High School Partial derivative of the harmonic complex function

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The discussion focuses on the harmonic function F(z) = 1/z and its representation as F(z) = f(z) + g(ȳ), where it satisfies the condition ∂xg = i∂yg. However, an alternative representation, F(z) = ȳ/(x²+y²), does not meet the same derivative condition. The conversation also highlights that if F is analytic, then ∂F/∂ȳ = 0. This raises questions about the conditions under which these representations hold true. The implications for harmonic and analytic functions are significant in complex analysis.
Adel Makram
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For a harmonic function of a complex number ##z##, ##F(z)=\frac{1}{z}##, which can be put as ##F(z)=f(z)+g(\bar{z})##and satisfies ##\partial_xg=i\partial_yg##. But this function can also be put as ##F(z)=\frac{\bar{z}}{x^2+y^2}## which does not satisfy that derivative equation!

Sorry, I should have put this thread in homework section.
 
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Actually, if F is analytic, \frac{\partial F}{\partial\bar{z}}=0.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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