1. Activity 1

Let’s see how Maple can be used to find the residue of the function at the pole z=1.

> f:='f': s:='s': z:='z':

f := z -> 2^z/((z-1)*(z^2+9)):

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

s := series(f(z), z=1, 5):

`f(z) ` = s;

Thus this residue is .

2. Activity 2

Using the result of activity 1) , continue using Maple to evaluate , where C is the circle .

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

f := z -> 2^z/((z-1)*(z^2+9)):

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

s := series(f(z), z=infinity, 5):

res := 1/5:

`Res[F,1] ` = res; 

print(int(F(z),z=C..``) = `2*Pi*I*Res[f,1])`);

print(int(F(z),z=C..``) = 2*Pi*I*res);

3. Activity 3

We can have a Cauchy Residue Theorem to integrals that have a finite number of isolated singularities (not necessarily poles) that lie inside the contour C.  This example, where the integrand has an essential singularity, shows how we can use Maple to check the value of an integral. 

> f:='f': s:='s': z:='z':

f := z -> exp(1/z):

s := series(f(z), z=infinity, 5):

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

`f(z) ` = s;

Thus the residue is .   Now we augment this with an application of the Cauchy Residue Theorem.

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

f := z -> exp(1/z):

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

s := series(f(z), z=infinity, 5):

res := 2:

`Res[F,0] ` = res; 

print(int(F(z),z=C..``) = `2*Pi*I*Res[f,0])`);

print(int(F(z),z=C..``) = 2*Pi*I*res);

4. Activity 4

Do some Maple experimentation with other functions by finding their residues and integrals.