How can we prove f(x,y) is differentiable using induction?

[raphael3d]

Consider the real function f(x,y)=xy(x²+y²)^-N,in the respective cases N = 2,1, and 1/2. Show that in each case the function is differentiable (C^$$\omega$$) with respect to x, for any fixed y-value.

whats the strategy for proving C^$$\omega$$-differentiability here? i have to show with induction that f is still continuos after n+1 derivatives with respect to x, or am i wrong?

seems like, one has to look after a pattern in the continued derivatives of f, for write down the (n+1)nth derivative.

on the other hand, one might have the differential quotient in mind, taking the limit, meeting a boundary condition(in the complex domain the radius of convergence), but i could only think of the 1th derivative here.

 

[Eynstone]

It'd be easier to work with polar coordinates, where f takes the form
(1/2)sin 2theta /r^(N-2).

 

[Office_Shredder]

$$C^{\omega}$$ is the set of analytic functions. Not only do all its derivatives need to exist (proving that the n+1st derivative is continuous isn't enough, you need that it's differentiable since otherwise you don't have the next derivative existing) but it also has a power series expansion

On the other hand, if you prove that you have a power series expansion, then as long as it converges uniformly you know you can differentiate term by term so the differentiability portion is solved immediately. So the hard part most likely is just finding the power series

 

[Hurkyl]

If a(x) and b(x) are analytic functions of x, then so is a(x)+b(x).