Sunday, 20 October 2013

integration - Evaluate the integral intexp(fracz23z3)




evaluate the integral exp(z23z3) the unit circle oriented counterclockwise.



I think that we use the following theorem:



Suppose that f is holomorphic in an open set containing a circle C and its interiror, except for poles at the points z1,...,zN inside C. Then Cf(z)dz=2πiNk=1Res(f,zk)



we have only pole z0=0 for f(z)=exp(z23z3) and



Res(f,0)=12d2dz2limz0z3f(z)




Is there another way to evaluate this integral? Thanks.


Answer



The residue method you suggest is fine. But you may not calculate it by the stated formula. The problem is that there is an essential singularity at the origin.



Hint: Several ways to go, but you may e.g. write:
exp(z23z3)=exp(1z)exp(3z3)


Do a Laurent series expansion (if you know about those?) and pick up the coefficient to 1/z. [it is in fact quite simple]


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