A Padé approximation is the use of a ratio of polynomials to approximate a function. This can be seen as a generalization of the Taylor series which can better account for singularities in the function.

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### Continuation of functions beyond natural boundaries

The article Continuation of functions beyond natural boundaries by John L. Gammel states I am particularly interested in the convergence of the $[N/N+1]$ Padé approximants beyond the natural ...
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### What is the Pade approximation of the matrix logarithm?

I would like to use the Pade approximation in my numerical procedure and I would like to use it to approximate the logarithm of a matrix. However, I couldn't find the correct expression for it in the ...
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### Pade approximation of $\frac{1-e^{-x}I_0(x)}{x}$

I need to expand $\frac{1-e^{-x}I_0(x)}{x}$ in Pade approximation. The answer should be $\frac{1}{1+x}$. But I'm not sure how to reach the answer. Here $I_0(x)$ is modified Bessel function of order 0 ...
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### Implementation help for Extended Euclidean Algorithm

I'm not sure if this question is entirely on-topic here, please notify if not. I feel it is more a math related problem, than a programming problem. Following the advice in this answer I'm trying to ...
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### What algorithm is used in Matlab's pade function?

I try to find out for quite some time now, how Matlab implements the calculation of Padé Approximants using its symbolic pade function. (the code of is buried in a ...
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### roots of Padé approximating polynomials to the exponential function

I need to obtain (numerically) the roots of the denominator in the Padé approximation to the exponential function $e^{-x}$, in Python. I can calculate its coefficients in closed form (see below). But ...
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### Can this approximation be made more formal?

When considering oscillating systems in physics, we end up with some response function like $$F(\omega) = \frac{\omega^2}{(\omega_0^2 - \omega^2)^2 + (\omega/\tau)^2},$$ where $\omega_0$ and $\tau$ ...
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I want to calculate the following integral in Mathematica 10 $$f(x) = \frac2\pi\int_0^\infty\exp\left(-\kappa_2\frac{t^2}{2!}+\kappa_4\frac{t^4}{4!}-\dots\right)\cos\left(\kappa_1t-\kappa_3\frac{t^3}{... 6answers 190 views ### Approximating \log x with roots The following is a surprisingly good (and simple!) approximation for \log x+1 in the region (-1,1):$$\log (x+1) \approx \frac{x}{\sqrt{x+1}} Three questions: Is there a good reason why this ...
I'm trying to understand how one would understand the error of a given Padé approximation for a function. For instance, the $[2,1]$ approximant for $\log(1+x)$ is $\frac{x(6+x)}{6+4x}$. Is there a ...
On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...