Namespaces
Variants
Actions

Difference between revisions of "Ostrogradski method"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A method for isolating the algebraic part in indefinite integrals of rational functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706202.png" /> be polynomials with real coefficients, let the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706203.png" /> be less than the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706204.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706205.png" /> is a proper fraction, let
+
<!--
 +
o0706201.png
 +
$#A+1 = 32 n = 0
 +
$#C+1 = 32 : ~/encyclopedia/old_files/data/O070/O.0700620 Ostrogradski method
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706207.png" /></td> </tr></table>
+
A method for isolating the algebraic part in indefinite integrals of rational functions. Let  $  P( x) $
 +
and  $  Q( x) $
 +
be polynomials with real coefficients, let the degree of  $  P( x) $
 +
be less than the degree of  $  Q( x) $,
 +
so that  $  P( x)/Q( x) $
 +
is a proper fraction, let
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706208.png" /> are real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o0706209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062011.png" /> are natural numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062013.png" />, and let
+
$$ \tag{1 }
 +
Q( x)  = ( x - a _ {1} ) ^ {\alpha _ {1} } \dots ( x - a _ {r} ) ^ {\alpha _ {r} } \times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
\times
 +
( x  ^ {2} + p _ {1} x + q _ {1} ) ^ {\beta _ {1} } \dots
 +
( x  ^ {2} + p _ {s} x + q _ {s} ) ^ {\beta _ {s} } ,
 +
$$
  
Then real polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062016.png" /> exist, the degrees of which are respectively less than the degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062018.png" /> of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062020.png" />, such that
+
where  $  a _ {i} , p _ {j} , q _ {j} $
 +
are real numbers,  $  ( p _ {j}  ^ {2} /4)- q _ {j} < 0 $,  
 +
$  \alpha _ {i} $
 +
and  $  \beta _ {j} $
 +
are natural numbers,  $  i = 1 \dots r $,
 +
$  j = 1 \dots s $,
 +
and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{2 }
 +
\left . \begin{array}{c}
  
It is important that the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062023.png" /> can be found without knowing the decomposition (1) of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062024.png" /> into irreducible factors: The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062025.png" /> is the greatest common divisor of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062026.png" /> and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062027.png" /> and can be obtained using the [[Euclidean algorithm|Euclidean algorithm]], while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062028.png" />. The coefficients of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062030.png" /> can be calculated using the method of indefinite coefficients (cf. [[Undetermined coefficients, method of|Undetermined coefficients, method of]]). The Ostrogradski method reduces the problem of the integration of a real rational fraction to the integration of a rational fraction whose denominator has only simple roots; the integral of such a fraction is expressed through transcendental functions: logarithms and arctangents. Consequently, the rational fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062031.png" /> in formula (3) is the algebraic part of the indefinite integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070620/o07062032.png" />.
+
Q _ {1} ( x)  =  ( x - a _ {1} ) ^ {\alpha _ {1} - 1 } \dots ( x - a _ {r} ) ^ {\alpha _ {r} - 1 } \times
 +
\\
 +
 
 +
\times ( x  ^ {2} + p _ {1} x + q _ {1} ) ^ {\beta _ {1} - 1 } \dots ( x  ^ {2} + p _ {s} x + q _ {s} ) ^ {\beta _ {s} - 1 } ,
 +
\\
 +
 
 +
Q _ {2} ( x)  =  ( x - a _ {1} ) \dots ( x - a _ {r} ) \times
 +
\\
 +
 
 +
\times ( x  ^ {2} + p _ {1} x + q _ {1} ) \dots ( x  ^ {2} + p _ {s} x + q _ {s} ).
 +
 +
\end{array}
 +
\right \}
 +
$$
 +
 
 +
Then real polynomials  $  P _ {1} ( x) $
 +
and  $  P _ {2} ( x) $
 +
exist, the degrees of which are respectively less than the degrees  $  n _ {1} $
 +
and  $  n _ {2} = r + 2s $
 +
of the polynomials  $  Q _ {1} ( x) $
 +
and  $  Q _ {2} ( x) $,
 +
such that
 +
 
 +
$$ \tag{3 }
 +
\int\limits P(
 +
\frac{x)}{Q(}
 +
x)  dx  = \
 +
 
 +
\frac{P _ {1} ( x) }{Q _ {1} ( x) }
 +
+ \int\limits
 +
\frac{P _ {2} ( x) }{Q _ {2} ( x) }
 +
  dx.
 +
$$
 +
 
 +
It is important that the polynomials $  Q _ {1} ( x) $
 +
and $  Q _ {2} ( x) $
 +
can be found without knowing the decomposition (1) of the polynomial $  Q( x) $
 +
into irreducible factors: The polynomial $  Q _ {1} ( x) $
 +
is the greatest common divisor of the polynomial $  Q( x) $
 +
and its derivative $  Q  ^  \prime  ( x) $
 +
and can be obtained using the [[Euclidean algorithm|Euclidean algorithm]], while $  Q _ {2} ( x) = Q( x)/Q _ {1} ( x) $.  
 +
The coefficients of the polynomials $  P _ {1} ( x) $
 +
and $  P _ {2} ( x) $
 +
can be calculated using the method of indefinite coefficients (cf. [[Undetermined coefficients, method of|Undetermined coefficients, method of]]). The Ostrogradski method reduces the problem of the integration of a real rational fraction to the integration of a rational fraction whose denominator has only simple roots; the integral of such a fraction is expressed through transcendental functions: logarithms and arctangents. Consequently, the rational fraction $  P _ {1} ( x)/Q _ {1} ( x) $
 +
in formula (3) is the algebraic part of the indefinite integral $  \int P( x)/Q( x)  dx $.
  
 
The method was first published in 1845 by M.V. Ostrogradski (see ).
 
The method was first published in 1845 by M.V. Ostrogradski (see ).

Latest revision as of 08:04, 6 June 2020


A method for isolating the algebraic part in indefinite integrals of rational functions. Let $ P( x) $ and $ Q( x) $ be polynomials with real coefficients, let the degree of $ P( x) $ be less than the degree of $ Q( x) $, so that $ P( x)/Q( x) $ is a proper fraction, let

$$ \tag{1 } Q( x) = ( x - a _ {1} ) ^ {\alpha _ {1} } \dots ( x - a _ {r} ) ^ {\alpha _ {r} } \times $$

$$ \times ( x ^ {2} + p _ {1} x + q _ {1} ) ^ {\beta _ {1} } \dots ( x ^ {2} + p _ {s} x + q _ {s} ) ^ {\beta _ {s} } , $$

where $ a _ {i} , p _ {j} , q _ {j} $ are real numbers, $ ( p _ {j} ^ {2} /4)- q _ {j} < 0 $, $ \alpha _ {i} $ and $ \beta _ {j} $ are natural numbers, $ i = 1 \dots r $, $ j = 1 \dots s $, and let

$$ \tag{2 } \left . \begin{array}{c} Q _ {1} ( x) = ( x - a _ {1} ) ^ {\alpha _ {1} - 1 } \dots ( x - a _ {r} ) ^ {\alpha _ {r} - 1 } \times \\ \times ( x ^ {2} + p _ {1} x + q _ {1} ) ^ {\beta _ {1} - 1 } \dots ( x ^ {2} + p _ {s} x + q _ {s} ) ^ {\beta _ {s} - 1 } , \\ Q _ {2} ( x) = ( x - a _ {1} ) \dots ( x - a _ {r} ) \times \\ \times ( x ^ {2} + p _ {1} x + q _ {1} ) \dots ( x ^ {2} + p _ {s} x + q _ {s} ). \end{array} \right \} $$

Then real polynomials $ P _ {1} ( x) $ and $ P _ {2} ( x) $ exist, the degrees of which are respectively less than the degrees $ n _ {1} $ and $ n _ {2} = r + 2s $ of the polynomials $ Q _ {1} ( x) $ and $ Q _ {2} ( x) $, such that

$$ \tag{3 } \int\limits P( \frac{x)}{Q(} x) dx = \ \frac{P _ {1} ( x) }{Q _ {1} ( x) } + \int\limits \frac{P _ {2} ( x) }{Q _ {2} ( x) } dx. $$

It is important that the polynomials $ Q _ {1} ( x) $ and $ Q _ {2} ( x) $ can be found without knowing the decomposition (1) of the polynomial $ Q( x) $ into irreducible factors: The polynomial $ Q _ {1} ( x) $ is the greatest common divisor of the polynomial $ Q( x) $ and its derivative $ Q ^ \prime ( x) $ and can be obtained using the Euclidean algorithm, while $ Q _ {2} ( x) = Q( x)/Q _ {1} ( x) $. The coefficients of the polynomials $ P _ {1} ( x) $ and $ P _ {2} ( x) $ can be calculated using the method of indefinite coefficients (cf. Undetermined coefficients, method of). The Ostrogradski method reduces the problem of the integration of a real rational fraction to the integration of a rational fraction whose denominator has only simple roots; the integral of such a fraction is expressed through transcendental functions: logarithms and arctangents. Consequently, the rational fraction $ P _ {1} ( x)/Q _ {1} ( x) $ in formula (3) is the algebraic part of the indefinite integral $ \int P( x)/Q( x) dx $.

The method was first published in 1845 by M.V. Ostrogradski (see ).

References

[1a] M.V. Ostrogradski, Bull. Sci. Acad. Sci. St. Petersburg , 4 : 10–11 (1845) pp. 145–167
[1b] M.V. Ostrogradski, Bull. Sci. Acad. Sci. St. Petersburg , 4 : 18–19 (1845) pp. 286–300
How to Cite This Entry:
Ostrogradski method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ostrogradski_method&oldid=18118
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article