So far we have supposed that \(y = \phi(x)\) is a purely real function of \(x\). If \(y\) is a complex function \(\phi(x) + i\psi(x)\), then we define the derivative of \(y\) as being \(\phi'(x) + i\psi'(x)\). The reader will have no difficulty in seeing that Theorems (1)–(5) above retain their validity when \(\phi(x)\) is complex. Theorems (6) and (7) have also analogues for complex functions, but these depend upon the general notion of a ‘function of a complex variable’, a notion which we have encountered at present only in a few particular cases.
| $\leftarrow$ 113. General rules for differentiation | Main Page | 115. The notation of the differential calculus $\rightarrow$ |