Page:EB1911 - Volume 11.djvu/325

The function $\exp(z)$ is used also to define a generalized form of the cosine and sine functions when z is complex; we write, namely, $\cos z={\frac {1}{2}}[\exp(iz)+\exp(-iz)]$ and $\sin z=-{\frac {1}{2}}i[\exp(iz)-\exp(-iz)]$. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, $\cos i=1+1/2!+1/4!+\dots$ is obviously greater than unity.

§4. Of Functions of a Complex Variable in General.—We have in what precedes shown how to generalize the ordinary rational, algebraic and logarithmic functions, and considered more general cases, of functions expressible by power series in z. With the suggestions furnished by these cases we can frame a general definition. So far our use of the plane upon which z is represented has been only illustrative, the results being capable of analytical statement. In what follows this representation is vital to the mode of expression we adopt; as then the properties of numbers cannot be ultimately based upon spatial intuitions, it is necessary to indicate what are the geometrical ideas requiring elucidation.

Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be counterclockwise; another square, also of side a, may be added to this, so that there is a side common; this common side being erased we have a composite region with a definite direction of perimeter; to this a third square of the same size may be attached, so that there is a side common to it and one of the former squares, and this common side may be erased. If this process be continued any number of times we obtain a region of the plane bounded by one or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of description, which is such that the region is on the left of the describing point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, it being understood that there is assigned an upper limit for the greatest side of a triangle, and a lower limit for the smallest angle. In the former method, each square may be divided into four others by lines through its centre parallel to its sides; in the latter method each triangle may be divided into four others by lines joining the middle points of its sides; this halves the sides and preserves the angles. When we speak of a region of the plane in general, unless the contrary is stated, we shall suppose it capable of being generated in this latter way by means of a finite number of triangles, there being an upper limit to the length of a side of the triangle and a lower limit to the size of an angle of the triangle. We shall also require to speak of a path in the plane; this is to be understood as capable of arising as a limit of a polygonal path of finite length, there being a definite direction or sense of description at every point of the path, which therefore never meets itself. From this the meaning of a closed path is clear. The boundary points of a region form one or more closed paths, but, in general, it is only in a limiting sense that the interior points of a closed path are a region.

There is a logical principle also which must be referred to. We frequently have cases where, about every interior or boundary, point z₀ of a certain region a circle can be put, say of radius r₀, such that for all points z of the region which are interior to this circle, for which, that is, $|z-z_{0}|\ <r_{0}$, a certain property holds. Assuming that to r₀ is given the value which is the upper limit for z₀, of the possible values, we may call the points $|z-z_{0}|\ <r_{0}$, the neighbourhood belonging to or proper to z₀, and may speak of the property as the property (z, z₀). The value of r₀ will in general vary with z₀; what is in most cases of importance is the question whether the lower limit of r₀ for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z, z₀) is of a kind which we may call extensive; such, namely, that if it holds, for some position of z₀ and all positions of z, within a certain region, then the property (z, z₁) holds within a circle of radius R about any interior point z₁ of this region for all points z for which the circle $|z-z_{1}|=R$ is within the region. Also in this case r₀ varies continuously with z₀. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (1) for some point z₀ within or upon the boundary of the sub-region, (2) for every point z within or upon the boundary of the sub-region.

We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (z, z₀) holds for all points z and some point z₀ of the region, be called suitable: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the preceding, which converge to a point, say ζ; after a certain stage all these will be interior to the proper region of ζ; this, however, is contrary to the supposition that they are all unsuitable.

We now make some applications of this result (B). Suppose a definite finite real value attached to every interior or boundary point of the region, say $f(x,y)$. It may have a finite upper limit H for the region, so that no point (x, y) exists for which $f(x,y)>H$, but points (x, y) exist for which $f(x,y)>H-\epsilon$, however small ε may be; if not we say that its upper limit is infinite. There is then at least one point of the region such that, for points of the region within a circle about this point, the upper limit of $f(x,y)$ is H, however small the radius of the circle be taken; for if not we can put about every point of the region a circle within which the upper limit of $f(x,y)$ is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function $f(x,y)$ is the radius r₀ of the neighbourhood proper to any point z₀, spoken of above. We can hence prove the statement (A) above.

Suppose the property (z, z₀) extensive, and, if possible, that the lower limit of r₀ is zero. Let then ζ be a point such that the lower limit of r₀ is zero for points z₀ within a circle about ζ however small; let r be the radius of the neighbourhood proper to ζ; take z₀ so that $|z_{0}-\zeta |\leq {\frac {1}{2}}r$; the property (z, z₀), being extensive, holds within a circle, centre z₀, of radius $r-|z_{0}-\zeta |$, which is greater than $|z_{0}-\zeta |$, and increases to r as $|z_{0}-\zeta |$ diminishes; this being true for all points z₀ near ζ, the lower limit of r₀ is not zero for the neighbourhood of ζ, contrary to what was supposed. This proves (A). Also, as is here shown that $r_{0}\geq r-|z_{0}-\zeta |$, may similarly be shown that $r\geq r_{0}-|z_{0}-\zeta |$. Thus r₀ differs arbitrarily little from r when $|z_{0}-\zeta |$ is sufficiently small; that is, r₀ varies continuously with z₀. Next suppose the function $f(x,y)$, which has a definite finite value at every point of the region considered, to be continuous but not necessarily real, so that about every point z₀, within or upon the boundary of the region, η being an arbitrary real positive quantity assigned beforehand, a circle is possible, so that for all points z of the region interior to this circle, we have $|f(x,y)-f(x_{0},y_{0})|<{\frac {1}{2}}\eta$, and therefore (x', y') being any other point interior to this circle, $f(x',y')-f(x,y)|<\eta$. We can then apply the result (A) obtained above, taking for the neighbourhood proper to any point z₀ the circular area within which, for any two points (x, y ), (x', y'), we have $|f(x',y')-f(x,y)|<\eta$. This is clearly an extensive property. Thus, a number r is assignable, greater than zero, such that, for any two points (x, y), (x', y') within a circle $|z-z_{0}|=r$ about any point z₀, we have |$|f(x',y')-f(x,y)|<\eta$, and, in particular, $|f(x,y)-f(x_{0},y_{0})|<\eta$, where η is an arbitrary real positive quantity agreed upon beforehand.

Take now any path in the region, whose extreme points are z₀, z, and let $z_{1},\dots ,z_{n-1}$ be intermediate points of the path, in order; denote the continuous function $f(x,y)$ by $f(z)$, and let ƒ_r denote any quantity such that $|f_{r}-f(z_{r})|\leq |f(z_{r+1})-f(z_{r})|$; consider the sum

$$(z_{1}-z_{0})f_{0}+(z_{2}-z_{1})f_{1}+\dots +(z-z_{n-1})f_{n-1}$$

By the definition of a path we can suppose, n being large enough, that the intermediate points $z_{1},\dots ,z_{n-1}$ are so taken that if $z_{i},z_{i+1}$ be any two points intermediate, in order, to z_r and $z_{r+1}$, we have $|z_{i+1}-z_{i}|\leq |z_{r}-z_{r+1}|$; we can thus suppose $|z_{1}-z_{0}|,|z_{2}-z_{0}|,\dots ,|z-z_{n-1}|$ all to converge constantly to zero. This being so, we can show that the sum above has a definite limit. For this it is sufficient, as in the case of an integral of a function of one real variable, to prove this to be so when the convergence is obtained by taking new points of division intermediate to the former ones. If, however, $z_{r,1},z_{r,2},\dots ,z_{r,m-1}$ be intermediate in order to z_r and $z_{r+1}$, and $|f_{r,i}-f(z_{r,i})|<|f(z_{r,i+1})-f(z_{r,i})|$, the difference between $\sum (z_{r+1}-z_{r})f_{r}$ and

$$\sum \{(z_{r,1}-z_{r})f_{r,0}+(z_{r,2}-z_{r,1}f_{r,1}+\dots +(z_{r+1}-z_{r,m-1})f_{r,m-1}\}$$

which is equal to

$$\sum _{r}\sum _{i}(z_{r,i+1}-z_{r,i})(f_{r,i}-f_{r})$$

is, when $|z_{r+1}-z_{r}|$ is small enough, to ensure $|f(z_{r+1})-f(z_{r})|<\eta$, less in absolute value than

$$\sum 2\eta \sum |z_{r,i+1}-z_{i}|,$$

which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is $<2\eta S$, and is therefore arbitrarily small.

The limit in question is called $\int _{z_{0}}^{z}f(z)dz$. In particular when ƒ(z) = 1, it is obvious from the definition that its value is $z-z_{0}$; when $f(z)=z$, by taking $f_{r}={\frac {1}{2}}(z_{r+1}-z_{r})$, it is equally clear that its value is ${\frac {1}{2}}(z^{2}-z_{0}^{2})$; these results will be applied immediately.

Suppose now that to every interior and boundary point z₀ of a certain region there belong two definite finite numbers $f(z_{0})$, $F(z_{0})$, such that, whatever real positive quantity η may be, a real positive number ε exists for which the condition

$$\left|{\frac {f(z)-f(z_{0})}{z-z_{0}}}-F(z_{0})\right|\leq \eta$$

which we describe as the condition (z, z₀), is satisfied for every point z, within or upon the boundary of the region, satisfying the limitation $|z-z_{0}|\leq \epsilon$. Then $f(z_{0})$ is called a differentiable function of the complex variable z₀ over this region, its differential coefficient being $F(z_{0})$. The function $f(z_{0})$ is thus a continuous function of the real