The conditions for Harmonics functions in complex analysis

Question

Let $\sigma(u, v)$ and $\gamma(u, v)$ be harmonic functions on a region $D$ in $\Bbb C$.

What are the conditions on $\sigma$ and $\gamma$ such that $\sigma \gamma$ is harmonic on $D$.

And I want to prove that these conditions hold when $\sigma + i\gamma$ is analytic on $D$. Or I want to Show that when $\sigma \gamma$ is harmonic, it does not necessarily imply that $\sigma + i\gamma$ is analytic on D.

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I have just been studying the topics. Therefore, please explain this at the high level knowledge. And please be explanatory:) and also, Honestly I dont have enough correct idea. So I cannot wrote. Please help me solving the question. Thank you a lot.

Answer 1

If $w = f(z) = \sigma +i\gamma$ is holomorphic where $\sigma$ and $\gamma$ are (real-valued) harmonic functions on a region $D$ in $\mathbb{C}$, then $w^2$ will also be holomorphic on $D$. But $w^2 = \sigma^2 - \gamma^2 +2i\sigma\gamma$, and the real and imaginary parts of any holomorpic function are certainly harmonic. Thus we can conclude in this case that $\sigma\gamma$ is a (real) harmonic function in $D$.

Additionally, (changing your notation slightly), to show that $\sigma\gamma$ being harmonic does not necessarily imply that $\sigma + i\gamma$ is holomorphic in $D$, consider $\sigma(x,y) = x$ and $\gamma(x,y)=-y$. Both of these are harmonic, and $\sigma\gamma = -xy$ is also harmonic, as can be seen by finding $\Delta \sigma, \Delta\gamma, \Delta(\sigma\gamma)$. However $\sigma+i\gamma = x-iy$ which is a standard example of a non-analytic function.

For the conditions for $\sigma\gamma$ to be harmonic, given that $\sigma$ and $\gamma$ are harmonic without using Maple, you can do the following (I am simply expanding the answer by user64494 here!):

$$\frac{\partial}{\partial x}(\sigma\gamma) = \frac{\partial\sigma}{\partial x}\gamma+\sigma\frac{\partial\gamma}{\partial x}$$

so that

$$\frac{\partial^2}{\partial x^2}(\sigma\gamma) = \frac{\partial^2\sigma}{\partial x^2}\gamma+\sigma\frac{\partial^2\gamma}{\partial x2} + 2\frac{\partial\sigma}{\partial x}\frac{\partial\gamma}{\partial x}$$

Now do the same for $y$ and add the two results and use the fact that $\sigma$ and $\gamma$ are harmonic to get the result that $\sigma\gamma$ will be harmonic if

$$\frac{\partial\sigma}{\partial x}\frac{\partial\gamma}{\partial x} + \frac{\partial\sigma}{\partial y}\frac{\partial\gamma}{\partial y} = 0$$

Answer 2

The Maple code $$ with(VectorCalculus): local(gamma); simplify(Laplacian(sigma(u, v)*gamma(u, v), [u, v])=0, {Laplacian(gamma(u, v), u, v]) = 0, Laplacian(sigma(u, v), [u, v]) = 0}); $$ outputs $$2\, \left( {\frac {\partial }{\partial u}}\sigma \left( u,v \right) \right) {\frac {\partial }{\partial u}}{\it \gamma} \left( u,v \right) +2\, \left( {\frac {\partial }{\partial v}}\sigma \left( u,v \right) \right) {\frac {\partial }{\partial v}}{\it \gamma} \left( u, v \right)=0. $$ We find the Laplacian of $\sigma\gamma$, using the rules of the differentiation and taking into account that the Laplacians $\Delta \sigma=0$ and $\Delta \gamma=0.$ Maple helps to avoid a routine work to this end. The Cauchy-Riemann conditions $$ \frac {\partial\sigma}{ \partial u}=\frac {\partial\gamma}{ \partial v},\frac {\partial\sigma}{ \partial v}= -\frac {\partial\gamma}{ \partial u}$$ clearly imply that condition, but not vice versa. Therefore, if $\sigma+ i\gamma$ is analytical on $D$, then $\sigma\gamma$ is harmonic in $D$, but not vice versa.

PS. The Maple code $$local(gamma); diff(sigma(u, v)*gamma(u, v), u, u)+diff(sigma(u, v)*gamma(u, v), v, v) $$outputs $$ \left( {\frac {\partial ^{2}}{\partial {u}^{2}}}\sigma \left( u,v \right) \right) {\it \gamma} \left( u,v \right) +2\, \left( {\frac { \partial }{\partial u}}\sigma \left( u,v \right) \right) {\frac { \partial }{\partial u}}{\it \gamma} \left( u,v \right) +\sigma \left( u ,v \right) {\frac {\partial ^{2}}{\partial {u}^{2}}}{\it \gamma} \left( u,v \right) + \left( {\frac {\partial ^{2}}{\partial {v}^{2}}} \sigma \left( u,v \right) \right) {\it \gamma} \left( u,v \right) +2\, \left( {\frac {\partial }{\partial v}}\sigma \left( u,v \right) \right) {\frac {\partial }{\partial v}}{\it \gamma} \left( u,v \right) +\sigma \left( u,v \right) {\frac {\partial ^{2}}{\partial {v }^{2}}}{\it \gamma} \left( u,v \right) ,$$ i.e. the Laplacian of $\sigma\gamma.$ It remains to apply $\Delta \sigma=0$ and $\Delta \gamma =0$.