Holomorphic Functions

What accounts for these differences betweeen real and complex differentiable functions? First, when z and f ( z ) are complex, the difference quotient (1) involves subtraction and division of complex rather than real numbers. (We discuss these operations below.) Second, the limit (1) must be independent of the direction through which z approaches zO. If z and z0 are real, z can approach z0 from two directions (left and right). In the complex plane, however, there is room for z to approach z0 from any angle. Thus the complex derivative, when it exists, contains more directional information than the real derivative.