Domb Sykes plot Hinch

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Kraaiennest

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Estimating the radius of convergence of a power series using the Domb–Sykes plot. As an example, the function

${\displaystyle {\begin{aligned}f(\varepsilon )&={\frac {\varepsilon \,\left(1+\varepsilon ^{3}\right)}{\sqrt {1+2\varepsilon }}}=\sum _{n=1}^{\infty }c_{n}\,\varepsilon ^{n}\\&\approx \varepsilon -\varepsilon ^{2}+{\tfrac {3}{2}}\,\varepsilon ^{3}-{\tfrac {3}{2}}\,\varepsilon ^{4}+{\tfrac {27}{8}}\,\varepsilon ^{5}-{\tfrac {51}{8}}\,\varepsilon ^{6}+{\tfrac {191}{16}}\,\varepsilon ^{7}-{\tfrac {359}{16}}\,\varepsilon ^{8}\end{aligned}}}$

is analysed.
On the left, (a) is a straightforward plot of the ratio of the power-series coefficients ${\displaystyle c_{n-1}/c_{n}}$ as a function of index ${\displaystyle n}$; on the right, (b) is the Domb–Sykes plot of ${\displaystyle c_{n}/c_{n-1}}$ as a function of ${\displaystyle 1/n}$.
The green-line asymptote in the Domb–Sykes plot intercepts the vertical axis at −2 and has a slope +1. Consequently, in this example ${\displaystyle f(\varepsilon )}$ has a singularity at ${\displaystyle \varepsilon _{_{0}}=-{\tfrac {1}{2}},}$ and slope parameter ${\displaystyle \alpha =-{\tfrac {1}{2}}:}$

${\displaystyle f(\varepsilon )\sim \left(\varepsilon -\varepsilon _{_{0}}\right)^{\alpha }}$

near the singularity. And

${\displaystyle \lim _{n\to \infty }{\frac {c_{n}}{c_{n-1}}}={\frac {1}{\varepsilon _{_{0}}}}-{\frac {1+\alpha }{\varepsilon _{_{0}}}}\,{\frac {1}{n}}.}$

So the limiting behaviour for large ${\displaystyle n}$ in the Domb–Sykes plot is a straight line, intercepting the vertical axis ${\displaystyle (1/n\to {0})}$ at ${\displaystyle 1/{\varepsilon _{_{0}}}}$ thus providing the radius of convergence ${\displaystyle |\varepsilon _{_{0}}|;}$ and having a slope ${\displaystyle -(1+\alpha )/\varepsilon _{_{0}}}$ from which subsequently ${\displaystyle \alpha }$ can be derived.

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CC BY-SA 4.0

Credit:

Vlastní dílo; using the example on page 146 (Fig. 8.1) in: E.J. Hinch (1991) "Perturbation Methods", Cambridge Texts in Applied Mathematics, Vol. 6, Cambridge University Press, ISBN 0521378974.

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