Proving that a "composition" is harmonic

In summary, PeroK is having trouble with a harmonic function and needs help from an expert to solve the problem.
  • #1
kmitza
17
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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?
 
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  • #2
kmitza said:
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.
 
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  • #3
kmitza said:
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).
 
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FAQ: Proving that a "composition" is harmonic

1. What is a composition in relation to music?

A composition in music refers to a piece of music that has been created by combining different musical elements such as melody, harmony, and rhythm. It is typically written by a composer and can be performed by musicians.

2. What does it mean for a composition to be harmonic?

A composition is considered harmonic when all the individual musical elements, such as chords and melodies, work together in a pleasing and balanced way. This creates a sense of unity and coherence in the music.

3. How can one prove that a composition is harmonic?

To prove that a composition is harmonic, one must analyze the musical elements and their relationships within the piece. This can include studying the chord progressions, melody lines, and overall structure of the composition to determine if they are balanced and cohesive.

4. Are there any specific techniques or theories used to prove a composition is harmonic?

Yes, there are several techniques and theories that can be used to prove a composition is harmonic. These include harmonic analysis, which involves breaking down the music into its individual parts and examining how they work together, and the study of music theory, which provides a framework for understanding how different musical elements interact.

5. Can a composition be considered harmonic if it breaks traditional harmonic rules?

Yes, a composition can still be considered harmonic even if it breaks traditional harmonic rules. This is because music is constantly evolving and new techniques and styles are constantly being developed. As long as the elements within the composition work together in a pleasing and balanced way, it can be considered harmonic.

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