Timeline for Inverse of a function defined by an integral
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Aug 30, 2012 at 19:25 | comment | added | Jacques Carette | You should ask your 'exact question' as a question here on MO, you will likely get all sorts of interesting answers. The answer would not fit in a comment. | |
| Aug 29, 2012 at 19:43 | comment | added | Sam Nead | Perhaps I was too hasty in abandoning the first order differential equation. It has some unpleasant exponents, but perhaps I can persuade it to give me a power series expansion for $g$... | |
| Aug 29, 2012 at 19:27 | comment | added | Sam Nead | To be more precise - my exact question is about inverting the incomplete beta function (mathworld.wolfram.com/IncompleteBetaFunction.html) $B(z,a,b)$ for fixed (rational) $a$ and $b$. This is motivated by wanting to find the inverse to the Schwarz-Christoffel mapping from the upper-half plane to a triangle. | |
| Aug 29, 2012 at 19:15 | comment | added | Sam Nead | I don't see how to get a linear second order ode. For my example I get something like $p(g) g′′ = q(g) (g′)^2$, where $p,q$ are low-order polynomials and $g$ is the inverse of $f$. The dependence of the later coefficients of the power series of $g$ on the first non-zero coefficient is not linear. I've been trying to use SAGE, but they have not yet implemented power series solutions to ODEs (or if they have, I haven't found it). | |
| Aug 12, 2012 at 16:53 | comment | added | Jacques Carette | The first order ODE is non-linear, but for your pattern the second order ODE is linear. Getting a power series from that is 'classical'. The easiest way is to use the recurrence relation for the coefficients (use any decent CAS to get that). The dependence on the first non-zero coefficient should be 'clear' from the recurrence. A lot of this information seems to be only found in (much) older texts on ODEs which focus on computation rather than on existence proofs; look at Forsyth's for example. | |
| Aug 10, 2012 at 5:21 | comment | added | Sam Nead | Question - how can I use the non-linear, second order ODE for $g = f^{−1}$ to get the first, say, 100 terms of the power series expansion for $g$ about zero? Just messing around with an example, it seems that the later coefficients depend, in a highly non-linear fashion, on the first non-zero coefficient... Is this standard? Is there a canonical reference? Best, | |
| Sep 7, 2010 at 9:24 | vote | accept | Mermoz | ||
| Sep 6, 2010 at 17:04 | history | answered | Jacques Carette | CC BY-SA 2.5 |