27 c an In fact, each formula can be deduced from the other one. The whole theory of functions of a complex variable can be based on Cauchy's Integral Formula as well as on the Taylor's development.
Conversely, let + cis + +. . . (4) be any power series converging for values of z +O. Then it can be shown that this series converges within a certain circle of radius R and (excepting the special but important case that it converges for all values of Z and thus represents a so-called integral transcendental function of z) diverges outside of the circle. Within this circle, called the true circle of con vergence, the series represents an analytic func tion and forms in fact the Taylor's (or Mac laurin's) development of this function about the point z= a =O.
9. Other (a) Laurent's Series. If f(z) is single-valued and analytic within a cir cular ring of radii ri and r, about the point a, then f(z) can be developed into a Laurent's Series: f(z)=Ica(s—a)s, convergent for all values of within this ring: n < I < rs. Here I Al)di —n p I cn I < Mr , 27i c (t--a)n+i where C denotes a closed curve lying within the ring and encircling the point a, and M, r have the same meaning as in § 8.
(b) Lagrange's Series. Kepler's problem of developing a root w of the equation w=a+z sin w (or more generally a function 4, (w) of this root) according to powers of z led Lagrange to formulate the following problem: To develop a root w of the equation w a -I- zf(w), or more generally a function 4) (w) of this root, according to powers of z. He thus obtained the series that bears his name: 4)(w)=44 (a) + ( a) f (a) + (a)f (a)nj.
n=2ts!dan-1 Example: Let f (w) = Jew' —1), 1— V I— 2as s2 w Then atv/8a= (1 —2az +23)-1 and 1zn dn r(a2— 1)1 1 +1/1-2as+ ZI - n! dan L 2n Thus the foftn of the coefficients P. in the ex ample of § 8 is determined: pnw 1 dn [(pi— 1)1 n!dlin L 2n These funCtions are lcnown as Zonal Harmonics.
10. Roots and Singular Points.—A function f(z) is said to have a root or zero at an in terior point a of the region S of § 8 if f(a)=0. It follows that f(z) can then be written in the form f(s)=4—a)In[cm+ cm+I(s—a) ...], where m is a positive integer and cm+ 0 (un less Az) 1=70). The point a is called a root or
zero of order m; f(z) is sometimes said to have m roots in z=a. A circle of definite radius can always be drawn about the point a containing no further root of f(z).
If f(z) is single-valued and analytic through out the neighborhood of a point a with the exception of this point itself, and if f(z) be comes infinite no matter how z approaches a, then a is called a pole of f(z). Since 1/ f(z) then has a zero in a, f(z) can be written in the form 1"(z) where m is a positive integer and (z) remains finite and different from 0 at a; f(z) is said to have a pole of order m, or to have m (simple) poles in z = a.
If, however, f(z) does not approach a limit when z approaches a, and does not have a pole there either, a is called an essential singularity. In the neighborhood of such a point Weierstrass has shown that the function comes as near as one pleases to any arbitrary preassigned value.
11. Linear The linear transformation az b Z— ad—bc4-.0,plays an important role in the theory of func tions. If c=- 0, we have an integral linear transformation. Interpreted geometrically as a transformation of the plane into itself, it comprises as special cases: (a) Z—z b, i.e., a translation; (b) Z=eniz, " a rotation; (c) Z=--- Az, A >0," a stretching.
The general integral linear transformation can be generated by a succession of these trans formations.
If, however, c 0, we need to consider one further transformation: 1 (d) Z=---s' This corresponds to an inversion in the unit R= lir, followed by a reflection in the axis of reals: Y=—y.
The general linear transformation carries circles over into circles, straight lines being re garded, as in the geometry of inversion, as cir cles with infinite radii. The infinite region of the plane is considered as a point (not, as in projective geometry, as a line), and this is the convention which it is desirable, for reasons that cannot be named here, to make in the theory of functions. We spealc of the point z=c0.—Cf. an article by F. N. Cole, 'Linear Functions of a Complex Variable,' Annals of Math., Ser. 1, Vol. 5 (1890), p. 121.