gtm245 Complex Analysis Introduced in the Spirit of Lipman BersGilman Jane P Kra Irwin Rodriguez R

# The verification is left as an exercise remark 48

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The verification is left as an exercise. Remark 4.8 . Three pictures in R 2 are naturally associated with each path γ = x + ı y : the picture of the curve and the graphs of the functions x and y . 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. Fix ζ D , and let E = { z D ; z can be joined to ζ 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 in C . (1) A function f on D is of class C p , with p Z 0 , if f has partial derivatives (with respect to x and y ) up to and including order p and these are continuous on D . It is of class C if it is of class C p for all p Z 0 . The vector space of functions of class C p on D is denoted by C p ( D ). (2) A differential form ω = Pdx + Qdy is of class C p if and only if P and Q are of that class.

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64 4. THE CAUCHY THEORY–A FUNDAMENTAL THEOREM (3) For a given function f , we have the (real) partial derivatives f x and f y as well as the formal (complex) partial derivatives f z and f z introduced in Exercise 2.8, where it was also shown that for a C 1 -function f = u + ı v , the Cauchy–Riemann equations hold for the pair u and v if and only if f z = 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) If f is a C 1 -function on D , then we define df , the total differ- ential of f , by either of the two equivalent formulas df = f x dx + f y dy = f z dz + f z d z, where dz = dx + ı dy and d z = dx ı dy. Thus in addition to the differential operator d , we have two other important differential operators and defined by ∂f = f z dz and ∂f = f z d z as well as the formula d = + ∂. We have defined the three differential operators on spaces of C 1 -functions. They can be also defined on spaces of C 1 - differential forms, and it follows from these definitions that, for example, on C 2 -functions the equality d 2 = 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 C 1 -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 function f if F is a primitive for the differential ω = fdz .
4.2. CLOSED AND EXACT FORMS 65 (6) A differential form ω on D is called closed 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.

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