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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 {\int_\gamma f(z)\ dz} even without knowing any explicit antiderivative of {f}. 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 {U} be an open subset of {{\bf C}}, and let {\gamma_0: [a,b] \rightarrow U}, {\gamma_1: [a,b] \rightarrow U} be two curves in {U}.

  • (i) If {\gamma_0, \gamma_1} have the same initial point {z_0} and terminal point {z_1}, we say that {\gamma_0} and {\gamma_1} are homotopic with fixed endpoints in {U} if there exists a continuous map {\gamma: [0,1] \times [a,b] \rightarrow U} such that {\gamma(0,t) = \gamma_0(t)} and {\gamma(1,t) = \gamma_1(t)} for all {t \in [a,b]}, and such that {\gamma(s,a) = z_0} and {\gamma(s,b) = z_1} for all {s \in [0,1]}.
  • (ii) If {\gamma_0, \gamma_1} are closed (but possibly with different initial points), we say that {\gamma_0} and {\gamma_1} are homotopic as closed curves in {U} if there exists a continuous map {\gamma: [0,1] \times [a,b] \rightarrow U} such that {\gamma(0,t) = \gamma_0(t)} and {\gamma(1,t) = \gamma_1(t)} for all {t \in [a,b]}, and such that {\gamma(s,a) = \gamma(s,b)} for all {s \in [0,1]}.
  • (iii) If {\gamma_2: [c,d] \rightarrow U} and {\gamma_3: [e,f] \rightarrow U} are curves with the same initial point and same terminal point, we say that {\gamma_2} and {\gamma_3} are homotopic with fixed endpoints up to reparameterisation in {U} if there is a reparameterisation {\tilde \gamma_2: [a,b] \rightarrow U} of {\gamma_2} which is homotopic with fixed endpoints in {U} to a reparameterisation {\tilde \gamma_3: [a,b] \rightarrow U} of {\gamma_3}.
  • (iv) If {\gamma_2: [c,d] \rightarrow U} and {\gamma_3: [e,f] \rightarrow U} are closed curves, we say that {\gamma_2} and {\gamma_3} are homotopic as closed curves up to reparameterisation in {U} if there is a reparameterisation {\tilde \gamma_2: [a,b] \rightarrow U} of {\gamma_2} which is homotopic as closed curves in {U} to a reparameterisation {\tilde \gamma_3: [a,b] \rightarrow U} of {\gamma_3}.

In the first two cases, the map {\gamma} will be referred to as a homotopy from {\gamma_0} to {\gamma_1}, and we will also say that {\gamma_0} can be continously deformed to {\gamma_1} (either with fixed endpoints, or as closed curves).

Example 2 If {U} is a convex set, that is to say that {(1-s) z_0 + s z_1 \in U} whenever {z_0,z_1 \in U} and {0 \leq s \leq 1}, then any two curves {\gamma_0, \gamma_1: [0,1] \rightarrow U} from one point {z_0} to another {z_1} are homotopic, by using the homotopy

\displaystyle \gamma(s,t) := (1-s) \gamma_0(t) + s \gamma_1(t).

For a similar reason, in a convex open set {U}, any two closed curves will be homotopic to each other as closed curves.

Exercise 3 Let {U} be an open subset of {{\bf C}}.

  • (i) Prove that the property of being homotopic with fixed endpoints in {U} is an equivalence relation.
  • (ii) Prove that the property of being homotopic as closed curves in {U} is an equivalence relation.
  • (iii) If {\gamma_0, \gamma_1: [a,b] \rightarrow U} are closed curves with the same initial point, show that {\gamma_0} is homotopic to {\gamma_1} as closed curves if and only if {\gamma_0} is homotopic to {\gamma_2 + \gamma_1 + (-\gamma_2)} with fixed endpoints for some closed curve {\gamma_2} with the same initial point as {\gamma_0} or {\gamma_1}.
  • (iv) Define a point in {U} to be a curve {\gamma_1: [a,b] \rightarrow U} of the form {\gamma_1(t) = z_0} for some {z_0 \in U} and all {t \in [a,b]}. Let {\gamma_0: [a,b] \rightarrow U} be a closed curve in {U}. Show that {\gamma_0} is homotopic with fixed endpoints to a point in {U} if and only if {\gamma_0} is homotopic as a closed curve to a point in {U}. (In either case, we will call {\gamma_0} homotopic to a point, null-homotopic, or contractible to a point in {U}.)
  • (v) If {\gamma_0, \gamma_1: [a,b] \rightarrow U} are curves with the same initial point and the same terminal point, show that {\gamma_0} is homotopic to {\gamma_1} with fixed endpoints in {U} if and only if {\gamma_0 + (-\gamma_1)} is homotopic to a point in {U}.
  • (vi) If {U} is connected, and {\gamma_0, \gamma_1: [a,b] \rightarrow U} are any two curves in {U}, show that there exists a continuous map {\gamma: [0,1] \times [a,b] \rightarrow U} such that {\gamma(0,t) = \gamma_0(t)} and {\gamma(1,t) = \gamma_1(t)} for all {t \in [a,b]}. 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 {\gamma_1: [a,b] \rightarrow U} is a reparameterisation of {\gamma_0: [a,b] \rightarrow U}, then {\gamma_0} and {\gamma_1} are homotopic with fixed endpoints in U.
  • (viii) Prove that the property of being homotopic with fixed endpoints in {U} up to reparameterisation is an equivalence relation.
  • (ix) Prove that the property of being homotopic as closed curves in {U} 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 {U} be an open subset of {{\bf C}}, and let {f: U \rightarrow {\bf C}} be holomorphic.

  • (i) If {\gamma_0: [a,b] \rightarrow U} and {\gamma_1: [c,d] \rightarrow U} are rectifiable curves that are homotopic in {U} with fixed endpoints up to reparameterisation, then

    \displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.

  • (ii) If {\gamma_0: [a,b] \rightarrow U} and {\gamma_1: [c,d] \rightarrow U} are closed rectifiable curves that are homotopic in {U} as closed curves up to reparameterisation, then

    \displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.

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 {U} thereof). Setting {\gamma_1} 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 {U} be an open subset of {{\bf C}}, and let {f: U \rightarrow {\bf C}} be holomorphic. Then for any closed rectifiable curve {\gamma} in {U} that is contractible in {U} to a point, one has {\int_\gamma f(z)\ dz = 0}.

Exercise 6 Show 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 {f}. The conclusion of Cauchy’s theorem only involves the values of a function {f} on the images of the two curves {\gamma_0, \gamma_1}. However, in order for the hypotheses of Cauchy’s theorem to apply, the function {f} must be holomorphic not only on the images on {\gamma_0, \gamma_1}, but on an open set {U} 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 {U := {\bf C} \backslash \{0\}}, and let {f: U \rightarrow {\bf C}} be the holomorphic function {f(z) := \frac{1}{z}}. Let {\gamma_{0,1,\circlearrowleft}: [0,2\pi] \rightarrow {\bf C}} be the closed unit circle contour {\gamma_{0,1,\circlearrowleft}(t) := e^{it}}. Direct calculation shows that

\displaystyle \int_{\gamma_{0,1,\circlearrowleft}} f(z)\ dz = 2\pi i \neq 0.

As a consequence of this and Cauchy’s theorem, we conclude that the contour {\gamma_{0,1,\circlearrowleft}} is not contractible to a point in {U}; note that this does not contradict Example 2 because {U} is not convex. Thus we see that the lack of holomorphicity (or singularity) of {f} at the origin can be “blamed” for the non-vanishing of the integral of {f} on the closed contour {\gamma_{0,1,\circlearrowleft}}, even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of {f}, not just the behaviour in the local neighbourhood of {\gamma_{0,1,\circlearrowleft}}, 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 {\gamma_0, \gamma_1: [0,\pi] \rightarrow U} denote the half-circle contours {\gamma_0(t) := e^{it}} and {\gamma_1(t) := e^{-it}}, then {\gamma_0,\gamma_1} are both contours in {U} from {+1} to {-1}, but one has

\displaystyle \int_{\gamma_0} f(z)\ dz = +\pi i

whereas

\displaystyle \int_{\gamma_1} f(z)\ dz = -\pi i.

In order for this to be consistent with Cauchy’s theorem, we conclude that {\gamma_0} and {\gamma_1} are not homotopic in {U} (even after reparameterisation).

In the specific case of functions of the form {\frac{1}{z}}, or more generally {\frac{f(z)}{z-z_0}} for some point {z_0} and some {f} that is holomorphic in some neighbourhood of {z_0}, 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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