COMPLEX ANALYSIS: Maple Worksheets, 2001
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

Complimentary software to accompany the textbook:

COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
Jones and Bartlett Publishers, Inc., 40 Tall Pine Drive, Sudbury, MA 01776
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 CHAPTER 8 RESIDUE THEORY

Section 8.7 Integrands with Branch Points

We now show how to evaluate certain improper real integrals involving the integrand z^alpha*P(x)/Q(x) .
The complex function z^alpha is multivalued, so we must first specify the branch that we will be using.

Let alpha be a real number with `0 <`*alpha < 1 . In this section we will use the branch of z^alpha corresponding with the branch of the logarithm log[0]*`(z)` (see Equation 5-20) as follows:

 

z^alpha = exp(alpha*log[0]*`(z)`) = exp(alpha*(ln*abs(z)+i*arg[0]*z)) = exp(alpha*ln*r+i*theta) = r^alpha*(cos*alpha*theta+i*sin*alpha*theta) ,

 

where z = r*exp(i*theta) and `0 <`*theta < 2*pi . Notice that this is not the traditional principal branch of z^alpha , and that as defined the function z^alpha is analytic in the domain {r*exp(i*theta)*`: r > 0, 0 <`*theta < 2*pi} .

 

 

Theorem 8.7 Let P(z) and Q(z) are polynomials of degree m and n , respectively, where m+2 <= n .

If Q(x) <> 0 for 0 < x and Q has a zero of order at most 1 at the origin and f(z) = z^alpha*P(z)/Q(z) where `0 <`*alpha < 1 , then

 

`P.V. `*Int(x^alpha*P(x)/Q(x),x = 0 .. infinity) = 2*pi*i/(1-exp(i*alpha*2*pi)) sum(`Res[P/Q, `*z[j]*`]`,j = 1 .. k) ,

 

where z[1] , z[2] , ..., z[k] are the nonzdro poles of P(z)/Q(z) .

 

Load Maple's "residue" procedure.
Make sure this is done only ONCE during a Maple session.

> readlib(residue):


 Example 8.23, Page 341. Use residues to find the P.V. for the integral
P. V. int(x^alpha/(x*(x+1)),x = -infinity .. infinity) = ... .

> a:='a': f:='f': F:='F': z:='z':
f := z -> z^a/(z*(z+1)):
`f(z) ` = f(z);

`f(z) ` = z^a/z/(z+1)

> `F(x)` = f(x);
z1 := -1:
`The singularity is at z = ` = z1;
r1 := residue(f(z), z=z1): `Res[f`,z1,`] ` = r1;
r1 := -exp(I*a*Pi): `Res[f`,z1,`] ` = r1;
val := res*2*Pi*I/(1 - exp(I*a*2*Pi)):
print(int(F(x),x=-infinity..infinity) = val);
val := Pi/( (exp(I*a*Pi)-exp(-I*a*Pi))/(2*I)):
print(int(F(x),x=-infinity..infinity) = val);
val := evalc(val):
print(int(F(x),x=-infinity..infinity) = val);

`F(x)` = x^a/x/(x+1)

`The singularity is at z = ` = -1

`Res[f`, -1, `] ` = -exp(-I*a*csgn(I*(z-1))*Pi)

`Res[f`, -1, `] ` = -exp(I*a*Pi)

int(F(x),x = -infinity .. infinity) = 2*I*res*Pi/(1...

int(F(x),x = -infinity .. infinity) = 2*I*Pi/(exp(I...

int(F(x),x = -infinity .. infinity) = Pi/sin(a*Pi)

 

End of Section 8.7.