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 .flow { margin: 0; font-size: 1em; } .flow .pagebreak { page-break-before: always; } .flow p { text-align: left; text-indent: 0; margin-top: 0; margin-bottom: 0.5em; } .flow .gstxt_sup { font-size: 75%; position: relative; bottom: 0.5em; } .flow .gstxt_sub { font-size: 75%; position: relative; top: 0.3em; } .flow .gstxt_hlt { background-color: yellow; } .flow div.gtxt_inset_box { padding: 0.5em 0.5em 0.5em 0.5em; margin: 1em 1em 1em 1em; border: 1px black solid; } .flow div.gtxt_footnote { padding: 0 0.5em 0 0.5em; border: 1px black dotted; } .flow .gstxt_underline { text-decoration: underline; } .flow .gtxt_heading { text-align: center; margin-bottom: 1em; font-size: 150%; font-weight: bold; font-variant: small-caps; } .flow .gtxt_h1_heading { text-align: center; font-size: 120%; font-weight: bold; } .flow .gtxt_h2_heading { font-size: 110%; font-weight: bold; } .flow .gtxt_h3_heading { font-weight: bold; } .flow .gtxt_lineated { margin-left: 2em; margin-top: 1em; margin-bottom: 1em; white-space: pre-wrap; } .flow .gtxt_lineated_code { margin-left: 2em; margin-top: 1em; margin-bottom: 1em; white-space: pre-wrap; font-family: monospace; } .flow .gtxt_quote { margin-left: 2em; margin-right: 2em; margin-top: 1em; margin-bottom: 1em; } .flow .gtxt_list_entry { margin-left: 2ex; text-indent: -2ex; } .flow .gimg_graphic { margin-top: 1em; margin-bottom: 1em; } .flow .gimg_table { margin-top: 1em; margin-bottom: 1em; } .flow { font-family: serif; } .flow span,p { font-family: inherit; } .flow-top-div {font-size:83%;} variables x0, y0, where zosro-Hiyo, over the £: it will appear : F(zo) is also continuous and in fact also a differentiable function ol-fo. Supposing 7 to be retained the same for all points roof the region, and go to be the upper limit of the possible values of e for the point zo, it is to be presumed that go will vary with zo, and it is not obvious as yet that the lower limit of the values of ao as zo varies 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, zo), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position #: 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 f(z), the integraffold: has the same value by whatever path within the region we pass from z1 to z. This we prove by showing that when taken round a closed path in the region the integral f(z)ds vanishes. Consider first a triangle over which the condition (z, zo) holds, for some position of 20 and every position of z, within or upon the boundary of the triangle. Then as f(z) = f(z)+(z-z)F(x)+"6(2-2), where!6|<1, we have ff(z)dz=[f(z)-z.F(2)]/d2+F(z)/zdz-Hm/9(3-z)ds, which, as the path is closed, is n/9(z-zo)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 map, where a is the greatest side of the triangle and p is its perimeter; if A be the area of the triangle, we have A = }ab sin C= (a/r)ba, where a is the least angle of the triangle, and hence a(a+b+c) <2a(b+c) <4*Ala; the integral ff(z) dz round the perimeter of the triangle is thus <45 mala. *: consider any region made up of triangles, as before explained, in each of which the condition (z, to) holds, as in the triangle just taken. The integral J f(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 4xnk/a, where K is the whole area of the region, and a is the smallest angle of the component triangles. However small y 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 f(z) at any interior point of a region is expressible in terms of the value of # at the boundary points. For consider in the original region the function f(z)/(z-z), where * is an interior point: this satisfies the same conditions as f(z) except in the immediate neighbourhood of z. Taking out then from the original ion a small regular »lygonal region with 20 as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon mes a circle, it appears that the integral/# round the boundary of the original region is equal to the same integral taken counterclockwise round a small circle having zo as centre; on this circle, however, if z-zo=rE(ió), dz/(z-zo) = ido, and f(z) differs arbitrarily little from f(zo) if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to - r 2rif(x). Hence f(x)=# boundary of the original region. From this it appears that * -tim&l=/el--! f# F(z) = lim: ==#--5: (l-zo also round the boundary of the original # This form shows, however, that F(zo) is a continuous, finite, differentiable function of zo over 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 #. where this integral is round the coefficient. W(z) -£6. 3)ds, for interior points zo, z, is a differentiable function of g, having for its differential coefficient the function f(x, y), which is therefore also a differentiable function of z at interior points. (3) Hence if the series us(z)+*(x)+... to do be uniformly convergent over a region, its terms being differentiable functions of 2. then its sum S(z) is a differentiable function of z, whose differential coefficient, given by #f#. 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 !' include the region bounded by two concentric circles of radii r, R, with centre at the origin, and zo be an interior point of this region, circle, centre the origin, of radius intermediate between r and R. Particular cases are: (a) when the region of definition of the function includes the whole-interior of the outer circle; then we may take r=o, the coefficients A" for which n R, then the function is representable for all such points # a power series in 2" or t, in such case we say that the region of existence of the function contains the point z = oo. A series in z* has a finite limit when |z|= oo; a series in 2 cannot remain finite for all points z for which lz|> R; for if, for £ the sum of a power series Xa.z" in z is in absolute value less than M, we have |a. 14 Mr", and therefore, if M remains finite for all values of r however great, an=o. Thus the region of existence of a function if it contains all finite points of the plane cannot contain the point z = co, such is, for instance, the case of the function exp (z) =Xz"/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 series 2.2", where m = n”, the series 2.2" 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 a be a point interior to the circle of convergence of a series representing the function, the series may be £ in powers of z-zo; as to #: to a singular point of the function, lying on the circle convergence, the radii of convergence of these derived series in z--& diminish to zero; when, however, a circle can be put about zo, 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-zo gives the value of the function # these external points. If the function be supposed to be given : for the interior of the original circle, by the original power series, the series inz-zo converging beyond the 9riginal circle gives what is known as, an analytical continuation of the function. It appears from what has been 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-zo, about any point zo interior to its circle of convergence, and the new series converges certainly for |z-zo|a z" as n increases (see the bibliography below under Integral 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 MittagLeffler's theorem, relating to the expression as a sum of simpler functions of a function whose singular points have the point 2 = 20 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 r £ generalized as follows: 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 a, there be associated a polynomial in (2-a.)-', say gr; then there exists a single valued function whose region of existence excludes only the points (a) and the points, (c), having in a point a, a pole whereat the expansion consists of the terms g., together with a power series in z-ax: the function is expressible as an infinite series of terms gr-yi, where y, is also a rational function. II. With a similar aggregate (a), with limiting points (c), suppose with every point at there is associated a £ integer r. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order r, at the point a, but not elsewhere, expressible in the form n _* - #) "n ** 1 (' #) exp (ga), where with every point a. is associated a proper point c. of (c), and - # 1. (# ) s *-r-,+,+ U+T). u, being a properly chosen positive integer. "If 'hould appen that the points £etermine a path dividin the plane into separated regions, as, for instance, if £5 exp (in v 2.n), when(c) consists of the points of the circle |2|= 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 Ro of 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 Co. 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 Co; let a, b, ... be finite points upon C or outside R. Then when b is near enough to a, the fraction (a-b)/(2-b) is arbitrarily small for all positions of 2; say we can thus build a rational function differing, in value, in Ro, as little as may be desired from a given rational function J =2A,0t-a)", and differing, outside R or upon the boundary of R, from f, in the fact that while f 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 Ro arbitrarily little from the former, whose poles are all at the point c. Now any monogenic function f(t) whose region of definition includes C and the interior of R can be represented at all points z in Ro by I (t)dt f(-)-;###. where the path of integration is C This integral is the limit of a where the points t, are upon C; and the proof we have given of the existence of the limit shows that the sum S converges to f(z) uniformly in regard to 2, when z is in Ro, so that we can suppose, when the subdivision of C into intervals t, -t, has been carried sufficiently far, that IS-f(z)| |t|, are without the region Re, £ not infinitely near to its boundary points. Taking then an arbitrary real positive e we can determine a polynomial in xt", say P(xt"). such that for all points x in Rowe have 1(1-xt-:)--P(xt-1)|c. Hence, if g be any real quantity, since the range (-g, ... g) contains only a finite number of intervals of i£ e, and there cannot two periods w=e-Hia such that we'e 3 (a+1)e, veea 4 (v-H1)e, where u, v are integers, it follows that there is only a finite number of periods for which both p and * are in the interval (-g .. .g). Considering then all the periods of the function which are real multiples of one period w, and in particular those periods Aw whereino ~A*1, there is a lower limit for A, greater than zero, and therefore, since there only a finite number of such periods for which the real and £ parts both lie between-g and g, a least value of A, say No. If in=A* and A-MA2+A', where is an integer and oSA'4x2, a period Aw is of the form Mo-HA'o'; since, however, Q, Mo and N are periods, so also is \'o, and hence, by the construction of No. we have X =o, thus all periods which are real multiples of a expressible in the form Mo, where M is an integer, and 9 a £: If beside a the functions have a period to which is not a real multiple of w, consider all existing periods of the ####. wherein u, v are real, and of these those for which deper?' that is « ՆախորդըՇարունակել »