Analytic Combinatorics/Cauchy's coefficient formula and Cauchy's inequality

Introduction Two important techniques we use in analytic combinatorics are Cauchy's coefficient formula and Cauchy's inequality. Cauchy's coefficient formula Contour integration Because a complex number has two components, real and imaginary, complex integration involves integrating around a curve in the two-dimensional plane. This is called contour integration. We denote this: where denotes the contour. It is not necessary to know how to compute contour integrals in order to understand the later material in this book, but it is useful to know a few properties of contour integration. One property of an analytic function is that when performing contour integration on a closed contour we can continuously deform the contour into another closed contour without changing the value of the integral (as long as in deforming the contour we do not pass through any singularities).Lang 1999, pp. 116-117. Another property, called Cauchy's theorem, states that if is analytic (i.e. has no singularities) inside and on the closed contour then . Titchmarsh 1939, pp. 75. A final property states that, if is any contour and is the same contour but going in the opposite direction, . Titchmarsh 1939, pp. 73. Cauchy's integral formula Cauchy's integral formula states:Titchmarsh 1939, pp. 80-81. where is a contour, is a point inside and is analytic on and inside the contour. Proof: Because is analytic, we can replace the integral around with a contour with centre and radius As is analytic it is also continuous. This means for any there exists a such that . We can do this by setting . Finally, as . Taylor series If is analytic inside and on a contour , the Taylor series expansion of around the point inside : Cauchy's coefficient formula Cauchy's coefficient formula states that: Proof: Cauchy's integral formula states: If you differentiate both sides with respect to times, you get: The Taylor series expansion of around : Therefore: Cauchy's inequality Theorem Theorem due to TitchmarshTitchmarsh 1939, pp. 84.. If is the radius of convergence of , for all and Proof Proof due to TitchmarshTitchmarsh 1939, pp. 84.. By Cauchy's coefficient formula: We haveTitchmarsh 1939, pp. 74.: and Therefore: Pictorially, we are estimating the contour integral by taking always at its maximum around the entire contour, shown by the green ring below. 400px Notes References Lang Serge Complex Analysis 4th Springer Science+Business Media, LLC 1999 Titchmarsh E. C. The Theory of Functions 2nd Oxford University Press 1939