Partial Fractions: Simplifying Square Root Fractions

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There is no general method for converting fractions involving square roots, like (1+x)/(1-x)^(1/2), into simpler fractions akin to partial fractions for rational functions. Such expressions yield irrational values for certain rational inputs, making them incompatible with the concept of partial fractions. A potential approach is to use a substitution, such as x = cosθ, to simplify the square root. Alternatively, one can derive a Taylor series for the function, which is convergent within specific intervals where the function is infinitely differentiable. This discussion highlights the limitations of applying traditional partial fraction techniques to square root fractions.
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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 ?
 
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hi phymatter! :smile:

not that i know of, except that sometimes it may help to substitute x = cosθ, so that √(1 - x) = (√2)sin(θ/2) :wink:
 
phymatter said:
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).
 
Thank You , Jarle and tiny-tim for your help :smile: !
 

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