# chain rule in complex analysis

Consider the function f : C → R given by f(z) = |z|2. For the usual real derivative, there are several rules such as the product rule, the chain rule, the quotient rule and the inverse rule. Worked example: Derivative of ∜(x³+4x²+7) using the chain rule. Having inspired from this discussion, I want to share my understanding of the subject and eventually present a chain rule for complex derivatives. c FW Math 321, 2012/12/11 Elements of Complex Calculus 1 Basics of Series and Complex Numbers 1.1 Algebra of Complex numbers ... all integer n6= 1. De nition 1.1 (Chain Complex). 3.2 Cauchy’s theorem Worked example: Derivative of sec(3π/2-x) using the chain rule. (3) (Chain Rule) d dz f(g(z)) = f0(g(z))g0(z) whenever all the terms make sense. So much for similarity. This line passes through the point . Two days ago in Julia Lab, Jarrett, Spencer, Alan and I discussed the best ways of expressing derivatives for automatic differentiation in complex-valued programs. Since z = To see the diﬀerence of complex derivatives and the derivatives of functions of two real variables we look at the following example. The chain rule is a method for finding the derivative of composite functions, or functions that are made by combining one or more functions.An example of one of these types of functions is \(f(x) = (1 + x)^2\) which is formed by taking the function \(1+x\) and plugging it into the function \(x^2\). Practice: Chain rule capstone. Suppose is differentiable at , and is differentiable at , and that is a limit point of . All the usual rules of di erentiation: product rule, quotient rule, chain rule,..., still apply for complex di erentiation … 10.13 Theorem (Chain Rule.) The chain rule gives us that the derivative of h is . Then the composition is differentiable at , and It is then not possible to differentiate them directly as we do with simple functions.In this topic, we shall discuss the differentiation of such composite functions using the Chain Rule. Example 2. Contents Preface 6 1 Algebraic properties of complex numbers 8 2 Topological properties of C 18 3 Di erentiation 26 4 Path integrals 38 5 Power series 43 Complex Analysis Mario Bonk Course notes for Math 246A and 246B University of California, Los Angeles Fall 2011 and Winter 2012. Thus, the slope of the line tangent to the graph of h at x=0 is . Click HERE to return to the list of problems. Next lesson. This is the currently selected item. For example, if = a+ biis a complex number, then applying the chain rule to the analytic function f(z) = ez and z(t) = t= at+ (bt)i, we see that d dt e t= e t: 3. Sort by: Top Voted. Fortunately, these carry over verbatim to the complex derivative, and even the proofs remain the same (although … Let be complex functions, and let . Using the point-slope form of a line, an equation of this tangent line is or . Derivative rules review. Proving the chain rule. Chain rule capstone. Sometimes complex looking functions can be greatly simplified by expressing them as a composition of two or more different functions. A chain complex is a set of objects fC ngin a category like vector spaces, abelian groups, R-mod, and graded R-mod, with d n: C n!C n 1 maps such that the kernel of d n is Z n, the n-cycles of C, the image of d n+1 is B n, the n-boundaries of C and H n(C) = Z n=B nis the kernel Implicit differentiation. This is sometimes called the chain rule for analytic functions.