Derivatives of complex functions. A differentiable function is continuous. Derivative formulas for the sum, product, ratio, composition of differentiable functions. Derivative of the inverse function. Cauchy-Riemann equations. Sufficient conditions for the existence of the derivative. Analytic functions. Singular points. If f is analytic and \(f'=0\) in D open and connected then f is constant in D. If f and \(\overline {f}\) are analytic in D open and connected then f is constant in D. If f is analytic in D open and connected and \(\left | f \right |\) is constant in D then f is constant in D. Derivatives of complex functions of real variables. Conformal transformations. An analytic function f is conformal at all points where \(f'\neq 0\) .