The Harmonic Functions[Definitions, Theorem And Proof ]

The Harmonic Functions

The Harmonic functions is a Function That satisfy the Laplace equation.In order words the
harmonic function is defined as:

"A complex valued function $f(z)$ in a domain $D$ is said to be harmonic if all its
second partial functions are continuous in the domain $D$ and if at each point of $D$,
$f$ is satisfied.this is represented by the equation
$$\begin{equation*} \frac{\mathfrak{d}^2{u}}{\mathfrak{d}x^2}+\frac{\mathfrak{d}^2{u}}{\mathfrak{d}y^2}=0 \end{equation*}$$

similarly

$$\begin{equation*} \frac{\mathfrak{d}^2{v}}{\mathfrak{d}x^2}+\frac{\mathfrak{d}^2{v}}{\mathfrak{d}y^2}=0 \end{equation*}$$

therefore $u$ and $v$ are harmonic functions.
Now lets take a look at a theorem and proof of the harmonic function.

theorem:
Let $f(z)=u+iv$ be an analytic function, then $u$ and $v$ are both harmonic
functions.

Proof:since $f(z)=u+iv$ is defined as an anlytic function, then by the conditions of
analytic functions we have
$$\begin{equation} \frac{\mathfrak{d}u}{\mathfrak{d}x}=\frac{\mathfrak{d}v}{\mathfrak{d}y} \end{equation}$$ and
$$\begin{equation} \frac{\mathfrak{d}u}{\mathfrak{d}y}=\frac{\mathfrak{-d}v}{\mathfrak{d}x} \end{equation}$$
which is the cauchy-riemann equation.
Now differentiate (1) with respect to x.
$$\begin{equation} \frac{\mathfrak{d}^2{u}}{\mathfrak{d}x^2}=\frac{\mathfrak{d}^2v}{\mathfrak{d}x\mathfrak{d}y} \end{equation}$$
differentiate (2) with respect to x too.
$$\begin{equation} \frac{\mathfrak{d}^2{u}}{\mathfrak{d}y^2}=\frac{\mathfrak{-d}^2v}{\mathfrak{d}x\mathfrak{d}y} \end{equation}$$
Add (3) and (4)
$$\begin{equation*} \frac{\mathfrak{d}^2{u}}{\mathfrak{d}x^2}+\frac{\mathfrak{d}^2{u}}{\mathfrak{d}y^2}=\frac{\mathfrak{d}^2v}{\mathfrak{d}x\mathfrak{d}y}-\frac{\mathfrak{d}^2v}{\mathfrak{d}x\mathfrak{d}y} \end{equation*}$$
hence
$$\begin{equation*} \frac{\mathfrak{d}^2{u}}{\mathfrak{d}x^2}+\frac{\mathfrak{d}^2{u}}{\mathfrak{d}y^2}=0 \end{equation*}$$ similarly
$$\begin{equation*} \frac{\mathfrak{d}^2{v}}{\mathfrak{d}x^2}+\frac{\mathfrak{d}^2{v}}{\mathfrak{d}y^2}=0 \end{equation*}$$
Therefore both $u$ and $v$ are harmonic function,where $u$ and $v$ are called conjugate
harmonic functions if $u+iv$ is also analytic function.