Laplacian and harmonic functions

[cummings12332]

Homework Statement

The hyperbolic coordinate sysem onthe first quadrant in R^2 is defined by the change of variables K(u,v)=(x(u,v),y(u,v))=(ve^u,ve^(-u)) u is in R,and v>0, find all harmonic functions on the first quadrant in R^2 which are constant on all rectangular hyperbolas xy=c , c is a positive (arbitrary) constant


2. The attempt at a solution
now i had solved the laplacian of a function in hyperbolic coordinates, then i let it equal to 0 then tried to get f,but how should i apply the given condition to get the f, cause the laplacian of it is quite complecated.

 

[mighty2000]

Hello,

Thank you for your question. It seems like you are on the right track with solving the Laplacian and setting it equal to 0. This is a common approach when trying to find harmonic functions.

To apply the given condition, you can use the definition of the hyperbolic coordinate system and substitute it into the Laplacian. This will give you a new expression that you can then manipulate to get an expression for f.

Another approach you can take is to use the fact that a function is harmonic if and only if it satisfies the mean value property. This means that the value of the function at any point is equal to the average of its values on any circle centered at that point. You can use this property to find a general form of the harmonic function that satisfies the given condition.

I hope this helps. Let me know if you have any further questions or need clarification on any of the steps. Good luck with your problem!