partial_fractions

partial_fractions - Partial Fractions Math 227 A function...

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Partial Fractions Math 227 A function that can be expressed as a ratio of polynomials is called a rational function . Any rational function may be expressed as a sum of a polynomial and another rational function whose numerator has lower degree that its denominator. This is accomplished by long division of polynomials. For example, x 3 + 2 x + 1 x 2 + x + 1 = ( x 1) + 2 x + 2 x 2 + x + 1 : The rational function on the far right is said to be in proper form since its numerator has lower degree than its denominator. For the purpose of integration, and for many engineering applications, it is useful to represent rational functions as sums of simple rational functions. The simple types that we strive for are of the following forms: A ( ax + b ) n and Ax + B ( ax 2 + bx + c ) n (0) where n is a positive integer , and where the quadratic polynomial ax 2 + bx + c is irreducible ( further factored). The condition of irreducibility is equivalent to the condition that b 2 4 ac < 0 : The method of expressing a rational function as a sum of the above types is called the method of partial fractions. Brie±y outlined, for a rational function r ( x
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This note was uploaded on 10/16/2011 for the course MATH 227 taught by Professor Cheung during the Spring '09 term at S.F. State.

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partial_fractions - Partial Fractions Math 227 A function...

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