Just a quick note because I realised I hadn't ever defined this properly -- a function is {\it analytic\/} when it has a complex derivative. For the most part we are only interested in functions which are analytic at every point in some region. This can be checked by checking whether the real and imaginary parts satisfy the Cauchy-Riemann equations and have continuous first-order partial derivatives.

See Section 4 (p. 8) of Goursat.

Let me know if you have any further questions! I am working on typing up solutions to the practice problems now.

Nathan