Complex Analysis/Complex Functions/Analytic Functions

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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 z_0 = x_0 + iy_0. Then the limit,

\lim_{\Delta z\rightarrow 0} \frac {f(z_0+ \Delta z) - f(z_0)}{\Delta z}

can be determined by letting \Delta z_0 (= \Delta x_0 + i \Delta y_0) approach zero from any direction in \mathbb{C}.

If it approaches horizontally, we have 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 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:

\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, \nabla^2 (\phi) := \frac {\partial^2 (\phi)}{\partial x^2}+\frac {\partial^2 (\phi)}{\partial y^2} = 0

Definition: A real valued function, \phi (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, \phi 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)

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