Calculating Residium of Complex Integral |z|=1

In summary, the conversation discusses an integral with no upper bound, representing a path on the complex plane. The objective is to calculate a residue and two methods are suggested: expanding \exp (1/z) as a power series and using a change of variables w = 1/z.
  • #1
nhrock3
415
0
[tex]\int_{|z|=1}^{nothing } \frac{1}{z}e^{\frac{1}{z}}[/tex]
in this integral there is no upper bound
its around |z|=1

there are no poles here
only singular significant
what to do here
when calclating the residium
??
 
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  • #2
I believe, and don't trust me on this, that it's asking you to calculate a path integral around the unit circle on the complex plane. There's no "upper bound" because the integral is describing a path, not just a starting point and an ending point. Incidentally, I believe that starting and ending points are the same.
 
  • #3
Start by expanding [tex]\exp (1/z)[/tex] as a power series, multiply by 1/z and look for the z^{-1} term. That will be your residue.

Mat
 
  • #4
A change of variables [itex]w = 1/z[/itex] will also do the trick.
 

1. What is the formula for calculating the residuum of a complex integral with |z|=1?

The formula for calculating the residuum of a complex integral with |z|=1 is Res(f(z),1) = 2πi * f(1), where f(z) is the function being integrated.

2. How do you determine the value of f(1) in the residuum formula?

The value of f(1) in the residuum formula can be determined by substituting z=1 into the function being integrated. This will give you a complex number as the result.

3. Can the residuum of a complex integral with |z|=1 be negative?

No, the residuum of a complex integral with |z|=1 cannot be negative. It is always a complex number with a positive real part and a zero imaginary part.

4. How is the residuum of a complex integral with |z|=1 related to the Cauchy Residue Theorem?

The Cauchy Residue Theorem states that the residuum of a complex integral with |z|=1 is equal to the sum of the residues at all the poles enclosed by the contour of integration. In other words, the residuum of the integral is the sum of the residues at all the singularities within the unit circle.

5. Is it necessary to use the residuum formula when calculating a complex integral with |z|=1?

Yes, the residuum formula is necessary when calculating a complex integral with |z|=1. It allows you to evaluate the integral without having to evaluate the function at every point along the unit circle, which can be a tedious and time-consuming process.

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