Proving that a "composition" is harmonic

[kmitza]

I am learning some complex analysis as it is a prerequisite for the masters program that I was accepted into and I didn't take it yet during my bachelors. I am using some lecture notes in Slovene and I have run into a problem that has proven troublesome for me :

If $g: D \rightarrow \mathbb{C}$ is a harmonic function and $f: D' \rightarrow D$ is holomorphic. If $f= u +iv$ prove that
$$h = g(u(x,y),v(x,y)) $$ is harmonic.

My attempt was to just calculate the derivatives and obtain that it is zero but I got stuck in the calculation. It is entirely possible that this is an easy problem as I am not an analysis person but I would like to know if there is a simpler way of proving this than straight calculation?

 

[PeroK]

I am learning some complex analysis as it is a prerequisite for the masters program that I was accepted into and I didn't take it yet during my bachelors. I am using some lecture notes in Slovene and I have run into a problem that has proven troublesome for me :

If $g: D \rightarrow \mathbb{C}$ is a harmonic function and $f: D' \rightarrow D$ is holomorphic. If $f= u +iv$ prove that
$$h = g(u(x,y),v(x,y)) $$ is harmonic.

My attempt was to just calculate the derivatives and obtain that it is zero but I got stuck in the calculation. It is entirely possible that this is an easy problem as I am not an analysis person but I would like to know if there is a simpler way of proving this than straight calculation?

It should work out by taking the appropriate derivatives. You will need to be careful to apply the chain rule correctly.

 

[WWGD]

I am learning some complex analysis as it is a prerequisite for the masters program that I was accepted into and I didn't take it yet during my bachelors. I am using some lecture notes in Slovene and I have run into a problem that has proven troublesome for me :

If $g: D \rightarrow \mathbb{C}$ is a harmonic function and $f: D' \rightarrow D$ is holomorphic. If $f= u +iv$ prove that
$$h = g(u(x,y),v(x,y)) $$ is harmonic.

My attempt was to just calculate the derivatives and obtain that it is zero but I got stuck in the calculation. It is entirely possible that this is an easy problem as I am not an analysis person but I would like to know if there is a simpler way of proving this than straight calculation?

As PeroK says, it's all a matter of computing $$h_{xx}, h_{yy}$$ and showing $$ h_{xx}+h_{yy} =0 $$ through the chain rule (and, I believe you'll need the product rule too, to take a partial of a partial).