The relation here pointed out between the theory of functions of a complex variable and certain problems of mathematical physics was employed by Riemann (1851-66) as a means of investigation in the former subject, and his method has been further developed by Klein (since 1881) and others.
7. Cauchy's Theorem.—Let f(s) be single valued and continuous throughout a region S, and let C be a curve lying in S. The integral of f(z) taken along C is defined much as a curvilinear integral is in the calculus. Let C be divided by the points z.= a, .. . , b into n arcs; set Then if, as n increases, the lengths of all the arcs approach 0, is— lim Z f(sx).111c=f s=0 a The following theorem, due to Cauchy (1825), is fundamental.
Cauchy's Integral Theorem. Let f(z) be single-valued and continuous throughout a finite region S, inclusive of the boundary, and let f(z) be analytic within S. Then the integral of f(z), extended over the entire boundary C of S, has the value 0: O.
Writing f(z)=u+ vi, z = x+ yi, we have + if + udy).
But each of these latter integrands is an exact differential (§ 6), since u and v are conjugate functions. Hence, by the theorem of § 6, each integral vanishes.
By means of this theorem many real definite integrals may be computed. In fact, it was through the attempt to obtain a rigorous deduc t • PIG. 2 Lion of such formulas that Cauchy was led to his theorem (memoir of 1814). We will give one example: Consider the integral of extended along the boundary of the region indi cated. Its value is O. Hence Vim +fAB+lso)fis)ds = 4).
Now let OA= R increase without litnit. The first integral thus becomes the well-Imown in tegral of the Theory of Probability: f: e_zsdx V-; 2 The second integral can readily be shown to approach the limit O. In the third, 1 r (cos 45° + i sin 45°)-=—=-- (1 + i)r.
V 2 Hence (1+1) re_oi dr_o. 1,/ 2 co Setting e—rn= cos e + i sin r' and separating reals and pure imaginaries, we obtain the eval uation of the Fresnel integrals: 1 ro ridr --41*cos r2dr= 4 2 The residuum of a function f(z) in a point a in whose neighborhood, with the exception of the point a itself, the function is single-valued and analytic, is defined as 1 r 27:Tri cud's, the path of integration C being a closed path surrounding the point a and containing.no other
singularity of the function. If f(!) is single valued and analytic in a finite region S except at a finite number of points lying within the region, then the sum of the residua of f(z) at these points is given by 1 r Fri) dlis)di' where C refers to the boundary of S.
8. Development into a Taylor's
In 1831 Cauthy extended Tayloes Theorem to functions of a complex variable and at the same time discovered a simple test for the range of values of the variable for which the series con verges. Let f(z) be single-valued and analytic in a region S, and let a be an interior point of S. Let K be the largest circle that can be drawn in S with a as centre so as to include in its interior no point of the boundary of S. Then, for all points z within this drcle, f(z) can be represented by the series f(z)--=c• +ci(z—a)-f-
fig)dt f(n)(a) where cm — 4dt— apt+
n! and I en
Example. Let f(z)= (1-2 pz-1- — where p is a real constant numerically less than 1. Here either value of the function is con tinuous and analytic except where the radicand vanishes: 1-2 /4+10=0- , iV 1—p2, i.e., at two points on the unit circle about z =O. Hence we can develop either value of the radical into a TaYlor's series: 1 —Po+ Po+Po2+ • • • ,V 1-2ps+e where the coefficients Pn depend on IA alone, and the series will converge for all values of s within the unit circle.
The proof of the theorem is based on Cauchy's Integral Formula, by means of which the value of the function f(z) at any interior point of S can be expressed solely in terms of the value of f(z) along the boundary, C, of S: I f Mre .f(z)-= — 2113 c 1—z This formula is deduced directly from Cauchy's Integral Theorem, and is analogous to the theorem in the Theory of the Potential by which the value of a harmonic function u at any point within a region is expressed in terms of its value U(s) on the boundary, and of a function peculiar to the shape of the region, namely, the Green's Function G: f u8G ds.