Complex Analysis/Complex Functions/Complex Derivatives

Complex differentiability Let us now define what complex differentiability is. 0 100% #FFFFFF Definition 2.3.1: Let , let be a function and let . is called complex differentiable at if and only if there exists a such that: Example 2.3.2 The function is nowhere complex differentiable. Proof Let be arbitrary. Assume that is complex differentiable at , i.e. that exists. We choose Due to lemma 2.2.3, which is applicable since of course is open, we have: But a contradiction. The Cauchy–Riemann equations We can define a natural bijective function from to as follows: In fact, is a vector space isomorphism between and . The inverse of is given by 0 100% #FFFFFF Theorem and definitions 2.3.3: Let be open, let be a function and let . If is complex differentiable at , then the functions are well-defined, differentiable at and satisfy the equations These equations are called the Cauchy-Riemann equations. Proof 1. We prove well-definedness of . Let . We apply the inverse function on both sides to obtain: where the last equality holds since is bijective (for any bijective we have if ; see exercise 1). 3. We prove differentiability of and and the Cauchy-Riemann equations. We define Then we have: From these equations follows the existence of , since for example exists due to lemma 2.2.3. The proof for and the existence of we leave for exercise 2. Holomorphic functions 0 100% #FFFFFF Definitions 2.3.4: Let and let be a function. We call holomorphic if and only if for all , is differentiable at . In this case, the function is called the complex derivative of . We write for the set of holomorphic functions defined on . Exercises Let be sets such that , and let be a bijective function. Prove that . Let be open, let be a function and let . Prove that if is complex differentiable at , then and exist and satisfy the equation . Next