coelho said:Ok... from what I've understood, you want write
[tex]\frac{1}{(ax_1+1)(ax_2+1)\cdots(ax_L+1)}[/tex]
as
[tex]\frac{c_1}{ax_1+1}+\frac{c_2}{ax_2+1}+\cdots+\frac{c_L}{ax_L+1}[/tex]
where the c's are constants. My question is, what is the variable the c's must be independent of? a or x?
coelho said:Ok... from what I've understood, you want write
[tex]\frac{1}{(ax_1+1)(ax_2+1)\cdots(ax_L+1)}[/tex]
as
[tex]\frac{c_1}{ax_1+1}+\frac{c_2}{ax_2+1}+\cdots+\frac{c_L}{ax_L+1}[/tex]
where the c's are constants. My question is, what is the variable the c's must be independent of? a or x?
mathman said:Problems involving partial fractions usually have one variable and many constants. From the appearance of your expression, I assume a is the variable and the x's are constants. In that case the c's will be determined by the x's.
A partial fraction is a mathematical expression that represents a rational function as a sum of simpler fractions. It is used to simplify integration and solve algebraic equations.
To decompose a rational function into partial fractions, you need to first factor the denominator into linear and irreducible quadratic factors. Then, you set up a system of equations using the coefficients of the partial fractions and solve for each unknown coefficient.
The purpose of using partial fractions is to simplify complex rational functions into smaller, easier to manage fractions. This allows for easier integration and solving of equations.
Yes, all rational functions can be decomposed into partial fractions. However, some may result in fractions with complex coefficients.
Partial fractions are used in many areas of science and engineering, including signal processing, control systems, and fluid dynamics. They are also commonly used in solving differential equations and calculating areas under curves in calculus.