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...