74 VARIABLE these must be excluded by surrounding them by curves as near them as AVC please and taking these curves as part of the boundary ; it can then be integrated in the multiply con nected region so found. The integral 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 Fig. 5. cease to be finite or to be an analytic function. 16. If Ave know the value of any function and those of all its derived functions at any one point, and if Ave knoAv that it is an analytic function Avithin 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 AVI thin the region ; and for each point within the region the function can be developed in a similar series. In other Avords, every function Avhich is analytic Avithout exception in a connected region can be expanded around each point of the region in a series of ascending positive integer poAvers. The method in Avhich Ave shoAv this is due to Cauchy. Let f(z) be an analytic function with out any exception in a simply connected region (including the boundary) ; then, if t = u + iv be any point within the f( z region, the quotient -^~ is likewise an analytic function Avithin 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, inte grating the quotient along both boundaries, the sum of both integrals vanishes. In regard to the small circle put z-t = pe w ; therefore Ave get dz = ipe i9 dO, and for it f (t + pe w }dO. NOAV/ is continuous near z t. Hence p can be taken so small that for all values of the difference of f(t) and f(t + pe iB ) may be less than 8, where the modulus of 8 is as small as Ave please. Hence differ by a quan- o o tity whose modulus is less than that of 27r8 ; thus, what ever be the value of p, we have i I f(t + pe iff )dO = 2i-nf(t). ff(z) Hence Ave have / -dz-2iirf(t) = 0, or 1 , 2 - t J ^ 2nrJ z -t 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 excep tion within this curve, the value of /(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 interal. In fact whence > t converge to zero, f (t) = 9^773^, where the integral may be taken along any path round t, provided it remain within the original region. But it follows that through out this f (t) also is an analytic function, for, as before, /"(?) = r. In like manner f (ll t) = - 7- I -^ 2iirJ (z - 1) 3 - w 2iirJ (z 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 mocl(Z - (t) <mod(.r - a) and therefore z t z-a. t-a 1 z-a gent series. z - a Now, substituting is a conver- this in the equation 1 /7 (V) /(O = :y~ ~^nfi z anc ^ taking the circle as curve of inte- gration, we find f(t) = v) ) 1 ^dz + &c. V , which by the equations already a) j established may also be written f(f) =/() + (t - a]f (a) (t .) 2 (t ) 3 H ; ~-/ ( a ) + 1 o Q/ () + .... This expansion con- 1 . .j L . a . O 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 shoAvs definitely hoAV 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 Avhose value, as well as those of all its derived functions, is knoAvn for a point is analytic in the neighbourhood of this point only when a finite circle, IIOAV- ever small, can be assigned Avithin which this series con verges. The series arrived at will, in general, not converge for all values of t Avithin the original region in Avhich f(z) Avas assumed to be analytic. But Ave can vary the centre a so as to arrive at a circle, and thereby at a development, Avhich embraces any such required point. If t be at a finite distance, however small, from the circle, draAv any curve from a to t Avhich shall be ahvays at a finite distance from the boundaries. Let the circle round a meet this curve in the point a betAveen a and t : then by means of the series Ave can calculate the function / and its deriveds for a point on the curve and within the circle as near as Ave please to a, and thus take this as the centre of a IICAV development all whose coefficients are knoAvn. The new circle of convergence cuts the curve in a point a", Avhich 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 beloAv a finite assignable quantity, the path a a a" . . . t being always at finite dis tance from the boundaries. By this process Ave can also carry on an analytic function that is defined by a series of powers beyond its circle of convergence into a region Avhich does not contain any singular point. Each such series of poAvers 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 arc 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, hoAvever defined Avithin given regions, are to be regarded as belong ing to the same function only Avhen the elements of one function can be derived from those of the others.
18. We conclude with illustrating by a feAv applica-