f00[s,x]

f0=(1/(s-x)x (-16 Imf0 s-36 Imf1 s-20 Imf2 s+2 Imf0 (s^2)+27 Imf1 (s^2)-5 Imf2 (s^2)+4 Imf0 x+36 Imf1 x+20 Imf2 x+3 Imf0 s x-9 Imf1 s x+15 Imf2 s x-2 Imf0 (x^2)-18 Imf1 (x^2)-10 Imf2 (x^2)+(s-x) (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[4-s-x]-(s-x) (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[x])-1/(s-x)((-4+s+x) ((s-x) (-5 Imf2 (4+s-2 x)+9 Imf1 (-4+3 s+2 x))+Imf0 (2 (s^2)-2 (-2+x) x+s (-16+3 x)))+(s-x) x (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[-x]-(s-x) x (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[-4+s+x]))/(3 π (-4+s) (-4+x) x) /. Log[__]-> 0 // Simplify

f0y=f0/s /. s-> 1/y;

Asympf0=-12Pi/(4)(x(x-4))Limit[f0y,y->0] // Simplify

f11[s,x]

f1=((1/3) (-4+s) (-2 Imf0 (s-x) ((((-4+s))^2)-48 x+12 (x^2))+5 Imf2 (s-x) ((((-4+s))^2)-48 x+12 (x^2))+9 Imf1 (-3 (((-4+s))^2) s+(16+(40-23 s) s) x+12 (-4+s) (x^2)+12 (x^3)))+2 (s-x) x (-4+s+2 x) (2 Imf0 (-4+x)-5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) (Log[4-s-x]-Log[-x]))/(6 π (((-4+s))^2) (s-x) (-4+x) x) /. Log[__]->0 // Simplify

f1y=f1/s /. s-> 1/y;

Asympf1=-72Pi/(4)(x(x-4))Limit[f1y,y->0] // Simplify

f20[s,x]

f2=((-4+s) (2 Imf0 (4+s-2 x) (s-x)+9 Imf1 (s-x) (-4+3 s+2 x)+Imf2 (-5 (s^2)+s (28-3 x)+2 (-2+x) x))+(s-x) x ((2 Imf0 (-4+x)+Imf2 (-4+x)-18 Imf1 (-4+2 s+x)) Log[4-s-x]+(-2 Imf0 (-4+x)-Imf2 (-4+x)+18 Imf1 (-4+2 s+x)) Log[-x]-(2 Imf0+Imf2) (-4+x) (Log[x]-Log[-4+s+x])))/(6 π (-4+s) (s-x) (-4+x) x) /. Log[__]->0 // Simplify;

f2y=f2/s /. s-> 1/y;

Asympf2=24Pi/(4)(x(x-4))Limit[f2y,y->0] // Simplify

K0000=(1/(s-x)x (-16 Imf0 s-36 Imf1 s-20 Imf2 s+2 Imf0 (s^2)+27 Imf1 (s^2)-5 Imf2 (s^2)+4 Imf0 x+36 Imf1 x+20 Imf2 x+3 Imf0 s x-9 Imf1 s x+15 Imf2 s x-2 Imf0 (x^2)-18 Imf1 (x^2)-10 Imf2 (x^2)+(s-x) (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[4-s-x]-(s-x) (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[x])-1/(s-x)((-4+s+x) ((s-x) (-5 Imf2 (4+s-2 x)+9 Imf1 (-4+3 s+2 x))+Imf0 (2 (s^2)-2 (-2+x) x+s (-16+3 x)))+(s-x) x (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[-x]-(s-x) x (Imf0 (-4+x)+5 Imf2 (-4+x)+9 Imf1 (-4+2 s+x)) Log[-4+s+x]))/(3 π (-4+s) (-4+x) x) /. {Imf0->1, Imf1->0, Imf2->0 } // Simplify;

Series[(K0000 /. x-> 1/y),{y,0,3} ] // Normal // Expand