Padé Exponential Poles
Match exponential series with rational functions and reveal approximation-created poles.
Free theory articleMatch exponential series with rational functions and reveal approximation-created poles.
Concept & interpretation
The Padé approximant uses a rational function—a ratio of two polynomials—to approximate a function. For the exponential function, matching its Taylor series produces rational functions with improved global behavior or specific desirable properties. However, introducing a denominator can create poles where the approximation becomes undefined, even though the original function is completely smooth and finite everywhere.
Experiment
We can explore the poles of these approximations using the interactive module.
- Set Approximation to
Pade 1 1. - Set Probe x to
2. Observe that the denominator becomes exactly 0. The approximation is unavailable, indicating a real pole, even though e² is perfectly well-defined. - Set Probe x to
3. The approximation gives a negative value (-5.0), whereas the true exponential function is always positive. - Change Approximation to
Pade 2 2and observe how its denominator 1 - x/2 + x²/12 has no real roots, thereby avoiding real poles.