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Zooming in

Image

Zooming in on the iterated exponential map…

The representation of the solution of:

y' =\frac{y^2}{1-xy}, y(o) =1

as a power series that has radius of convergence of \approx 1.21, whereas the cardioid above delimits the convergence of the solution representation defined by the implicit equation, y =e^{xy} approximated by the nested/iterated exponential:

y_0=1

y_n=e^{y_{n-1}x}

The zone of convergence is larger than series representation (note the graphic represents [-2,2]x[-2,2]

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Categories: Mathematica, Mathematics
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