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 $f(z)=u(x,y)+v(x,y)i$ is differentiable at $z_0=x_0+y_{0}i$ . Then the limit

$\underset{\mathrm{\Delta }z\to 0}{\lim }\frac{f(z_0+\mathrm{\Delta }z)-f(z_0)}{\mathrm{\Delta }z}$

can be determined by letting $\mathrm{\Delta }{z}_0(=\mathrm{\Delta }{x}_0+\mathrm{\Delta }{y}_{0}i)$ approach zero from any direction in $ℂ$ .

If it approaches horizontally, we have $f^{\prime }(z_0)=\frac{\partial u}{\partial x}(x_0,y_0)+i\frac{\partial v}{\partial x}(x_0,y_0)$ . Similarly, if it approaches vertically, we have $f^{\prime }(z_0)=\frac{\partial v}{\partial y}(x_0,y_0)-i\frac{\partial u}{\partial y}(x_0,y_0)$ . By equating the real and imaginary parts of these two equations, we arrive at:

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

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

Theorem: Let a function $f(z)=u(x,y)+v(x,y)i$ be defined on an open set $G$ containing a point, $z_0$ . If the first partials of $u,v$ exist in $G$ and are continuous at $z_0$ and satisfy the Cauchy-Riemann equations, then f is differentiable at $z_0$ . 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, ${\mathrm{\nabla }}^2(\varphi ):=\frac{{\partial }^2(\varphi )}{\partial x^2}+\frac{{\partial }^2(\varphi )}{\partial y^2}=0$

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

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