Partial Fractions: Simplifying Square Root Fractions

[phymatter]

is there a general way of converting fractions involving square root like (1+x)/(1-x)^1/2 to simpler fractions , like we have a method of converting rational fractions into sum of partial fractions ?

 

[tiny-tim]

hi phymatter!

not that i know of, except that sometimes it may help to substitute x = cosθ, so that √(1 - x) = (√2)sin(θ/2)

 

[disregardthat]

is there a general way of converting fractions involving square root like (1+x)/(1-x)^1/2 to simpler fractions , like we have a method of converting rational fractions into sum of partial fractions ?

You wouldn't get this expression equal to a sum of partial fractions, since a sum of partial fraction is a rational function and would therefore for rational number input yield rational values. Your function involving a square root will yield irrational values for some rational numbers, which clearly is contradictory. Allowing infinite sums you could however find the Taylor series for your expression. http://en.wikipedia.org/wiki/Taylor_series

This will however normally only be convergent for some open interval where your function is infinitely differentiable, but for your function you could find a Taylor series around every point where it is defined (since your particular function is infinitely differentiable whenever it is defined).

 

[phymatter]

Thank You , Jarle and tiny-tim for your help !

 

[CellCoree]

Yes, there is a general method for converting fractions involving square roots to simpler fractions using partial fractions. This method is known as the "method of undetermined coefficients" and involves breaking down the fraction into smaller, simpler fractions that can be easily simplified. This method is similar to the process of converting rational fractions into partial fractions, but it involves using square root terms instead of linear terms.

To simplify a fraction involving a square root, you would first factor the denominator into two distinct linear factors. For example, in the fraction (1+x)/(1-x)1/2, the denominator can be factored into (1-x)(1+x). Then, you would set up a system of equations using the factored denominator and the original fraction. The unknown coefficients in these equations represent the constants in the partial fractions.

Solving this system of equations will give you the values of the unknown coefficients, which can then be used to rewrite the original fraction as a sum of partial fractions. These partial fractions can then be simplified using algebraic techniques, resulting in a simpler form of the original fraction.

Overall, the method of undetermined coefficients provides a systematic way of simplifying fractions involving square roots, making it a useful tool for solving mathematical problems in various fields of science.