*5*

*Complex-variable theory*

**5.1 Analytic functions**

A complex-valued function *f*(*z*) of a complex variable *z* is **differentiable** at *z* with derivative *f*′(*z*) if the limit

(5.1) |

exists as *z*′ approaches *z* from **any direction** in the complex plane. The limit must exist no matter how or from what direction *z*′ approaches *z*.

If the function *f*(*z*) is differentiable in a small disk around a point *z*_{0}, then *f*(*z*) is said to be **analytic** at *z*_{0} (and at all points inside the disk).

**Example 5.1** (Polynomials) If *f*(*z*) = *z ^{n}* for some integer

Start Free Trial

No credit card required