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Author G. N. Watson begins by reviewing various propositions of Poincaré's Analysis Situs, upon which proof of the theorem's most general form depends. Subsequent chapters examine the calculus of residues, calculus optimization, the evaluation of definite integrals, and expansions in series. A historical summary concludes the text, which is supplemented by numerous challenging exercises.
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Complex Integration and Cauchy's Theorem - G. N. Watson
CHAPTER I
ANALYSIS SITUS
§ 3. Problems of Analysis situs to be discussed.—§ 4. Definitions.—§ 5. Properties of continua.—§ 6. Theorems concerning the order of a point.—§ 7. Main theorem; a regular closed curve has an interior and an exterior.—§ 8. Miscellaneous theorems; definitions of counterclockwise and orientation.
3. The object of the present chapter is to give formal analytical proofs of various theorems of which simple cases seem more or less obvious from geometrical considerations. It is convenient to summarise, for purposes of reference, the general course of the theorems which will be proved:
A simple curve is determined by the equations x = x(t), y = y(t) (where t varies from t0 to T), the functions x(t), y(t) being continuous; and the curve has no double points save (possibly) its end points; if these coincide, the curve is said to be closed. The order of a point Q with respect to a closed curve is defined to be n, where 2πn is the amount by which the angle between QP and Ox increases as P describes the curve once. It is then shewn that points in the plane, not on the curve, can be divided into two sets; points of the first set have order ±1 with respect to the curve, points of the second set have order zero; the first set is called the interior of the curve, and the second the exterior. It is shewn that every simple curve joining an interior point to an exterior point must meet the given curve, but that simple curves can be drawn, joining any two interior points (or exterior points), which have no point in common with the given curve. It is, of course, not obvious that a closed curve (defined as a curve with coincident end points) divides the plane into two regions possessing these properties.
It is then possible to distinguish the direction in which P describes the curve (viz. counterclockwise or clockwise); the criterion which determines the direction is the sign of the order of an interior point.
The investigation just summarised is that due to Ames¹; the analysis which will be given follows his memoir closely. Other proofs that a closed curve possesses an interior and an exterior have been given by Jordan², Schoenflies³, Bliss⁴, and de la Vallée Poussin⁵. It has been pointed out that Jordan’s proof is incomplete, as it assumes that the theorem is true for closed polygons; the other proofs mentioned are of less fundamental character than that of Ames.
4. D
EFINITIONS
. A simple curve joining two points z0 and Z is defined as follows:
Let⁶ x = x (t), y = y (t),
where x(t), y(t) are continuous one-valued functions of a real parameter t for all values of t such that⁷ t0 ≤ t ≤ T; the functions x(t), y(t) are such that they do not assume the same pair of values for any two different values of t in the range t0 < t < T ; and
z0 = x (t0) + iy (t0), Z = x (T) + iy (T).
Then we say that the set of points (x, y), determined by the set of values of t for which t0 ≤ t ≤ T, is a simple curve joining the points z0 and Z. If z0 = Z, the simple curve is said to be closed⁸.
To render the notation as simple as possible, if the parameter of any particular point on the curve be called t with some suffix, the complex coordinate of that point will always be called z with the same suffix; thus, if
t0 ≤ tr(n) ≤ T,
we write zr(n) = x(tr(n)) + iy (tr(n)) = xr(n) + iyr(n).
Regular curves. A simple curve is said to be regular⁹, if it can be divided into a finite number of parts, say at the points whose parameters are t1, t2, … tm where t0 ≤ t1 ≤ t2 ≤ … ≤ tm ≤ T, such that when tr − 1 ≤ t ≤ tr, the relation between x and y given by the equations x = x(t), y = y(t) is equivalent to an equation y = f(x) or else , where f or denotes a continuous one-valued function of its argument, and r takes in turn the values 1, 2, … m + 1, while tm + 1 = T.
It is easy to see that a chain of a finite number of curves, given by the equations
(where b2 = f1(a2), a3 = f2(b3), … and f1, f2, … are continuous one-valued functions of their arguments), forms a simple curve, if the chain has no double points; for we may choose a parameter t, such that
If some of the inequalities in equations (A) be reversed, it is possible to shew in the same manner that the chain forms a simple curve.
Elementary curves. Each of the two curves whose equations are (i) y = f(x), (x0 ≤ x ≤ x1) and (ii) , (y0 ≤ y ≤ yl), where f and denote one-valued continuous functions of their respective arguments, is called an elementary curve.
Primitive period. In the case of a closed simple curve let ω = T − t0; we define the functions x(t), y(t) for all real values of t by the relations
,
where n is any integer; ω is called the primitive period of the pair of functions x(t), y(t).
Angles. If z0, z1 be the complex coordinates of two distinct points P0, P1, we say that ‘P0 P1 makes an angle with the axis of x’ if satisfies both the equations¹⁰
where κ is the positive number . This pair of equations has an infinite number of solutions such that if , be any two different solutions, then is an integer, positive or negative.
Order of a point. Let a regular closed curve be denned by the equations x = x(t), y = y(t), (t0 ≤ t ≤ T) and let ω be the primitive period of x(t), y(t). Let Q be a point not on the curve and let P be the point on the curve whose parameter is t. Let be the angle which QP makes with the axis of x; since every branch of arc cos and of arc sin is a continuous function of t, it is possible to choose so that is a continuous function of t reducing to a definite number when t equals t0. The points represented by the parameters t and t + ω are the same, and hence , are two of the values of the angle which QP makes with the axis of x; therefore