Complex Analysis/Complex differentiable, holomorphic, Cauchy–Riemann equations

Open sets in the complex plane Since the complex numbers are nothing but with a multiplicative structure, the notion of open sets carries over from to . Hence, we consider a set open if and only if is open as a subset of (that is, around each point there exists a ball around completely contained within ) w.r.t. Euclidean norm (or any other norm, due to norm equivalence). Complex differentiability Recall that a function is differentiable at iff the limit exists, and in this case the derivative is defined as the value of that limit. By analogy, we infer the analogous definition for functions : 0 100% #FFFFFF Definition 2.1: Let be open, and let be a function. Let . We say that is complex differentiable in iff , where this time is complex, exists, and in this case we write as an abbreviation for that limit. If the function is complex differentiable on all its domain of definition (in this case ), we say that is holomorphic. The derivative is linear in the following sense: 0 100% #FFFFFF Theorem 2.3: Since a complex number is a tuple in , a map is tantamount to a map . The complex differentiability of at a certain point implies its real differentiability, in the sense that the directional derivatives exist. In fact, we will later prove that in case of holomorphy, even continuous differentiability of (in the sense of existence of partial derivatives) will follow, and hence, we have a Jacobian matrix which equals the differential of . However, the converse is not true: If in the sense of real numbers (that is, considering as a map : All partial derivatives exist and are continuous), we don't know yet whether is holomorphic or not. The next section will make that precise. The Cauchy–Riemann equations If in the real sense, then there is a precise criterion for when is complex differentiable. This is given by the Cauchy–Riemann equations: 0 100% #FFFFFF Theorem 2.3: Let be continuously differentiable at , that is, all partial derivatives exist and are continuous, both at . Then as a function is complex differentiable in if and only if the Cauchy–Riemann equations, given by and , are satisfied, where . Proof: For the whole proof, note that if , then for . If is continuously differentiable in , then , , where means . Thus, in this case, If the Cauchy–Riemann equations are satisfied, we may replace by and by in the latter expression and obtain (whereby we also obtained another formula for the complex derivative). On the other hand, if the limit does exist, then in particular we are free to choose for real positive to get and similarly for real, positive to get and hence the Cauchy–Riemann equations. Rules for the complex derivative For the usual real derivative, there are several rules such as the product rule, the chain rule, the quotient rule and the inverse rule. Fortunately, these carry over verbatim to the complex derivative, and even the proofs remain the same (although we will repeat them for the sake of completeness). 0 100% #FFFFFF Theorem 2.4 (chain rule): Let be complex differentiable at and let be complex differentiable at (this means of course that must lay in ). Then is complex differentiable at and . Proof: Set Then by the continuity of at (which can be easily proven by multiplying the limit definition of complex differentiability by and observing that the limit is then by multiplicativity of limits). 0 100% #FFFFFF Theorem 2.5 (product rule): Let be complex differentiable at . Then the product function is complex differentiable at and . Proof: 0 100% #FFFFFF Theorem 2.6 (quotient rule): Assume that are complex differentiable at and . By continuity of , the function exists on a small ball around . It is then complex differentiable at and the derivative equals . Proof: The derivative of the function is given by ; for Hence, product and chain rule imply 0 100% #FFFFFF Theorem 2.7 (inverse rule): Assume is a bijective function that is complex differentiable at . Then is complex differentiable at and . Proof: Derivatives of polynomials 0 100% #FFFFFF Theorem 2.8: Let . Then a derivative of at any is given by . Proof: by the binomial theorem. Due to the linearity of the complex derivative, we can now compute the complex derivative of any polynomial, even with complex coefficients: . Exercises Compute the complex derivative of the polynomial .