# 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 ${\displaystyle f(z)}$ is holomorphic on an open set ${\displaystyle G}$ if it has a derivative at every point in ${\displaystyle G}$ .

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 ${\displaystyle f(z)=u(x,y)+v(x,y)i}$ is differentiable at ${\displaystyle z_{0}=x_{0}+y_{0}i}$ . Then the limit

${\displaystyle \lim _{\Delta z\to 0}{\frac {f(z_{0}+\Delta z)-f(z_{0})}{\Delta z}}}$

can be determined by letting ${\displaystyle \Delta z_{0}(=\Delta x_{0}+\Delta y_{0}i)}$ approach zero from any direction in ${\displaystyle \mathbb {C} }$ .

If it approaches horizontally, we have ${\displaystyle f'(z_{0})={\frac {\partial u}{\partial x}}(x_{0},y_{0})+i{\frac {\partial v}{\partial x}}(x_{0},y_{0})}$ . Similarly, if it approaches vertically, we have ${\displaystyle f'(z_{0})={\frac {\partial v}{\partial y}}(x_{0},y_{0})-i{\frac {\partial u}{\partial y}}(x_{0},y_{0})}$ . By equating the real and imaginary parts of these two equations, we arrive at:

${\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},{\frac {\partial v}{\partial x}}=-{\frac {\partial u}{\partial y}}}$

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

Theorem: Let a function ${\displaystyle f(z)=u(x,y)+v(x,y)i}$ be defined on an open set ${\displaystyle G}$ containing a point, ${\displaystyle z_{0}}$ . If the first partials of ${\displaystyle u,v}$ exist in ${\displaystyle G}$ and are continuous at ${\displaystyle z_{0}}$ and satisfy the Cauchy-Riemann equations, then f is differentiable at ${\displaystyle z_{0}}$ . Furthermore, if the above conditions are satisfied, ${\displaystyle f}$ is analytic in ${\displaystyle G}$ . (proof).

2.4.3 Harmonic Functions

Now we move to Harmonic functions. Recall the Laplace equation, ${\displaystyle \nabla ^{2}(\phi ):={\frac {\partial ^{2}(\phi )}{\partial x^{2}}}+{\frac {\partial ^{2}(\phi )}{\partial y^{2}}}=0}$

Definition: A real valued function ${\displaystyle \phi (x,y)}$ is harmonic in a domain ${\displaystyle D}$ if all of its second partials are continuous in ${\displaystyle D}$ and if at each point in ${\displaystyle D}$ , ${\displaystyle \phi }$ is analytic in a domain ${\displaystyle D}$ , then both ${\displaystyle u(x,y),v(x,y)}$ are harmonic in ${\displaystyle D}$ . (proof)