VARIABLE, Complex. 1. The solution of a quadratic equation involves the extraction of the square root of a quantity which may be negative. In that case the solution is of the form a + √ - b 2 , which may be written a + b √-l, or a + ib, putting, with Gauss, for brevity, √-1= i . Analysis was, therefore, at a very early stage compelled to contemplate the possibility that symbols of magnitude may represent combinations of dissimilar constituents; but it is only within the 19th century, mainly owing to the initiative of Cauchy and of Gauss, that the whole domain of analysis has been explored from this point of view. A quantity, a + ib, consisting of a real part containing a units and an imaginary part containing b units, was called by Gauss a complex number, and the conception of it includes those of each kind, the real and the purely imaginary as they are found from it by making in succession b = 0 and a = 0. Quantities thus expressed are subject to all the ordinary algebraic rules applicable to real quantities, and in addition to the relation i 2 = - 1. The occurrence of this symbol i in an identity thus singles out two distinct identities: for instance, if the product of a + ib and a' + ib' be A + iB, we have A = aa' - bb' and B = ab' + ba'; and in this sense Cauchy, as lately as 1844, still maintained that every imaginary equation was only the symbolic representation of two equations between real quantities. If either or both of the real quantities be variable, the quantity is called a complex variable; adopting the usual notation, it is then written x + iy = z. In order that z may vanish, both x and y must do so, as it is not possible for the real part to cancel the imaginary. But either part becoming infinite renders z infinite, and a breach of continuity in the change of the variable z arises when either x or y is discontinuous; but, as long as both change continuously, z is also a continuous complex variable.
2. As long as a variable is conceived to admit of real values only, the distances from a fixed point, measured along a right line, are sufficient to represent it. These may be taken positively or negatively, and in both directions, through all magnitudes, from a vanishing amount to values large beyond conception, i.e. , infinite. But, inasmuch as a complex variable z = x + iy depends on two perfectly independent real variables, x and y, its geometrical realization demands a field of two dimensions. When this is assumed to be a plane, to each value z = x + iy corresponds a point P of the plane whose rectangular coordinates referred to an arbitrarily assumed pair of axes in the plane have the real values x and y. Thus to all real quantities in the old sense correspond the points of the axis of abscissae and to pure imaginaries those of the axis of ordinates. The coordinates of the point P may change quite independently, just as the variables x and y do. If the point P be given, the corresponding value of z is known, and also for a given value of z the corresponding point P is known. The continuous variation of z is exhibited by the motion of P on a curve. The complex number may also be written in the following form, which is found as early as Euler. Joining the point P to the origin O, the line OP=ρ is called the modulus, and the angle XOP=θ which this line forms with the positive axis of x, or the prime axis, is called the a rgumen t (or amplitude) of the complex x + iy. Thus p = + √ x 2 + y 2 , cos θ = x∕ρ, sin θ = y∕ρ, and z = x + iy = p(cos θ + i sin θ) = ρe iθ . This, again, shows that the variable vanishes only when x = 0 and y = 0, i.e., when ρ = 0, since no real angle makes cos θ + i sin θ= 0. In this way the quantity is represented by rotating through the angle θ from the prime axis a line whose length is p. The value of the radical √ x 2 + y 2 is always taken positively; the difference of sign arises from a reversal of direction and is indicated by the angle in the factor e iθ . When P is given, p is unique and θ is determined by cos θ and sin θ : that is to say, it admits only of a certain value or of those which differ from it by integer multiples of 2π ’. Thus two quantities are equal whose moduli are equal and their directions parallel. The modulus varies continuously with z, and so does the argument, except when z vanishes: that is, when the curve passes through the origin, the argument then undergoes generally a sudden change by π.
3. Complex numbers form a system complete in themselves, and any process of calculation on complex numbers always reproduces a complex number. Thus, as any such quantity is represented by a point approached from a given point, the result of calculation also has a point or points to represent it. This is illustrated as follows.
The addition of the quantity z=x + iy to z'=x' + iy' produces z 1 =x+x'+i(y + y') ; the point z 1 has the coordinates x+x', y + y': that is, it is the extremity of the diagonal of the parallelogram included by the lines Oz and Oz'. If a third quantity z"=x"+iy" were added on, the result would be represented by the diagonal of the parallelogram included by Oz 1 and Oz"; and so on. Subtraction is the addition of a line with opposite sign. The modulus of the sum or difference of two quantities is found by assuming ρ 1 cos θ 1 =ρ cos 0± ρ 'cos θ' and ρ 1 sin θ 1 = ρ sin θ ± ρ ' sin θ', whence ρ 1 2 =ρ 2 + ρ ' 2 ± 2ρρ' cos (θ - θ'). Thus the modulus of the sum is never greater than the sum of the moduli and the modulus of the difference never less than their difference. The line zz 1 in fig. 2 is found by subtracting z from z 1 , and is thus seen to be equal to the parallel line Oz '. The multiplication of z=ρe iθ by z=ρ'e iθ gives Z=zz'=ρρ'e i ( θ+θ' ) : that is, we have to measure along a line inclined to the prime axis at an angle equal to the sum of the arguments a length representing the product of the moduli; in order to determine this, the unit of length must be introduced. If this be OU measured along the prime axis, Z is constructed by placing on Oz' a triangle z'OZ similar and similarly placed to UOz. From this we can proceed at once to division; as an example, when z" — z ’ z" - z'
-r 1 =~7 , the triangle formed by the three points represent-
1 <c j z z ing z 1 , z' 1 , z " 1 is homothetic with that formed by z, z', z".
In like manner we might proceed to powers of complex quantities with real exponents: e.g., every complex number has n distinct nth roots included in the form n √ z = n √ ρ . e i (θ+2kπ)∕n ; then to p0wers with complex exponents; to logarithms of complex numbers: for example, every number has infinitely many logarithms to the base e, which differ by multiples of 2 iπ , in accordance with the formula log z = log ρ + i(θ ± 2 kπ ); and, finally, to powers with complex base and complex exponent. All these in turn lead again to complex numbers, and to such only.
4. In dealing with real numbers any definite range of values is represented by the points of a finite right line; but to realize a definite range of complex numbers we must have recourse in general to a “region” of two dimensions of the plane bounded by some curve. For instance, all complex numbers whose modulus is less than p and greater than ρ' occupy a region bounded by two concentric circles with radii ρ and ρ' round the origin. All those whose moduli are equal to p form a linear region, namely, the circumference of the circle round the origin with that radius. A region of two dimensions is called connected when we can pass from any point within it to any other point within it without crossing the boundary curve. A quantity is said to be unrestrictedly variable in a region when it can assume all numerical values in this region. It is said to be continuously variable when all values which it assumes always belong to a finite connected region. Thus a variable is said to be unrestrictedly continuous for a certain value when it can take all the values which belong to a finite region, however small, which includes this value. The variable is restrictedly continuous for this value when the values it takes near this one form a region on whose boundary the value itself occurs, or it may be a region of one dimension. It is discontinuous for this value when the point is isolated, and does not belong to any region. When we know two definite values of a real variable, we know all the intermediate values which it must assume in passing from one to the other; but in the case of the complex variable there is an essential difference: it can pass continuously from one given value to another by infinitely numerous series of continuously consecutive values. In geometric terms, with the real variable we can only travel along the axis of x, with the complex we can travel along any path whatever leading from one point to the other.
5. When the values of one complex variable w = u + iv are determined by the values of another z = x + iy, in such a way that for each value of z within a definite region one or more values can be assigned to w by performing a definite set of mathematical operations on z, then w is said to be a function of the complex variable z. Functions are distinguished as one-valued and many-valued according to the number of values of w which belong to one value of 2, as algebraic and transcendental according to the form in which the variables appear, and as implicit and explicit according as the relation is solved for w or not. The course of the function w will accordingly be illustrated by points in another plane or planes having the rectangular coordinates u and v, and its dependence on 2 will be geometrically a transformation from one plane to the other. A very simple instance is when w = 1∕z, where the transformation is equivalent to geometric inversion. If the explicit function w = f (z ) is to be unrestrictedly continuous in the region for which it has definite values, there must be a finite connected area, however small, around any admissible value of 2, to which corresponds a connected region of w : that is, u and v must vary continuously when x and y do so. Let 2 be increased by Δ z = Δ x + i ∆ y and put w + ∆ w = (u + ∆ u ) + i(v + ∆ v ) =f(z + Δ z ); then in whatever way ∆x and ∆ y may converge to zero we must have lim∆ u = 0 and lim∆ v = 0: in other words, w is continuous for a value 2 when this value can be included in a region such that for every value 2 + Δ2 within it the modulus of the difference mod(∆ w ) = mod[ f (z + Δ z ) - f (z )] = √∆ u 2 + ∆ υ 2 is less than any assignable small number.
6. We proceed to give some elementary examples of functions.
Ex. 1. The power with positive integer exponent. w = z m is a onevalued (monotrope) function which is continuous for the entire plane, since u =ρ m cos mθ , v=ρ m sinmθ are continuous functions of p and θ which do not alter when the argument θ is increased by multiples of 2π. When 2 becomes infinite so does w, and thus this value z is a singular one (pole) for the function, as contrasted with a regular one, which is such that a function has a finite value continuously changing with z.
Ex. 2. Any rational integer function of the nth degree in z. f(z)=a 0 + a 1 z + a 2 z 2 + ... + a n z n is one-valued and continuous for the entire plane and has no singularity, except for z= ∞, for which it becomes infinite (pole).
Ex. 3. The power with a rational fractional index. w==z p/q [(cos(p/q )(θ + 2 kπ ) + i sin (p/q )+ 2 kπ )], where k takes all integer values from 0 to q- 1. Each value of k determines at each spot the value of a branch of the function, which is therefore a many-valued (polytrope) one. At the points 2 = 0 and z= ∞ all branches have the same value. Thus, if p∣q>0, at z =0 the value of w is 0; at z= ∞ its value is ∞ . These two points are called branch (critical) points of the function. If we wish to contemplate the values of the function which belong to one branch, so as to have each branch by itself in general a continuous function, we employ the method of Cauchy: that is, we draw from the origin to the point infinity a curve which does not enclose any space, for instance, the positive part of the axis of x. Suppose we let p = 1, q = 2, then w=ρ ½ e ½(θ+2kπ)i , where k=0 or 1. Taking k =0, the values of the independent variable and of the function along the prime axis for θ = 0 are z=ρ and w=ρ ½ ; along the axis of y when θ = π∕2 they are z=ρi and w= ρ ½ e ¼πi ; along the negative axis of x when θ = π they are 2= - ρ and w = ρ ½ e ½πi ; along the negative axis of y, when θ=3π∣2, z = -ρi and w=ρ ½ e¾ πi ; and along the prime axis, on completing a rotation, when θ = 2π∙, z=ρ and w = - ρ½. Hence, for any value of p, w is continuous all round the circle, but its ultimate value is different from its initial value.
Thus, as will happen in general with many-valued functions along their branching sections, the values belonging to points alongside of the prime axis having positive ordinates, however small, differ by finite amounts from those for the corresponding points, however near that line, having negative ordinates. The branch of the function so constructed is therefore discontinuous along the prime axis. Accordingly we conceive the plane cut through along this prime axis, and the branch of the function is easily seen to be continuous in the connected surface, which consists of the infinite plane so cut through. Riemann perfected this method of Cauchy and enabled us to keep all the branches simultaneously in view, and thus regard the function as unique and continuous upon all admissible paths without restriction. His process is in the case of a q -valued function to let the variable 2 move upon q different plane leaves.
Thus, as above, when p=l, q=2, taking the case k =l, let us suppose 2 to move in a second plane; proceeding as before for the points of the prime axis, we have now θ = 0, z=ρ, w=ρ ½ e πi = - pi ; and, when θ = 2π, z=p and w=ρ ½ e 2πι =ρ⅛. Thus, though the rotation has brought w round continuously from the values for small positive ordinates to those for small negative ones, these last differ finitely from the first. Now these final values in the second plane are the same as the initial ones in the first plane; moreover, the final values in the first leaf are the same as the initial ones in the second; hence, if the second leaf be cut through all along the prime axis, as the first was, and if we conceive the leaves to cross one another (as indicated in profile in fig. 3) all along the cut, we have a connected two-leaved surface, called by Riemann a winding surface of the first order, to each point of which corresponds a definite value of the function w=z ½ , and to each continuous curve in it, whether in the same leaf or passing through both, correspond values of the function which vary continuously; every closed curve which crosses the branching section either an even number of times or not at all leads finally back to the initial value. This conception can evidently be extended to surfaces with a greater number of leaves corresponding to functions admitting of a greater number of values.
7. In considering infinite series we must explain the con ception of absolute convergence introduced by Cauchy: an infinite series of real quantities is absolutely convergent when the sum of its positive terms taken apart, and likewise that of its negative terms taken by themselves, are each finite. When these are not separately finite, a series which converges for some value of the variable is semi-convergent. A complex series ∑(u + iv ) converges absolutely when the u series and the v series are both absolutely convergent. Hence it can be shown that the necessary and sufficient condition that a complex series may converge absolutely is that the series of the moduli of its terms converges. Such a series presents the same value if the sequence of its terms is rearranged according to any law whatever. Thus an infinite series of ascending powers of 2 the complex variable, a 0 + a 1 z + a 2 z 2 + ... + a n z n +... = f(z ), with complex coefficients, is absolutely convergent for any value of 2 for which the series A o + A 1 ρ + A 2 ρ 2 +... + A n p n +... formed by the moduli of its terms is convergent; and vice versa. In geometric language, if such a series be absolutely convergent for the point corresponding to any value of 2, it is absolutely convergent for all other points equally distant from the origin of coordinates; these are therefore situated on a circle. It is also absolutely convergent at all points within this circle, and therefore the region of convergence of this series is always a circle round the origin. The radius of the circle of convergence of a series is the greatest value of p for which the series of moduli converges. It is found by the condition that as n increases indefinitely we must have < 1
or ρ<(A n / A n +1 ). As the terms of an infinite series of powers are one-valued, the series itself, so long as it converges, is a one-valued function of the complex variable, which does not become infinite anywhere. It can also be proved that this function is continuous: that is, if z and z ± δ be values of 2 for which it converges, then lim[ f (z ±δ) - f(z )] = 0, when δ = 0.
8. When a variable quantity depends upon two others in such a way that, if their values are given, its value is known, it is of course a function of them; but, inasmuch as for real values of x and y, when z( = x + iy ) is given, we know both x and y, Cauchy considered any function F(x, y) to be a function of z. On the other hand, Riemann considered a function F(x, y) to be one of z when it contains x and y only in the combination x + iy. In fact, if we have 8F 8F dF = 8χ dx + ^y d V for any indefinitely small increments dx and dy, inasmuch as this may be written ∖(8F .8F∖. . 7 . lf8F .8F∖ ., x j dF= 2 ⅛ + *~⅛) (dx ~ id ri + 2 ⅛" τ ⅛)(dx + ld ^ and if <7# must always be zero as long as d(x + iy) vanishes, 8F 8F then we must have ÷ i = 0. This, which is the condition that, when x and y are perfectly unrestricted (and so may each be complex), they may enter only in the combination x + iy, must also hold when they can both admit only real values, in order that F may be in Riemann’s sense a function of z . Cauchy called the function monogène when it satisfies this relation. The further significance of this relation is seen if we put dx = p cosθ and dy =ρ sinθ, whence dz — pe iθ and dx - idy = ρe -iθ ; for then 1 dF 1 ∕8F . 8F∖ n . 0 1 ∕8F .8F∖ 1
1n general -. τ f depends not only on the value of z but also on the direction θ of the adjacent point. When +1 = 0 we have = fø = J that is, for each point z , in case the function is monogeneous, there is a derived function with regard to the complex variable, independent of the direction in which this variable is supposed to change. If the relation do not hold, there is a different derived for each such direction. When we separate F into its real and imaginary parts, we may write F—u + iv, and the preceding relation becomes 8u 8v .∕8v 8u∖ λ , 8u 8v ^ .8u 8v ⅛ - ¾ + ∖⅛ + ¾) = 0 ∙ whence S ^ ¾i =0 a,lt ⅛ + ⅛= 0 ∙
rr r <-v λ λ
Hence further γ - 2 + f -≡ = 0 and =- 5 + τ - r , = 0.
ox-oy i oχ i oy-
9. A function of a complex variable which is continuous, one-valued, and has a derived function when the variable moves in a certain region of the plane is called by Cauchy synectic in this region, and by Briot and Bouquet holomorphe, to indicate that it is like an integer function, for which this property holds throughout the entire plane. Weierstrass styles it an analytic function. The property that dw∣dz is independent of dy∣dx furnishes a geometrical relation between the figure which w describes in its plane and that which z describes. In fact, let values w 1 , w 2 , w 3 correspond to the values z 1 , z 2 , z 3 „ then, if ———Û
1 2 3, ’ w 2 - w 1 z 2 - z 1 ’ as already observed, the triangle for the values w is similar and similarly placed to that for the values z ; in that case w 3 -w. w. 2 -w.
∕ = ~. But, if we suppose z 3 to close up to z 1 , ⅞ z ι 2 2 2 ι
then w 3 closes up to w 1 , and, if z 2 close up to z 1 , w 2 closes up to w 1 . In the limit the equality of these quotients for all directions of variation of z shows that the figure which w describes in its plane is in its infinitely small parts similar to that described by z , and two intersecting curves in the plane of w cut at the same angle as the corresponding curves in the plane of 2, provided always that dw∣dz is neither zero nor infinity.
10. Let z + h be a complex value within the circle of convergence of the series f (z ) = a 0 + a 1 z + a 2 z 2 +... + a n z n +..., then we have f (z + h ) = a 0 + a 1 (z + h ) + a 2 (z + h ) 2 + ... +a n (z+λ) w + .. . Now, if we arrange this absolutely convergent series by powers of h, the coefficients are infinite series which can be proved to be the successive derived functions of f (z); thus, when a one-valued function f (z ) can be expanded in a series of powers of z , the expansion can be made only in a single manner. We have therefore
∕(s + A)=∕(z) +j-∕'(z) + ∣2-∕"(z)+ . . . +∣^^∕ i, (2 )+ · ∙∙, which is Taylor’s theorem for a function given by an infinite series of powers of a complex variable.
11. We proceed to examine at how many points within a circle of convergence of radius R the function f (z ) vanishes, i.e., has a root. In the first place it can be shown that, if R be finite, the function cannot vanish at infinitely many points in the circle without vanishing identically, that is, being zero everywhere in the circle. Hence we can only have a finite number of roots in any finite portion of the plane, and we may suppose none of these to lie upon the boundary circle. If the function vanish at any point z 1 , then, when we take the expansion Λ*) =Λ⅞ + -Λ⅞) + 1 ⅛V(⅞) + ⅛^∕ "(⅞) + ie., we have ∕(z 1 ) = 0. We may also have ∕'(z 1 ) = 0, ∕"(z 1 ) — θ, &c. Suppose that the first of these which does not vanish is ∕^(2 ι)> or that the function f(z) vanishes for z = z 1 to the order α, then ∣∣∕W(z 1 ) + ^∕< α+1 >(¾)+ &c.
Now let this quotient be rearranged according to powers of z , and suppose the function to vanish at z 2 to the order β, we can expand by powers of z - z 2 , whence we get- —-3—, ultimately = ϕ(z),—a series not
(z - z 1 ) (z - z 2 ) p · . ·
vanishing anywhere within the region. If f(z ) were a rational integer algebraic function, we should find ϕ(z) a constant, and we see that a + β +. . . &c. = n : that is, such a function cannot have more than n roots in the region of convergence, i.e., in the entire plane. Taking logarithms, we have log f (z ) = a log(z - z 1 ) + βlog(z - z 2 ) + ... + logϕ(z). Hence, taking for each logarithm on the right one of its innumerable values, let z move from any point A round the circumference of the boundary circle, keeping its interior on the left. Since z - z 1 vanishes only once within the circle, namely, for z = z 1 , log(zz 1 ) = logρ + iθ will become log ρ + i (θ + 2π) when z returns to A. Similarly log(z - z 2 ) will differ from its initial value by 2iπ; and so for the others. But, as ϕ(z) does not vanish anywhere in the circle, logϕ(z) will return to its original value at A. Hence, if there be v vanishing points or roots within the circle of convergence, the value of log f(z) changes by 2iπv, when z travels all round the circumference. Conversely, if the logarithm alter by 2iπv as z travels round the circle of convergence, the number of roots within the circle is v.
12. This theorem of Cauchy’s at once leads to the fundamental theorem of algebra, that every equation of the nth degree f (z) = a 0 + a 1 z + a 2 z 2 + .. . + a n z n = 0 has n complex roots. For we can take the radius of the circle so great that the value of a n z n far exceeds that of the other terms; and, writing f(z) = a n z n (1 + P), the modulus of P can be made as small as we please. Hence log f (z) = log(a n z n ) + log(l + P). Now as we go round the circle log(l + P) differs as little as we please from log(l) = ± 2kiπ, and so, whatever value we begin with, as the corresponding logarithm varies continuously, the value of log(l + ) will not have changed by a multiple of 2πi when we return to the point we set out from. But, since the point z = 0 is 1
within the circle, log(a n z n ) = nlog(a n 1/n z ) changes its argument by n.2πi in consequence of the circuit. Hence log f(z ) changes by 2π in : that is, in the circle of convergence whose radius is as large as may be required there are always n values for which the rational function a 0 + a 1 z + a 2 z 2 + ... + a n z n vanishes. Some of these may be coincident; but in all cases the total number of the orders of the different vanishings is n.
13. If two given points z 0 and Z be joined by any curve of finite length, and if we take as many intermediate points as we please z 1 , z 2 , . . . z n -1 upon the curve and form the sum ∙∕(⅞)(2 1 -2 θ) ÷∕(3 1)(⅞7 2 1) ÷,y(2 2)(⅞^ -2r 2) +∙∙· ÷∕(2 n -l)(^~ z n -l)ι then the complex limiting value to which this sum tends as n is increased indefinitely is called the definite integral of the function f(z), formed from z 0 to Z along the path prescribed by the equation of the curve. Having first examined, after the manner of the theorems regarding real integrals, under what conditions this sum has a determinate limiting value, we have next to take into account that a definite complex integral differs essentially from a real one in that the path along which the variable z may travel from one limit to the other is perfectly arbitrary. We have therefore to investigate this problem: When is the integral of a complex function a unique function of its superior limit independent of the path of integration? In order to answer this we must recall an important theorem concerning the reduction of a double integral of a function to simple integrals along the boundary curve, which is in fact a case of Green’s theorem (see Infinitesimal Calculus, vol. xiii. p. 58) when confined to two variables.
14. Suppose P and Q are two real functions of x and y which are everywhere within a certain boundary finite and continuous, the theorem is then that the double integral ∬ (δr - ⅝∕ ) dxdy taken over the entire region is equal to the simple integral ∫ (Pdx + Qdy) taken along the entire boundary of the region.
Taking the positive directions of the axes as usual, we define the positive direction along the boundary to be that for which the bounded surface is on the left hand; then, if we have to exclude any portion, e.g., a circular space altogether within the external boundary, so that the points within it are to be regarded as not included in the region, the positive direction of the boundary is that indicated by the arrows in fig. 4. In all the simple integrals the integration must be effected in the positive direction thus explained.
To integrate JJ ~^~dxdy with respect to x, let us split up the region into elements by parallels to the axis of x. Select any one and, reading from left to right, denote the values of Q where the element crosses the boundary at the entrances by Q 1 , Q 2 , &c., and at the exits by Q', Q", &c. ; then = -Q 1 + Q' -Q 2 + Q" - &c., and thus ff⅛⅛xdy = i-Q χ dy + ic∕dy + i-Q 2 dy + I Q"dy-&c.
Now in each of these integrals y goes through all its values from the least to the greatest, therefore dy is always to be taken positively. But, denoting by dy 1 , dy 2 , &c., and by dy', dy", &c., the projections on the axis of y of the arcs of the boundary cut by the element as above, when we take the positive direction of the boundary into account, we have dy = - dy 1 = - dy 2 = &c. = + dy' = + dy" = &c.; thus J^ dxd V = ∕Q1 c ⅛1 + ∕Q' c ⅛' + ∕<3 2 ⅜2 + ∙⅛-≈fQdy, where this is taken along the entire boundary in the positive direction. In like manner, dividing the region into elements parallel to the axis of y, and denoting the values of P at the entrances, proceeding from below upwards, by P 1 , P 2 , &c., and at the exits by P', P", &c., we have J∕^ξ~dxdy ≈ ~ βP χ dx + ∣Pdx - ∣ P 2 dx + <kc., where dx is positive. Hence, as before, taking account of the positive direction of the boundary, dx = +dx χ = + dx 2 = <fcc. = - dx' = - dx" = <tc., and consequently --dxdy = - J P χ dx x - y Pdx' - Jp 2 dx 2 - <tc. = - ∣Pdx, taking the integral in the positive direction along the entire boundary. Accordingly, putting these both together, we have the surface integral ff(~^~^) dχ dy = j(Pdx + Qdy), taken positively along the entire boundary. This proposition may be extended to complex values: when P = P' + iP' and Q = Q' + iQ", as the proposition is already proved for real values, we can put ∬ ∕(^ - ^) *rfy -∕(⅛ - *⅛, . n γδQ" 8P ’ ∖ + W &T - ^8y) dxd ^’ t ^ ιen > applying it to each of these,
^ ⅛) dx ⅛ = ∕ <- p,ffa + e ' rf 3 , > + , '/ ^ p ' dx + 0 " rf 3 ’ > = y(Pdx + Qdy). We assumed that there were no branchpoints within the region, or other points at which P or Q are discontinuous. If there were such, we should have to surround them with actual closed lines, as small as we please, and thus exclude them, introducing these lines as parts of the boundary of the region.
15. We can now let P=f(z), Q=if(z). We see that, if f(z ) be an analytic function of z without exception in any simply connected region, ∫f(z)dz = 0 when taken all round the boundary of that region. If we take any two paths from z 0 to Z in this region, since these paths from z 0 through Λ to Z and back through B to z 0 include such a region, we have I (z 0 A ZBz 0 ) = 0 or I(z 0 A Z) + I(ZBz 0 )= 0. But, as the value of the integral I(ZBz 0 )≈ - I(z 0 BZ) along the same path, the last equation becomes I(z 0 AZ) = I(z 0 BZ), or the value with which the integral starting from z 0 arrives at Z is independent of the path travelled under the conditions supposed: it is a function of the upper limit only. The path z 0 BZ is said to be reconcilable with the path z 0 AZ . If the region within which f (z ) is an analytic function be multiply connected, ∫ (z)dz vanishes when taken positively all round the boundary. The integral has a definite value along each separate boundary curve. For instance, if within the region we take a closed curve which includes a simply connected region, as a in fig. 5, the integral along this curve vanishes. But for a curve which includes a region along with a boundary curve, as β in fig. 5, the in tegral has the same value as it has for this boundary curve. Whence, if a function fi(z) which has to be integrated lose the character of an analytic function in isolated places,.
these must be excluded by surrounding them by curves as near them as we please and taking these curves as part of the boundary; it can then be integrated in the multiply connected region so found. The iutegral for the boundary of such a place may vanish or not. If not, the integral function is many-valued and branches at this place. If it vanish, the integral remains one-valued, but the point may still be a singular one for the integral: that is, the integral may at it cease to be finite or to be an analytic function.
16. If we know the value of any function and those of all its derived functions at any one point, and if we know that it is an analytic function within a finite connected region,—that is, that it is unique, continuous, and has a determinate first derived,—then Taylor’s series for it converges throughout a circle round this point, which is included within the region; and for each point within the region the function can be developed in a similar series. In other words, every function which is analytic without exception in a connected region can be expanded around each point of the region in a series of ascending positive integer powers. The method in which we show this is due to Cauchy. Let f(z) be an analytic function without any exception in a simply connected region (including the boundary); then, if t = u + iυ be any point within the region, the quotient is likewise an analytic function within the region, only the point z = t is a singular one. Surrounding t by a circle whose radius p is as small as we please, we obtain a region bounded by two curves in which the quotient is analytic without exception. Hence, integrating the quotient along both boundaries, the sum of both integrals vanishes. In regard to the small circle put z-t=ρe iθ; therefore we get dz = iρe iθ dθ, and for it ∫ f(z) . i 2π h-±ddz== -il f(t +pe iff )dθ. Now f is continuous near J z-t Jo z = t. Hence p can be taken so small that for all values of θ the difference of f(t) and f(t + pe iθ ) may be less than δ, where the modulus of δ is as small as we please. Hence ? I Λ t ÷ pe lβ }dθ and i I f(t)dθ = 2iπf(f) differ by a quan- J o Jo tity whose modulus is less than that of 2πδ: thus, whatever be the value of p, we have i+ p( ie )dθ = 2⅛∕(i). Hence we have j⅛⅛dz - 2iπf(t) = 0, or ∕(0 = 2 ⅛ r ∕S^ when the integral is taken positively along the outer boundary, and so along any curve which includes the point t. This is what Cauchy called the residue of the quotient function relative to its infinite t. We thus see that, when the values of a function f (z) are given along a closed curve, and the function is analytic without exception within this curve, the value of f (z) can be found for each internal point by means of a definite integral.
It was assumed that f (z) has throughout a derived. This can be exhibited as a definite integral. In fact ∕(< + ΔQ-∕(Q If dz u ¾⅝√-δ^√ whence > lettιn s ∆i converge to zero, = where the integral may be taken along any path round t, provided it remain within the original region. But it follows that throughout this f'(t) also is an analytic function, for, as before, f "(t ) = ^-f ln like manner √' i ∖i) = i-
∙ 7 v7 2ιτr∕(z-<) 3 · x7 2tτrJ (z - t) n + 1
Thus for all points in the region the function has higher deriveds of all orders, and they are all analytic functions.
17. We can now develop the function f (t ) in a series of powers. Let us select a point a , otherwise arbitrary, such that the largest possible circle which can be drawn round it without going outside the region may include the point t, for which the function is to be expanded. For every point on the circumference of this circle we have mod(t - a) < mod(z - a) ; and therefore —- = ——
x 7 x 7 z-t z-a^ t-a
1 [ t - a (t-d) 2 (t-d) 3 I .
— < H ∙^ 7 ∖5 + 7 vi +... f is a conver-
z — a ( z - a {z-a) i (z-o) λ J
gent series. Now, substituting this in the equation ∕(^) ~ 2^πJ anC *· ta k* n g ^he c ι rc lθ as curve of inte-
gration, we find
+ (i — a ) 2 βζ^~~dpd ,z ÷ & c · j > which by the equations already established may also be written ∕(i) =∕(α) + (i - α)∕ v (α) + 2^ + 9⅞ ∕" ’ (α ) + ∙∙∙∙ This expansion con verges unconditionally for all values of t within the circle round a which does not go outside the original region. This is Taylor’s series for a complex function, and this deduction of it shows definitely how far its convergence extends, namely, to all points t which are less distant from a than the nearest point of discontinuity or of ramification. Thus a function whose value, as well as those of all its derived functions, is known for a point is analytic in the neighbourhood of this point only when a finite circle, however small, can be assigned within which this series converges. The series arrived at will, in general, not converge for all values of t within the original region in which f(z ) was assumed to be analytic. But we can vary the centre a so as to arrive at a circle, and thereby at a development, which embraces any such required point. If t be at a finite distance, however small, from the circle, draw any curve from a to t which shall be always at a finite distance from the boundaries. Let the circle round a meet this curve in the point a' between a and t : then by means of the series we can calculate the function f and its deriveds for a point on the curve and within the circle as near as we please to a', and thus take this as the centre of a new development all whose coefficients are known. The new circle of convergence cuts the curve in a point a", which is certainly nearer to t. The continuation of this process must ultimately lead to a circle which includes the point t, since the radii cannot diminish below a finite assignable quantity, the path a a' a" ... t being always at finite distance from the boundaries. By this process we can also carry on an analytic function that is defined by a series of powers beyond its circle of convergence into a region which does not contain any singular point. Each such series of powers defines the function for a determinate circular region of convergence, and is said to be an element of the function. Different elements of the function are got according to the centre chosen for the expansion, and even the same value of the argument belongs to different elements of the function; but for a unique function the different elements must lead to the same value for the same argument. Two or more analytic functions, however defined within given regions, are to be regarded as belonging to the same function only when the elements of one function can be derived from those of the others.
18. We conclude with illustrating by a few applications to elliptic functions of a complex variable. Calling R(z) = (z - a)(z - b)(z - c)(z - d), then √ R(z) will be expressed as a unique function by means of a two-leaved Riemann’s surface. Its branches take the same values at each of the points a , b , c , d. Let the leaves be pierced through and joined transversely along the lines a to b and c to d (fig. 6). Assume that in the upper leaf √ R(z) at the origin is + R and in the under leaf is - R. Every path which, when projected on the upper leaf, goes an odd number of times round an odd number of branchpoints leads from one leaf into the other; on the other hand, a path which goes an even number of times round an odd number of branch-points, or which goes round an even number of branch-points, ends in the leaf in which it originally began. Now let us investigate what changes in value accrue to ∣ an *^ tθ f *∕R( z }dz when z de scribes part of a vanishing circle round the branchpoint a. First, let z = a + pe iθ , then dz = iρe iθ dθ, R(z) = ρe iθ (a -b + ρe iθ )(a - c + ρe iθ )(a - d + ρe iθ S = ρe iθ S. Hence —⅞≈-= ipM#/ 2 —- f = and R(z) . dz = ipie% iff d0 *JS. But, V R∖~) ’ v S when p is evanescent, lim √ S= √[(a-b)(α-c)(α - d )] = T , and is therefore neither infinitely great nor evanescent; integrating therefore between θ 0 and θ 1 , we find ⅛ ∕"¾ . ï [^θχ limvp. -, p e i W 2 dθ and lim vp 3 ~l e⅛ iβ dθ. Since these J ⅛ -I J Θq vanish when p does, and the other factors are not infinite, we see that, when ∣and ∕ ’ ^R(z}dz are taken along paths surrounding indefinitely closely a single branchpoint, both these integrals vanish.
If now we confine our attention to the integral of the first kind, let z go from the origin in the upper leaf along a straight line to near the point a ; then the integral I— ·, putting y =mx in z=x+iy = (1 + im)x, J ∖∕R(z) may be calculated as for a real variable, and it increases ( a dz by I — 7⅛τr∖ ∙ Next let z go round α in a vanishing circle, J 0 V R(z) the integral gains nothing; then let z return to the origin along the same line, it proceeds now in the lower leaf; thus both √ R(z) and dz have signs opposite, but values equal to, what they had on the outward path. Thus on the whole by travelling this elementary contour (lacet, Schleife) from O to near a, round a , and back to O, the value of the in- ∕*ct dz tegral is increased by A = 2 ∕ — In like manner we
7 0 √Λ(z)
have for the elementary contours or loops round b, c, d rt> dz C c dz the three increments B = 2 ∕ —=, C = 21 ,- , and d rd dz JoVR(z) Joχ∕R(z)
19. Now, if the loop round a be described twice in succession, the integral changes by A - A=0 and so on. But if, after a is once described, b is described once, the change is A — B. If then z go straight to any arbitrary point for which the value of the rectilinear integral is w , the value along the entire path, or along any path reconcilable with it, is A - B + w. The two loops round a and b may be taken as often as we please before going by the straight course from O to z ; thus A - B is a period of the integral, which is accordingly an infinitely many-valued function of z . As there are three independent differences of any four quantities, it would seem as if there should be three periods; but we can show that A - D arises from a combination of A - B and A - C. In fact, describe round the origin a circle to include all the four points; then let z go straight from the origin to a point p on this circle, travel round it positively, and return from p to the origin. The integral has along this path the same value as along the four loops in succession, since between the two paths there is not any singular point and both have the same origin and goal. Thus the value of the integral along this path is A - B + C - D. But, when z goes round the circle from p, the radical changes sign four times. If loops be described from p round a, b, c, d, they are reconcilable with the circumference of the circle. Hence the radical has the same sign when it returns to p as that it set out with; accordingly, since along Op and pO the radical has the same sign, the integrals I(Op) and I(pO) cancel, and the integral along the four loops in succession has the same value as along the circle alone. To find this value, put z = ρ e iθ and it becomes
£ f β ° +2,r e iθ dθ τj λ ' √R÷-!)F3("-3' ' “the radical is unique along the circumference of the circle, the value of this integral is the same whatever be the value of p, provided it be large enough to include all the singular points. Hence we can let p increase indefinitely and then this has the limiting value zero. Consequently A-B+C-B — 0, or A-B = AC-(A-B); thus the third period is composed from the other two. Hence the periods reduce to A - B and A - C. It can thus be shown that all values of the integral for all possible paths of integration between O and z are included in the two formulae m(A - B) + n(A -C) + w and m(A - B) + n(A - C) +A - w.
20. If a new variable z' be introduced, homographically related to z by the equation a + βz' + γz + δzz ' = 0, it transforms the integralinto (αδ - βy) J∙ The coefficients of A 1 (z') can be determined so that it may differ only by a constant multiplier from (1 - z ' 2 )(l - k 2 z ' 2 ). As k is given by a reciprocal quadratic equation, it is always possible to choose such a value that its modulus may be less than 1. In other words, it is possible by a homographic transformation to convert the quadrilateral a, b, c, d into a parallelogram, the new origin being at the intersection of its diagonals (fig. 7). This leads to Jacobi’s form of the integral when k is supposed real. The interi dz grals along the loops are here A = 2 — z - ov ,⅞— τ ⅜-√⅜> J o v (1 - 2-)(1 -λ 2 z 2 ) nf 1 ' t ⅛ r <,i~' ½ "Jo √(l-^)(l-½≡) ’ J t √(l- 2 2)(l-i⅞≈) ’ r-Q∕ fe ) dz 7, = 2 ∫ 0 √(1-∕)(1-W)∙ lle ≡Λ-C = 2Λ.
A - B is the integral taken consecutively along the loops of a, b; but for these paths we may substitute the following: proceed straight from O to a near a, then round a back to a , then straight from a to β near b, next round b back to β, then again straight to a , and straight back to O. The integral takes the same value by this path as by the other, since there is no singular point between them and both paths have the same origin and goal. Now the radical changes sign as z goes round a , and again as it goes round b; hence it returns to a with the same sign. Thus Oα is travelled in the two directions with the root having the same sign; hence I (O a ) and I(a O) cancel. By decreasing indefinitely the radii of the circles, the circular integrals vanish; and, as the root is negative along aβ and positive along βa, the value of A - B reduces to the negative double value of the integral from a to β, that is, from a to b for vanishing radii; thus
« f 1/fc dz AB= ~ 2 J 1 √(1 _ 2 2 )(1 - ⅛¾ 2 ) ’ The rectilinear integrals required are now only between the limits 0, 1, and 1/ k , so that they do not cross any singular point. In case k is a positive proper fraction we have in Jacobi’s notation A = 2K, A - C = 4 K , A - B = 2iK' ; therefore all C z dz values of the elliptic integral∕ - — τo what-
J o √(1 -z 2 )(l -k 2 z 2 ) ever be the path taken between the limits 0 and z, are included in the two formulae 4 nK + 2miK' + w and 4 nK+ 2miK' + 2K-w. Hence the inverse function z = sin amw is doubly periodic for 4 K and 2iK', thus—
sin am(4nK + 2miK' + w) = sinamw sin am(2(2n + 1) K + 2miK' - w) = sin amw .
This last shows that sin amw takes the same value for two different values of w within the same period. In fact, in the plane of w measure a series of parallels (see fig. 8) to the axis of ordinates at distances 4A, and to the prime axis at distances 2 K ', then the plane will be divided into rectangles such that in each sin amw) goes through all its values and has equal values at corresponding points of different rectangles: thus, whatever it is at P for w = u + iv , it is the same again at P 1 , T ’ 2 , and P 3 ; at where w = 2K - u + i(2K' - v), it has the same value; at R, where w = 2K + u + iv , and at S, where w = 4K - u + i(2K' - v), it has the equal and opposite negative value. The value of sin amw vanishes at the points marked in fig. 8 by circles and is infinite at those denoted by crosses. (b. w.)
