Timeline for 'Dirichlet problem' along axis for harmonic functions
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 9, 2022 at 20:25 | comment | added | Leo Moos | Thanks again - I appreciate the clarification! | |
| Nov 9, 2022 at 20:00 | comment | added | user49822 | @LeoMoos I just searched for a solution of the form $u(x,y,z) = \sum_k \left(x^2+y^2\right)^k u_k(z)$ (so that it would be rotationally symmetric along the z axis) with $u_0 = f$, by expanding the equation $\Delta u=0$ to a recurrence relation on $u_k$. Also I'm not sure if a large enough radius of convergence is a necessary condition. | |
| Nov 9, 2022 at 19:20 | comment | added | Leo Moos | Oops, forgot to say that I used the shorthand notation where $\rho = (x^2 + y^2)^{1/2}$ in the comment above. | |
| Nov 9, 2022 at 19:19 | vote | accept | Leo Moos | ||
| Nov 9, 2022 at 19:19 | comment | added | Leo Moos | That's very neat - thanks! I checked the formal calculation, and it works out for $u$ to be harmonic. To be clear, if you want $u$ to be defined for $\rho < 1$, you want the radius of convergence of the series $\sum_k f^{(k)}(z) \rho^k / k!$ to be at least $1$ for each $z \in \mathbf{R}$, right? Is it obvious that this is also necessary - that there can't be a solution if the radius of convergence were strictly smaller? Also, would you mind explaining how you got the formula? I'll admit I haven't dealt with Taylor series much (or at all) in a while, so I'm not familiar with the expression... | |
| Nov 9, 2022 at 14:17 | history | answered | user49822 | CC BY-SA 4.0 |