Difference between revisions of "Formal derivative"
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The derivative of a polynomial, rational function or formal power series, which can be defined purely algebraically (without using the concept of a limit transition), and makes sense for any coefficient ring. For a polynomial | The derivative of a polynomial, rational function or formal power series, which can be defined purely algebraically (without using the concept of a limit transition), and makes sense for any coefficient ring. For a polynomial | ||
| − | + | $$ | |
| + | F ( X) = \ | ||
| + | \sum _ {i = 0 } ^ { n } | ||
| + | a _ {i} X ^ {i} | ||
| + | $$ | ||
| + | |||
| + | (or a power series $ A ( X) = \sum _ {i = 0 } ^ \infty b _ {i} X ^ {i} $) | ||
| + | the formal derivative $ F ^ { \prime } ( X) $ | ||
| + | is defined as $ \sum _ {i = 1 } ^ {n} ia _ {i} X ^ {i - 1 } $( | ||
| + | or $ \sum _ {i = 1 } ^ \infty ib _ {i} X ^ {i - 1 } $, | ||
| + | respectively), and for a rational function $ f ( X) = P ( X)/Q ( X) $ | ||
| + | it is the rational function | ||
| − | + | $$ | |
| + | f ^ { \prime } ( X) = \ | ||
| − | + | \frac{P ^ { \prime } ( X) Q ( X) - Q ^ \prime ( X) P ( X) }{[ Q ( X)] ^ {2} } | |
| + | . | ||
| + | $$ | ||
Formal derivatives of higher order and formal partial derivatives of functions of several variables are defined similarly. | Formal derivatives of higher order and formal partial derivatives of functions of several variables are defined similarly. | ||
| − | A number of properties of the ordinary derivative remain valid for the formal derivative. Thus, if | + | A number of properties of the ordinary derivative remain valid for the formal derivative. Thus, if $ F ^ { \prime } ( X) = 0 $, |
| + | then $ F ( X) $ | ||
| + | is a constant in the coefficient field (in the case of characteristic 0) and is equal to $ G ( X ^ {p} ) $( | ||
| + | in the case of characteristic $ p $). | ||
| + | If $ x _ {0} $ | ||
| + | is a root of multiplicity $ k $ | ||
| + | of a polynomial, then $ x _ {0} $ | ||
| + | is a root of multiplicity $ k - 1 $ | ||
| + | of the derivative. | ||
Latest revision as of 19:39, 5 June 2020
The derivative of a polynomial, rational function or formal power series, which can be defined purely algebraically (without using the concept of a limit transition), and makes sense for any coefficient ring. For a polynomial
$$ F ( X) = \ \sum _ {i = 0 } ^ { n } a _ {i} X ^ {i} $$
(or a power series $ A ( X) = \sum _ {i = 0 } ^ \infty b _ {i} X ^ {i} $) the formal derivative $ F ^ { \prime } ( X) $ is defined as $ \sum _ {i = 1 } ^ {n} ia _ {i} X ^ {i - 1 } $( or $ \sum _ {i = 1 } ^ \infty ib _ {i} X ^ {i - 1 } $, respectively), and for a rational function $ f ( X) = P ( X)/Q ( X) $ it is the rational function
$$ f ^ { \prime } ( X) = \ \frac{P ^ { \prime } ( X) Q ( X) - Q ^ \prime ( X) P ( X) }{[ Q ( X)] ^ {2} } . $$
Formal derivatives of higher order and formal partial derivatives of functions of several variables are defined similarly.
A number of properties of the ordinary derivative remain valid for the formal derivative. Thus, if $ F ^ { \prime } ( X) = 0 $, then $ F ( X) $ is a constant in the coefficient field (in the case of characteristic 0) and is equal to $ G ( X ^ {p} ) $( in the case of characteristic $ p $). If $ x _ {0} $ is a root of multiplicity $ k $ of a polynomial, then $ x _ {0} $ is a root of multiplicity $ k - 1 $ of the derivative.
Formal derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_derivative&oldid=46954