Activity 1 – Getting familiar with an example from Maple
>
f:=z^2/(z^2+1);
>
singular(f);
>
residue(f, z=I);
Activity 2. Confirm that Maple is right by justifying
what happens in activity 1) mathematically.
Activity 3 – Construct your example on a different
function to find the singularities and residue of the
poles in the upper semi plane.
Activity 4. Confirm that Maple is right by justifying
what happens in the following activity mathematically.
>
f:='f':
F:='F': z:='z':
F0 := z
-> z*sin(z)/(z^2 + 4):
f := z
-> z*exp(I*z)/(z^2 + 4):
`F(z) `
= F0(z);
`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;
>
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));
>
r1 := residue(f(z), z=z1): `Res[f`,z1,`] ` = r1;
>
`F(x)`
= F0(x);
val :=
2*Pi*Re(r1):
print(int(F(x),x=-infinity..infinity) = val);
Activity 5 –
Construct similar work to evaluate
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