# The particular approximation of a function

Suppose the function

f[x_,y_] := P[x/y]*Sqrt[1-x^2/y^2] + F[x/y]*Log[(1-Sqrt[1-x^2/y^2])/(1+Sqrt[1-x^2/y^2])]


Here $P[x,y], F[x,y]$ are polynomial of some definite degree in the argument $x/y$.

My question is, whether some manipulation exists providing "smart" manipulation with $f[x,y]$ which neglect positive degrees of $x/y$ inside $P,F$ (say, assuming it to be small), but leaving the argument of the logarithm completely unchanged?

For specific example, assume

f[x_,y_] := (a+b*x/y + c*x/y^2)*Sqrt[1-x^2/y^2] + (d + e*x/y + f*x^2/y^2)*Log[(1-Sqrt[1-x^2/y^2])/(1+Sqrt[1-x^2/y^2])]


I need the following output after the manipulation:

f[x_,y_] := a + d*Log[(1-Sqrt[1-x^2/y^2])/(1+Sqrt[1-x^2/y^2])]

• A more concrete demonstration might be helpful. – J. M.'s technical difficulties Jul 30 '17 at 9:05
• What do you mean by neglect? – Andrew Jul 30 '17 at 9:06
• @Andrew : I've added an example. – John Taylor Jul 30 '17 at 9:10
• @Andrew : I want to set them to zero. – John Taylor Jul 30 '17 at 9:11

g[x_,y_]=f[x, y] /. Log[A_] :> (B = A; C) /. x -> 0 /. C -> Log[B]

• First Log[A] changed on symbol C, then x is put to zero - it will nullify all the terms you want. Finally C is changed back to Log[A]. it is assumed here that symbols A, B, C are not used in the code. – Andrew Jul 30 '17 at 9:34
• Could you also please help me how to remain unchanged both the logarithm and the square root in the front of P[x,y] in my example? Add $$\text{Log[A_] -> (B = A; C) && Sqrt[1-x^2/y^2] -> (B1 = A1; C1)}$$ instead of $$\text{Log[A_] -> (B = A; C),}$$ right? – John Taylor Jul 30 '17 at 9:51