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.

''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.''
 * 2.4.1 Holomorphic functions

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)

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 $$z_0=x_0+y_0i$$. Then the limit
 * 2.4.2 The Cauchy-Riemann Equations
 * $$\lim_{\Delta z\to0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}$$

can be determined by letting $$\Delta z_0(=\Delta x_0+\Delta y_0i)$$ approach zero from any direction in $$\Complex$$.

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

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).

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

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 $$D$$, $$\phi$$ is analytic in a domain $$D$$ , then both $$u(x,y),v(x,y)$$ are harmonic in $$D$$. (proof)

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