Activity 1

Inspect carefully the Maple evaluation of the following rational function integral:

> f:='f': F:='F': z:='z':

f := z -> 1/((z^2 + 1)*(z^2 + 4)):

`f(z) ` = f(z);

> Zn := sort([solve(denom(f(z))=0, z)]):

`For  f(z) ` = f(z);

`The singularities are:`;

z1 := subs(z=Zn[1],z): z[1] = z1;

z2 := subs(z=Zn[2],z): z[2] = z2;

z3 := subs(z=Zn[3],z): z[3] = z3;

z4 := subs(z=Zn[4],z): z[4] = z4;

> print(`0 <`,Im(z[1]),`  `, Im(z1)>0, evalb(evalf(Im(z1))>0));

print(`0 <`,Im(z[2]),`  `, Im(z2)>0, evalb(evalf(Im(z2))>0));

print(`0 <`,Im(z[3]),`  `, Im(z3)>0, evalb(evalf(Im(z3))>0));

print(`0 <`,Im(z[4]),`  `, Im(z4)>0, evalb(evalf(Im(z4))>0));

> r1 := residue(f(z), z=z1): `Res[f`,z1,`] ` = r1;

r3 := residue(f(z), z=z3): `Res[f`,z3,`] ` = r3;

> `F(x)` = f(x);

val := 2*Pi*I*(r1 + r3):

print(int(F(x),x=-infinity..infinity) = val);

Activity 2

Follow a similar technique, as above, to evaluate the rational function integral:

.