Solve x=ytan(y): Analytical Technique Needed

[H_man]

I am trying to solve x=ytan(y)

x is an independent function of which I know the value. I wish to solve for y.

I can do this numerically. But I was really hoping someone out there had seen an analytical technique.

Anyone?

 

[tiny-tim]

Hi H_man!

I can do this numerically.

Sorry, numerically is the only way.

(But why do you want to solve it at all? It presumably isn't from trigonometry. )

 

[H_man]

It's to match the boundary conditions of a corrugated waveguide.

I guess I'll just have to do it the messy way :-(

Thanks tiny-tim

 

[Xyius]

You can always approximate tan(y) by a couple of terms in its Taylor Expansion ;]

 

[vela]

Do you need the actual solutions or just need to get an idea of how they behave? If it's the latter, you can analyze the equation graphically.

 

[AdrianZ]

well, maybe you could expand the Taylor series for ytan(y) and then find the inverse of this function using Lagrange inversion theorem and see if the series obtained for the inverse function has a positive radius of convergence? That's one way.

 

[H_man]

Hi Vela, I'm looking for exact solutions, I've already produced pretty graphical plots to get the general idea.

Hi AdrianZ, before your post I'd never heard of the Lagrange inversion theorem. I will try this method and let you know if it succeeds.

Thank you both for your suggestions.