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