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.: Analytic functions

Note: Analytic functions are also known as holomorphic functions.

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

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

2.4.2.: The Cauchy-Riemann Equations

The definition of analyticity of a function 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 $z_0=x_0+i{y}_0$. 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 $\mathrm{\Delta }{z}_0(=\mathrm{\Delta }{x}_0+i\mathrm{\Delta }{y}_0)$ 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:

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial v}{\partial x}=\frac{-\partial u}{\partial y}$. These are known as the Cauchy-Riemann Equations, and leads us to an important theorem.

Theorem: Let a function f(z) = u(x,y)+iv(x,y) be defined on an open set G containing a point, $z_0$. If the first partials of u and 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, $\varphi$ 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)

Next