A Padé approximant is a rational function that provides an approximation to a given function, often yielding better accuracy than a Taylor series by matching the function's value and derivatives at a particular point. It is particularly useful in handling functions with poles or other singularities, where Taylor series may not converge well.

Padé Approximant

Padé approximant

A rational function is a ratio of two polynomials, where the denominator is not zero. It is defined for all real numbers except those that make the denominator zero, which are called the function's vertical asymptotes or points of discontinuity.

rational function

approximation to a given function

better accuracy than a Taylor series

matching the function's value and derivatives

particular point

functions with poles

Singularities refer to points or regions in space-time where gravitational forces cause matter to have infinite density and zero volume, leading to undefined physics. They are most commonly associated with the centers of black holes and the initial state of the universe at the Big Bang.

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