Power, Laurent and Puiseux Series Rings

Magma contains an extensive package for formal power series. The fact that we may only work with a finite number of terms, n say, of a power series, i.e., a truncated power series, is made precise by noting that we are working in the quotient ring $R[[x]]/\langle x^{n+1} \rangle$, for some n, rather than in the full ring R[[x]]. Provided this is kept in mind, calculations with elements of a power series ring (though not field) are always precise.

Given a field K, a field of Laurent series K((x)) is regarded as the localization of the power series ring K[[x]] at the ideal $\langle 0 \rangle$. More simply, it is the field of fractions of K[[x]]. Since elements of such a field are infinite series, calculation is necessarily approximate.

A power series ring R[[x]] is regarded as the completion of the polynomial ring R[x] at the ideal $\langle 0 \rangle$.

Puiseux series with arbitrary fractional exponents are also supported (since V2.4).

• Arithmetic
• Inversion of units
• Derivative, integral
• Square root, valuation
• Exponentiation, composition, convolution, reversion
• Power series expansions of transcendental functions
• $R[[x]]/\langle x^{n+1} \rangle$ as an algebra over R