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C. Results from Analytic Function Theory

C.2 Independence of Path

Consider functions of two independent variables, and . (The reader can think of as the real axis and as the imaginary axis.)

Let and be two functions of and , continuous in some domain . Say we have a curve in , described by the parametric equations

 (C.2.1)

We can then define the following line integrals along the path from point to point inside .

 (C.2.2) (C.2.3)

Definition C.1  The line integral is said to be independent of the path in if, for every pair of points and in , the value of the integral is independent of the path followed from to .

We then have the following result.

Theorem C.1  If is independent of the path in , then there exists a function in such that

 (C.2.4)

hold throughout . Conversely, if a function can be found such that (C.2.4) hold, then is independent of the path.

Proof

Suppose that the integral is independent of the path in . Then, choose a point in and let be defined as follows

 (C.2.5)

where the integral is taken on an arbitrary path in joining and . Because the integral is independent of the path, the integral does indeed depend only on and defines the function . It remains to establish (C.2.4).

For a particular in , choose so that and so that the line segment from to in is as shown in Figure C.1. Because of independence of the path,

 (C.2.6)

We think of and as being fixed while may vary along the horizontal line segment. Thus is being considered as function of . The first integral on the right-hand side of (C.2.6) is then independent of .

Hence, for fixed , we can write

 (C.2.7)

The fundamental theorem of Calculus now gives

 (C.2.8)

A similar argument shows that

 (C.2.9)

Conversely, let (C.2.4) hold for some . Then, with as a parameter,

 (C.2.10) (C.2.11) (C.2.12)

Theorem C.2  If the integral is independent of the path in , then

 (C.2.13)

on every closed path in . Conversely if (C.2.13) holds for every simple closed path in , then is independent of the path in .

Proof

Suppose that the integral is independent of the path. Let be a simple closed path in , and divide into arcs and as in Figure C.2.

 (C.2.14) (C.2.15)

The converse result is established by reversing the above argument.

Theorem C.3  If and have continuous partial derivatives in and is independent of the path in , then

 (C.2.16)

Proof

By Theorem C.1, there exists a function such that (C.2.4) holds. Equation (C.2.16) follows by partial differentiation.

Actually, we will be particularly interested in the converse to Theorem C.3. However, this holds under slightly more restrictive assumptions, namely a simply connected domain.

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