# Complex Analysis/Complex Functions/Analytic Functions

From our look at complex derivatives, we now examine the analytic functions, the Cauchy-Riemann Equations, and Harmonic Functions.

**2.4.1.: Holomorphic functions**

*Note: Holomorphic functions are sometimes referred to as analytic functions. This equivalence will be shown later, though the terms may be used interchangeably until then.*

*Definition*: A complex valued function f(z) is **holomorphic** on an open set *G* if it has a derivative at every point in *G*.

Here, holomorphicity is defined over an open set, however, differentiability could only at one point. If f(z) is holomorphic over the entire complex plane, we say that f is *entire*. As an example, all polynomial functions of z are entire. (proof)

**2.4.2.: The Cauchy-Riemann Equations**

The definition of holomorphic suggests a relationship between both the real and imaginary parts of the said function. Suppose f(z) = u(x,y)+iv(x,y) is differentiable at . Then the limit,

can be determined by letting approach zero from any direction in .

If it approaches horizontally, we have . Similarly, if it approaches vertically, we have . By equating the real and imaginary parts of these two equations, we arrive at:

and . These are known as the **Cauchy-Riemann Equations**, and leads us to an important theorem.

*Theorem*: Let a function f(z) = u(x,y)+iv(x,y) be defined on an open set *G* containing a point, . If the first partials of u and v exist in *G* and are continuous at and satisfy the Cauchy-Riemann equations, then f is differentiable at . Furthermore, if the above conditions are satisfied, f is analytic in *G*. (proof).

**2.4.3.: Harmonic Functions**

Now we move to Harmonic functions. Recall the Laplace equation,

*Definition*: A real valued function, is **harmonic** in a domain *D* if all of its second partials are continuous in *D* and if at each point in *D*, satisfies the Laplace equation.

*Theorem*: If f(z) = u(x,y)+iv(x,y) is analytic in a domain *D*, then both u(x,y) and v(x,y) are harmonic in *D*. (proof)