Derivative of an Analytic Function

The derivative of a complex function is given by the obvious transcription of the definition used for real functions  

In the definition of a real derivative, such as f/dx or df/dy, there is only one way for Ax or Ay to approach zero. For Az = Ax + iAy in the complex plane, there are, however, an infinite number of ways to approach Az = 0. For an analytic function, these should all give the same result for the derivative. 

Equating the real and imaginary parts of Eqs. (13.21) and (13.22), we again arrive at the Cauchy-Riemann equations (13.10). 

All the familiar formulas for derivatives remain valid for complex variables, for example, d(z )/dz = nz and so forth. 

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Since the derivatives of an analytic function are still analytic functions, we can differentiate (25) to obtain ... 

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