The Cauchy Theory–A Fundamental Theorem
4.1. Line integrals and differential forms
We recall the definitions of the one-sided derivative for functions of a real variable.
Definition 4.1. Let [a, b] be a closed (finite) interval onRand let g : [a, b]→R be a function. As in calculus, for a≤c < b, we define
(D+g)(c) = lim
h→0 h>0
g(c+h)−g(c)
h ,
the right-sided derivative ofg at c(whenever this limit exists).
Similarly, for a < c ≤b, we define (D−g)(c) = lim
h→0 h<0
g(c+h)−g(c)
h ,
the left-sided derivative of g at c (whenever this limit exists), and for a < c < b, we define
g(c) = (Dg)(c) = (D+g)(c) = (D−g)(c),
the derivative of g at c, whenever the last two limits exist and are equal.
We say that g is differentiable on [a, b] if g exists on (a, b) and (D+g)(a) and (D−g)(b) exist (these defineg(a) andg(b), respectively);
g is called continuously differentiable on [a, b] if g is continuous on
[a, b]; in which case, we will write g ∈ C1R([a, b]) or, equivalently, g ∈ C0R([a, b]) =CR([a, b]).
Remark 4.2. The concepts we have been discussing are from real analysis; if f : [a, b] → C is a complex-valued function, they apply to the two real-valued functions of a real variable given by u = f and v = f, and we can set f = u +ı v. We abbreviate C1C([a, b]) and C0C([a, b])) byC1([a, b]) and C0([a, b])), respectively.
In what follows D is a domain in C.
Definition 4.3. A function γ ∈ C1([a, b]) with γ([a, b]) ⊂ D ⊆ C ∼=R2 will be called a differentiable path or curve in D, and we say that γ is parameterized by [a, b].
We will write
γ(t) = (x(t), y(t)) =z(t) =x(t) +ıy(t) for a≤t≤b.
We denote the image or range of γ by rangeγ.
The curve is called closed if γ(a) = γ(b). The closed curve γ is simple if γ is one-to-one except at the end points of the interval [a, b];
to be precise, if γ(t1) = γ(t2) for a ≤ t1 < t2 ≤ b, then a = t1 and t2 =b.
Definition4.4. LetDbe a domain inC. Adifferential (one-)form ω on D is an expression
ω =P dx+Q dy,
where P =P(x, y) and Q = Q(x, y) are continuous (complex-valued) functions on D, anddx and dy are symbols associated with the coor- dinate z =x+ıy and called the differentials of x and y, respectively.
If γ is a differentiable path in D and ω is a differential form on D, then we define the line orpath orcontour integral of ω along γ, by the formula
γ
ω = b
a
[P(x(t), y(t))x(t) +Q(x(t), y(t))y(t)]dt
= b
a
(p1(x(t), y(t))x(t) +q1(x(t), y(t))y(t))dt +ı
b a
(p2(x(t), y(t))x(t) +q2(x(t), y(t))y(t))dt, where P =p1+ı p2 and Q=q1+ı q2.
62 4. THE CAUCHY THEORY–A FUNDAMENTAL THEOREM
Remark 4.5. The above definition involves again only concepts from real analysis, even though the paths and functions involve the complex numbers.
Reparametrization: If t : [α, β] →[a, b] is one-to-one, onto, and differentiable, and γ : [a, b]→C is a differentiable path, thenγ =γ◦t is again a differentiable path, called a reparametrization of γ.
Since for any closed interval [a, b] there exists a one-to-one, onto, and differentiable map t : [0,1] →[a, b], we can always assume that a given path γ is parameterized by [0,1].
For all differential forms ω defined in a neighborhood of the range of the pathγ and all reparametrizations γ ofγ, the following equalities hold:
γ
ω =
γ
ω , if t(u)≥0 for allu∈[α, β] and
γ
ω =−
γ
ω , if t(u)≤0 for all u∈[α, β].
Note that there are no other possibilities for the sign of the deriva- tive of t.
Subdivision of interval: Let γ : [a, b] → C be a differentiable path and consider the partition of [a, b] defined by
a =t0 < t1 < . . . < tn+1 =b . (4.1) If
γj =γ|[tj,tj+1] , for j = 0, . . . , n, (4.2) then γj is a differentiable path and
γ
ω = n
j=0
γj
ω. (4.3)
For a set T contained in the domain of γ, γ|T, of course, denotes the restriction of γ toT.
Definition 4.6. Let γ : [a, b]→ C be a continuous path. We say that γ is a piecewise differentiable path (henceforth abbreviated pdp) if there exists a partition of [a, b] of the form given in (4.1) such that each path defined by (4.2) is differentiable. Then we use (4.3) to define the path integral
γω.
a b c
d
x y
(a) Picture of the curve
a b
t x
(b) The graph ofx
c d
t y
(c) The graph ofy
Figure 4.1. Three figures
Remark 4.7. The path integral is well defined (independent of the partition) and agrees with earlier definition for differentiable paths.
The verification is left as an exercise.
Remark 4.8. Three pictures in R2 are naturally associated with each path γ =x+ı y: the picture of the curve and the graphs of the functions xandy. Figure 4.1 illustrates this with a curve whose image is the rectangle with vertices (c, e),(d, e),(d, f), and (c, f).
Lemma 4.9. If D is a domain in C, then any two points in D can be joined by a piecewise differentiable path in D.
Proof. Fixζ ∈D, and let
E ={z ∈D;z can be joined toζ by a pdp in D}.
Then E is open in D,D−E is also open in D, and ζ ∈E.
Definition 4.10. Let D be a domain inC.
(1) A functionf onDisof class Cp, withp∈ Z≥0, iff has partial derivatives (with respect toxandy) up to and including order p and these are continuous on D. It is of class C∞ if it is of class Cp for allp∈Z≥0. The vector space of functions of class Cp onD is denoted by Cp(D).
(2) A differential form ω = P dx+Qdy is of class Cp if and only if P and Q are of that class.
64 4. THE CAUCHY THEORY–A FUNDAMENTAL THEOREM
(3) For a given function f, we have the (real) partial derivatives fxandfy as well as the formal (complex) partial derivativesfz
andfz introduced in Exercise 2.8, where it was also shown that for a C1-functionf =u+ı v, the Cauchy–Riemann equations hold for the pair u and v if and only if fz = 0.
The four partial derivatives just described may be regarded as directional derivatives.
Remark 4.11. At this point we recommend that all con- cepts and definitions that are formulated in terms of x and y be reformulated by the reader in terms of z and z (and vice versa).
(4) Iff is a C1-function on D, then we define df, thetotal differ- ential of f, by either of the two equivalent formulas
df =fxdx+fydy=fzdz+fzdz, where
dz =dx+ı dy and dz =dx−ı dy.
Thus in addition to the differential operatord, we have two other important differential operators ∂ and ∂ defined by
∂f =fzdz and ∂f =fzdz as well as the formula
d=∂ +∂.
We have defined the three differential operators on spaces of C1-functions. They can be also defined on spaces of C1- differential forms, and it follows from these definitions that, for example, on C2-functions the equality d2 = 0 holds. We shall not need these extended definitions.
(5) A differential form ω on an open set D is called exact if there exists a C1-function F on D (called a primitive for ω) such that
ω=dF.
If D is connected, then a primitive (if it exists) is unique up to addition of a constant.
By abuse of language we also say that a function F is a primitive for a functionf if F is a primitive for the differential ω =f dz.
(6) A differential formωonDis calledclosed if it is locally exact;
that is, if for each a∈ D, there exists a neighborhood U of a such that ω|U is exact.
4.2. The precise difference between closed and exact forms