Is There a General Formula for this Partial Fraction Function?

[EngWiPy]

Hello,

Is there any general formula for the partial fraction of the following function:

$$\frac{1}{(ax_1+1)(ax_2+1)\cdots (ax_L+1)}$$

I can work for L=3, but it get involved for larger L!

Thanks in advance

 

[coelho]

Ok... from what I've understood, you want write

$$\frac{1}{(ax_1+1)(ax_2+1)\cdots(ax_L+1)}$$

as

$$\frac{c_1}{ax_1+1}+\frac{c_2}{ax_2+1}+\cdots+\frac{c_L}{ax_L+1}$$

where the c's are constants. My question is, what is the variable the c's must be independent of? a or x?

 

[EngWiPy]

Ok... from what I've understood, you want write

$$\frac{1}{(ax_1+1)(ax_2+1)\cdots(ax_L+1)}$$

as

$$\frac{c_1}{ax_1+1}+\frac{c_2}{ax_2+1}+\cdots+\frac{c_L}{ax_L+1}$$

where the c's are constants. My question is, what is the variable the c's must be independent of? a or x?

a is a constant, and x's are the variables.

 

[mathman]

Ok... from what I've understood, you want write

$$\frac{1}{(ax_1+1)(ax_2+1)\cdots(ax_L+1)}$$

as

$$\frac{c_1}{ax_1+1}+\frac{c_2}{ax_2+1}+\cdots+\frac{c_L}{ax_L+1}$$

where the c's are constants. My question is, what is the variable the c's must be independent of? a or x?

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.

 

[EngWiPy]

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.

yes, right. a is the variable and x's are the constants. I got the general solution expression.

Thanks