Interesting property of Harmonic Functions（or more general) and a Novel Perspective on the Semi-Direct Product

Interesting property of Harmonic, or more general function.

Initial idea

I met this question in the tutorial on complex analysis.

If $u(x,y)$$u(x,y)$ is a harmonic function (for convenience, let $u\in C^{\mathrm{\infty }}({\mathbb{R}}^2)$$u\in C^{\infty}(\mathbb R^2)$), then $u_x-i{u}_y$$u_x-iu_y$ is harmonic as well.

To prove this statement, I consider $\mathbb{C}[{\partial }_x,{\partial }_y]$$\mathbb C[\part_x,\part_y]$, it is a subring of endomorphism of $C^{\mathrm{\infty }}({\mathbb{R}}^2)$$C^{\infty}(\mathbb R^2)$ in Category of vector space.

Since it is a commutative ring, thus

$$\begin{array}{}\text{(1)}& \mathrm{\forall }\delta \in \mathbb{C}[{\partial }_x,{\partial }_y],({\partial }_x^2+{\partial }_y^2)\delta =\delta ({\partial }_x^2+{\partial }_y^2)\end{array}$$

In particular

$$\begin{array}{}\text{(2)}& ({\partial }_x^2+{\partial }_y^2)({\partial }_x-i{\partial }_y)u=({\partial }_x-i{\partial }_y)({\partial }_x^2+{\partial }_y^2)u=0\end{array}$$

$Q.E.D.$$Q.E.D.$

In general, we could consider $\mathbb{C}[{\partial }_{x_1},...,{\partial }_{x_n}]$$\mathbb C[\part_{x_1},...,\part_{x_n}]$ as a subring of endomorphism of ${\mathbb{C}}^{\mathrm{\infty }}({\mathbb{R}}^n)$$\mathbb C^{\infty}(\mathbb R^n)$...

That tells us that if $u\in {\mathbb{C}}^{\mathrm{\infty }}({\mathbb{R}}^n),\mathcal{T}u=0,\mathcal{T},\delta \in \mathbb{C}[{\partial }_{x_1},...,{\partial }_{x_n}]$$u\in \mathbb C^{\infty}(\mathbb R^n),\mathcal Tu=0,\mathcal T,\delta\in \mathbb C[\part_{x_1},...,\part_{x_n}]$, then $\delta u$$\delta u$ belong to $\text{Ker}(\mathcal{T})$$\text{Ker}(\mathcal T)$ as well.

In particular, Let $\mathcal{T}$$\mathcal T$ be $\mathrm{\Delta }$$\Delta$, we get the harmonic version.

If we let $\mathcal{T}$$\mathcal T$ be the ${\partial }_x+i{\partial }_y$$\partial_x+i\part_y$, easy to check that Cauthy-Riemann Condition is equivalence to $({\partial }_x+i{\partial }_y)f=0$$(\part_x+i\part_y)f=0$

Thus we get the holomorphic function version.

Easy to see that in

$$\begin{array}{}\text{(3)}& \mathbb{C}[{\partial }_x,{\partial }_y],\mathrm{\Delta }=({\partial }_x+i{\partial }_y)({\partial }_x-i{\partial }_y)=({\partial }_x-i{\partial }_y)({\partial }_x+i{\partial }_y)\end{array}$$

Thus holomorphic function has to be a harmonic function, and we can see that for any harmonic function $u(x,y)$$u(x,y)$,

$$\begin{array}{}\text{(4)}& ({\partial }_x-i{\partial }_y)u=u_x-i{u}_y\end{array}$$

is an analytic function!

If you are farmilar with Cauthy-Riemann Condition, then you can observe that for an analytic function $u+iv$$u+iv$

$$\begin{array}{}\text{(5)}& u_x-i{u}_y\end{array}$$

is exactly the derivative of $u+iv$$u+iv$ !

That is why the real part of a holomorphic function is a harmonic function, and since the whole holomorphic function is harmonic, the imaginary part is harmonic as well.

Thus we find the primitive function of $u_x-i{u}_y$$u_x-iu_y$ on a simply connected domain!

i.e. extend a harmonic function to a holomorphic function on a simply connected domain.

Generalization

Indeed, we could generalize this idea to holomorphic function

$$\begin{array}{}\text{(6)}& h:D⟶\mathbb{C},D\in {\mathbb{C}}^n\text{ is an open set }\end{array}$$

But before that, we should give the definition of holomorphic function.

That is, $h$$h$ could be linearly approximate at a point $p\in D$$p\in D$.

In other words, $\mathcal{D}$$\mathcal D$ (i.e. differentiate )is a functor from the category of complex manifold with base point to ${\mathrm{Vect}}_{\mathbb{C}}$$\mathrm{Vect}_{\mathbb C}$ .

Thus each $\begin{array}{c}\frac{\partial h}{\partial z_k}\end{array}$$\frac{\part h}{\part z_k}\\$ should be complex number for all $p\in D$$p\in D$. i.e. $\mathrm{\forall }k,({\partial }_{x_k}+i{\partial }_{y_k})h=0$$\forall k,(\part_{x_k}+i\part_{y_k})h=0$

i.e. if you consider the inclusion map $i_k:\mathbb{C}\to {\mathbb{C}}^n:z_k\to (z_1,...,z_k,...,z_n)$$i_k:\mathbb C\to\mathbb C^n:z_k\to(z_1,...,z_k,...,z_n)$.

Then take the composition $\mathrm{\forall }k,h_k=h\circ i_k$$\forall k,h_k=h\circ i_k$ is holomorphic in the sense of one variable function.

Let each $z_k=x_k+i{y}_k$$z_k=x_k+iy_k$, then the Laplace operator is

$$\begin{array}{}\text{(7)}& \mathrm{\Delta }:={\partial }_{x_1}^2+{\partial }_{y_1}^2+...+{\partial }_{x_n}^2+{\partial }_{y_n}^2=({\partial }_{x_1}-i{\partial }_{y_1})({\partial }_{x_1}+i{\partial }_{y_1})+...+({\partial }_{x_n}-i{\partial }_{y_n})({\partial }_{x_n}+i{\partial }_{y_n})\end{array}$$

...

An observation

$$\begin{array}{}\text{(8)}& \begin{array}{c}\mathrm{\Delta }=det\left(\begin{array}{cc}{\partial }_x& -{\partial }_y\\ {\partial }_y& {\partial }_x\end{array}\right)\end{array}\end{array}$$

More every, if we consider the

$$\begin{array}{}\text{(9)}& \mathbb{C}[{\partial }_x,{\partial }_y]\end{array}$$

act on the space of harmonic space, the annihilator is the ideal $(\mathrm{\Delta })$$(\Delta)$.

Thus we get the coordinate ring of harmonic function.

$$\begin{array}{}\text{(10)}& \mathbb{C}[{\partial }_x,{\partial }_y]/({\partial }_x^2+{\partial }_y^2)\cong \mathbb{C}[X,Y]/[X^2+Y^2]\end{array}$$

Which is the coordinate ring of

$$\begin{array}{}\text{(11)}& X^2+Y^2=0\end{array}$$

i.e.

$$\begin{array}{}\text{(12)}& (z,\pm iz)\in {\mathbb{C}}^2\end{array}$$

But that is

$$\begin{array}{}\text{(13)}& \left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)\end{array}$$

or

$$\begin{array}{}\text{(14)}& \left(\begin{array}{cc}a& b\\ b& -a\end{array}\right)\end{array}$$

If we consider the coordinte ring of homomorphic function

$$\begin{array}{}\text{(15)}& \mathbb{C}[{\partial }_x,{\partial }_y]/({\partial }_x+i\partial y)\cong \mathbb{C}[X,Y]/(X+iY)\end{array}$$

i.e.

$$\begin{array}{}\text{(16)}& (z,iz)\in {\mathbb{C}}^2\end{array}$$

That is the matrix representation of complex number.

$$\begin{array}{}\text{(17)}& \left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)\end{array}$$

Or, we could consider

$$\begin{array}{}\text{(18)}& \mathbb{C}[{\partial }_z,{\partial }_{\overline{z}}]/({\partial }_{\overline{z}})\cong \mathbb{C}[{\partial }_z]\end{array}$$

 

Another Representation of semi-direct product

In the previous essay Math Essays: Semi-direct product and linear function (wuyulanliulongblog.blogspot.com)

We see we could use $h+kx$$h+kx$ to represent the semi-direct product $H⋊K$$H\rtimes K$.

We know we have a way of embedding affine transformation into projective transformation.

That is

$$\begin{array}{}\text{(19)}& \begin{array}{c}b+Ax⟼\left(\begin{array}{cc}1& 0\\ b& A\end{array}\right)\end{array}\end{array}$$

Then

$$\begin{array}{}\text{(20)}& \left(\begin{array}{cc}1& 0\\ b& A\end{array}\right)\left(\begin{array}{cc}1& 0\\ c& B\end{array}\right)=\left(\begin{array}{cc}1& 0\\ b+Ac& AB\end{array}\right)\end{array}$$