Complex Analysis/Cauchy's theorem for star-shaped domains, Cauchy's integral formula, Montel's theorem

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In the last section, we learned about contour integrals. A fundamental theorem of complex analysis concerns contour integrals, and this is Cauchy's theorem, namely that if is holomorphic, and the domain of definition of has somehow the right shape, then

for any contour which is closed, that is, (the closed contours look a bit like a loop). For this theorem to hold, surprisingly, the shape of the domain of definition is supremely important; for some it does hold, for some it doesn't. In this chapter, we will prove that the theorem holds for certain which are so-called star-shaped domains. Later on in the book, we will see that it even holds for a larger class of domains, namely the simply connected ones, which will require advanced tools which we will build up along the course of this book.

Star-shaped domains[edit | edit source]

Definition 5.1:

A set is called star-shaped if and only if there exists a such that for all other , the line connecting and lies completely in ; this line can be written down in set notation for instance as follows:


which is why the condition of star-shapedness may be phrased in precise mathematical terms as follows:


Primitives on star-shaped domains[edit | edit source]

The basis for the following considerations (and thus for almost every theorem of the remainder of the book, except for some stuff that has to do with cycles) is the following technical lemma.

Lemma 5.2 (Goursat, Pringsheim variant):

Let be holomorphic, and let be a triangle contained within . Then


where by we also denote the contour that arises when traversing the triangle sides successively, as indicated in the following picture:


Corollary 5.3:

Let be holomorphic, where is star-shaped. Then has a primitive (which we shall call ) on , and it is given by



is the straight line from to .

Cauchy's theorem on star-shaped domains[edit | edit source]

Theorem 5.4:

Let be holomorphic, where is a star-shaped domain. Then for any closed contour whose image is contained within

Cauchy's integral formula[edit | edit source]

Another important theorem by Cauchy, called Cauchy's integral formula, is almost as fundamental as Cauchy's integral theorem. We begin with the following lemma.

Lemma 5.5

Montel's theorem[edit | edit source]

Theorem 2.3 (Arzelà–Ascoli):

Let be a sequence of functions defined on an interval which is

  • equicontinuous (that is, for any there exists such that ) and
  • uniformly bounded (that is, there exists such that ).

Then contains a uniformly convergent subsequence.


Let be an enumeration of the set . The set is bounded, and hence has a convergent subsequence due to the Heine–Borel theorem. Now the sequence also has a convergent subsequence , and successively we may define in that way.

Set for all . We claim that the sequence is uniformly convergent. Indeed, let be arbitrary and let such that .

Let be sufficiently large that if we order ascendingly, the maximum difference between successive elements is less than (possible since is dense in ).

Let be sufficiently large that for all and .

Set , and let . Let be arbitrary. Choose such that (possible due to the choice of ). Due to the choice of , the choice of and the triangle inequality we get


Hence, we have a Cauchy sequence, which converges due to the completeness of .