2 The Cauchy–Riemann equations

After working through this section, you should be able to:

• find the partial derivatives of a function from to
• use the Cauchy–Riemann equations to show that a function is not differentiable at a given point
• use the Cauchy–Riemann equations to show that a function, such as the exponential function, is differentiable at a given point, and to find the derivative.

This section is challenging, so you may find that you do not appreciate some of the details on a first reading. Most importantly, you should try to understand the definitions, strategies and theorems, and apply them in the examples and exercises.