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We now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute a large number of contour integrals even without knowing any explicit antiderivative of . There are many forms and variants of Cauchy’s theorem. To give one such version, we need the basic topological notion of a homotopy:

Definition 1 (Homotopy)Let be an open subset of , and let , be two curves in .

- (i) If have the same initial point and terminal point , we say that and are
homotopic with fixed endpointsin if there exists a continuous map such that and for all , and such that and for all .- (ii) If are closed (but possibly with different initial points), we say that and are
homotopic as closed curvesin if there exists a continuous map such that and for all , and such that for all .- (iii) If and are curves with the same initial point and same terminal point, we say that and are
homotopic with fixed endpoints up to reparameterisationin if there is a reparameterisation of which is homotopic with fixed endpoints in to a reparameterisation of .- (iv) If and are closed curves, we say that and are
homotopic as closed curves up to reparameterisationin if there is a reparameterisation of which is homotopic as closed curves in to a reparameterisation of .In the first two cases, the map will be referred to as a

homotopyfrom to , and we will also say that can becontinously deformed to(either with fixed endpoints, or as closed curves).

Example 2If is a convex set, that is to say that whenever and , then any two curves from one point to another are homotopic, by using the homotopyFor a similar reason, in a convex open set , any two closed curves will be homotopic to each other as closed curves.

Exercise 3Let be an open subset of .

- (i) Prove that the property of being homotopic with fixed endpoints in is an equivalence relation.
- (ii) Prove that the property of being homotopic as closed curves in is an equivalence relation.
- (iii) If are closed curves with the same initial point, show that is homotopic to as closed curves if and only if is homotopic to with fixed endpoints for some closed curve with the same initial point as or .
- (iv) Define a
pointin to be a curve of the form for some and all . Let be a closed curve in . Show that is homotopic with fixed endpoints to a point in if and only if is homotopic as a closed curve to a point in . (In either case, we will callhomotopic to a point,null-homotopic, orcontractible to a pointin .)- (v) If are curves with the same initial point and the same terminal point, show that is homotopic to with fixed endpoints in if and only if is homotopic to a point in .
- (vi) If is connected, and are any two curves in , show that there exists a continuous map such that and for all . Thus the notion of homotopy becomes rather trivial if one does not fix the endpoints or require the curve to be closed.
- (vii) Show that if is a reparameterisation of , then and are homotopic with fixed endpoints in U.
- (viii) Prove that the property of being homotopic with fixed endpoints in up to reparameterisation is an equivalence relation.
- (ix) Prove that the property of being homotopic as closed curves in up to reparameterisation is an equivalence relation.

We can then phrase Cauchy’s theorem as an assertion that contour integration on holomorphic functions is a homotopy invariant. More precisely:

Theorem 4 (Cauchy’s theorem)Let be an open subset of , and let be holomorphic.

- (i) If and are rectifiable curves that are homotopic in with fixed endpoints up to reparameterisation, then
- (ii) If and are closed rectifiable curves that are homotopic in as closed curves up to reparameterisation, then

This version of Cauchy’s theorem is particularly useful for applications, as it explicitly brings into play the powerful technique of *contour shifting*, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to either compute or integrate. This formulation of Cauchy’s theorem also highlights the close relationship between contour integrals and the algebraic topology of the complex plane (and open subsets thereof). Setting to be a point, we obtain an important special case of Cauchy’s theorem (which is in fact equivalent to the full theorem):

Corollary 5 (Cauchy’s theorem, again)Let be an open subset of , and let be holomorphic. Then for any closed rectifiable curve in that is contractible in to a point, one has .

Exercise 6Show that Theorem 4 and Corollary 5 are logically equivalent.

An important feature to note about Cauchy’s theorem is the *global* nature of its hypothesis on . The conclusion of Cauchy’s theorem only involves the values of a function on the images of the two curves . However, in order for the hypotheses of Cauchy’s theorem to apply, the function must be holomorphic not only on the images on , but on an open set that is large enough (and sufficiently free of “holes”) to support a homotopy between the two curves. This point can be emphasised through the following fundamental near-counterexample to Cauchy’s theorem:

Example 7 (Key example)Let , and let be the holomorphic function . Let be the closed unit circle contour . Direct calculation shows thatAs a consequence of this and Cauchy’s theorem, we conclude that the contour is not contractible to a point in ; note that this does not contradict Example 2 because is not convex. Thus we see that the lack of holomorphicity (or

singularity) of at the origin can be “blamed” for the non-vanishing of the integral of on the closed contour , even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of , not just the behaviour in the local neighbourhood of , has an impact on the contour integral.

One can of course rewrite this example to involve non-closed contours instead of closed ones. For instance, if we let denote the half-circle contours and , then are both contours in from to , but one haswhereas

In order for this to be consistent with Cauchy’s theorem, we conclude that and are not homotopic in (even after reparameterisation).

In the specific case of functions of the form , or more generally for some point and some that is holomorphic in some neighbourhood of , we can quantify the precise failure of Cauchy’s theorem through the Cauchy integral formula, and through the concept of a winding number. These turn out to be extremely powerful tools for understanding both the nature of holomorphic functions and the topology of open subsets of the complex plane, as we shall see in this and later notes.

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