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 !