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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])] 
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  • $\begingroup$ A more concrete demonstration might be helpful. $\endgroup$ – J. M.'s technical difficulties Jul 30 '17 at 9:05
  • $\begingroup$ What do you mean by neglect? $\endgroup$ – Andrew Jul 30 '17 at 9:06
  • $\begingroup$ @Andrew : I've added an example. $\endgroup$ – John Taylor Jul 30 '17 at 9:10
  • $\begingroup$ @Andrew : I want to set them to zero. $\endgroup$ – John Taylor Jul 30 '17 at 9:11
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If there is only one logarithm, as in your example, this will do:

g[x_,y_]=f[x, y] /. Log[A_] :> (B = A; C) /. x -> 0 /. C -> Log[B]
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  • $\begingroup$ sorry, couly you please explain briefly what does this code mean? $\endgroup$ – John Taylor Jul 30 '17 at 9:30
  • $\begingroup$ 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. $\endgroup$ – Andrew Jul 30 '17 at 9:34
  • $\begingroup$ Thank you very much! $\endgroup$ – John Taylor Jul 30 '17 at 9:36
  • $\begingroup$ 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? $\endgroup$ – John Taylor Jul 30 '17 at 9:51
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    $\begingroup$ {Log[A_] :> (B = A; C), Sqrt[A1_] :> (B1 = A1; C1)} $\endgroup$ – Andrew Jul 30 '17 at 9:59

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