0
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26 views

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = ...
4
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1answer
47 views

Optimal series expansion for “invertability”

Motivation: Often when dealing with physical phenomena, deviations from the model must be considered, so a variable, say $x\in[0,1]$ will be replaced by a power series expansion: $$x'\ \to \ x(1+k ...
1
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1answer
67 views

Reversion of power series

So, I just heard about this method. How does one determine the coefficients, and what is it used for? For example, given $$ y = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$ reversion would give a ...
0
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1answer
147 views

What's the limit of coefficient ratio for a reciprocating power series?

I have a question about the coefficient in the inverse of the power series. Assume $$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$ where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...