos [beta], where R sin [beta] = r sin [alpha], and finally of a large circle z = RE(i[phi]), terminated at the points x = R cos [beta], y = [+-] R sin [beta]. We suppose [alpha] and [beta] both zero, and that the phase of z is zero for r cos a =< x =< R cos [beta], y = r sin [alpha] = R sin [beta]. Then on r cos [alpha] =< x =< R cos [beta], y = -r sin [alpha], the phase of z will be 2[pi], and z^([alpha] - 1) will be equal to x^([alpha] - 1) exp (2[pi]i(a - 1)), where x is real and positive. The two straight portions of the contour will thus together give a contribution
_ / R cos [beta] x^(a - 1) [1 - exp (2[pi]i[alpha])] | --------- dx. _/ r cos [alpha] 1 + x
It can easily be shown that if the limit of z[f](z) for z = 0 is zero, the integral [int] [f](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 z[f](z) for z = [oo] 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 [f](z) = z^([alpha] - 1)/(1 + z), we have z[f](z) = z^a/(1 + z), which, for 0 < a < 1, diminishes to zero both for z = 0 and for z = [oo]. Thus, finally the limit of the contour integral when r = 0, R = [oo] is _ / [oo] x^([alpha] - 1) [1 - exp (2[pi]i[alpha])] | --------------- dx. _/ 0 1 + x
Within the contour [f](z) is single valued, and has a pole at z = 1; at this point the phase of z is [pi] and z^(a - 1) is exp [i[pi](a - 1)] or - exp (i[pi]a); this is then the residue of [f](z) at z = -1; we thus have _ / [oo] x^(a - 1) [1 - exp (2[pi]ia)] | --------- dx = -2[pi]i exp (i[pi]a), _/ 0 1 + x
that is _ / [oo] x^(a - 1) | --------- dx = [pi] cosec (a[pi]). _/ 0 1 + x
S 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.