# A "Closed-Form" Solution to the Geometric Goat Problem

Posted 1 year ago
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Posted 1 year ago
 As an addendum to Paul's notes:The Luck-Stevens formula is useful if you are unwilling or unable to compute function derivatives. This, however, comes at the expense of having to compute a ratio of two integrals. $$z_0=\oint_\gamma \frac{z\, h^\prime(z)}{h(z)}\mathrm dz,$$on the other hand, only requires a single integral, but one needs to compute derivatives. If the function whose root is being sought is easily differentiated, this might be acceptable.Using Paul's examples above, here are the Delves-Lyness equivalents of some of his examples: (* first root of Bessel function BesselJ[0, u], diamond contour *) NIntegrate[u (-BesselJ[1, u])/(BesselJ[0, u]), {u, 1, 2 - I, 3, 2 + I, 1}, WorkingPrecision -> 20]/(2 Pi I) 2.4048255576957727688 (* second root of Bessel function BesselJ[0, u], circular contour *) With[{h = 5, r = 1}, Re[(r/(2 Pi)) NIntegrate[# (-BesselJ[1, #])/(BesselJ[0, #]) &[h + r Exp[I t]] Exp[I t], {t, 0, 2 Pi}, Method -> {"Trapezoidal", "SymbolicProcessing" -> 0}, WorkingPrecision -> 20]]] 5.5200781102863106496 (We use the trapezoidal rule here, as it is very efficient for numerically evaluating such integrands; see e.g. this and this) (* third root of Bessel function BesselJ[0, u] via FFT *) With[{h = 9, r = 1, n = 32}, Chop[r Fourier[Table[N[# (-BesselJ[1, #])/(BesselJ[0, #]) &[h + r Exp[I t]], 20], {t, 0, 2 Pi - Pi/n, Pi/n}], FourierParameters -> {-1, 1}]][]] 8.653727912911012217 As for the original "goat problem" solution, we have the following: Re[NIntegrate[# (# Sin[#])/(Sin[#] - # Cos[#] - Pi/2) &[Pi/2 + Pi Exp[I t]/4] Exp[I t], {t, 0, 2 Pi}, Method -> {"Trapezoidal", "SymbolicProcessing" -> 0}, WorkingPrecision -> 20]/8] 1.9056957293098838949  The survey paper of Austin, Kravanja, and Trefethen is of interest if you wish to delve further into these matters.
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