Cauchy integral also for b< e^(1/e)?

[Kouznetsov]

How did you evaluate Taylor series coefficients with Cauchy integral?

I have algorithm \( A \) that returns at least 14 decimal digits of function \( f \).
I express \( f(z) \approx \frac{1}{2\pi i} \oint \frac{ A(t) \mathrm{d}t}{t-z} \).
I differentiate this expression with respect to \( z \) so many times I need.

Usually, I use the circular contour of integration, centered at the point of expansion,
with radius slightly smaller than distance to the closest singularity.
This allows to work with good precision, close to the limit due to the rounding errors of the complex double variables.