I'm doing this example of Cauchy principle value
∫∞0dxx3+1=2π3√3
After some steps i got,
∫[0,R]+CRdzz3+1=2πi(B1) where B1=Resz=z01z3+1
also I got that |∫CRdzz3+1|→0 as R→∞
There is problem to to finding residue at z0=1+√3i2
Here i am considering the following contour:
please help me.thanks in advance.
Answer
The contour is good. Two things though:
1) You have to consider the integral along the angled line of the wedge contour. The angle of the contour was chosen to preserve the integrand. 2) Write z=ei2π/3x and get that the contour integral is
(1−ei2π/3)∫∞0dxx3+1=i2π13ei2π/3
The term on the right is the residue at the pole z=eiπ/3 times i2π. I used the fact that, if f(z)=a(z)/b(z), then the residue of a simple pole zk of f is a(zk)/b′(zk).
Note that ei2π/3−ei4π/3=i√3. The result follows.
No comments:
Post a Comment