Integral of f(z) dz Around C1 & C2: Complex Math Solutions

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Homework Help Overview

The discussion revolves around evaluating the integrals of the function f(z) = z²/sin²(z) around two specified contours, C1 (the circle |z| = 1) and C2 (the circle |z - pi| = 1). The context is complex analysis, particularly focusing on contour integration and residue theory.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants explore the use of the Cauchy Integral formula and the residue theorem for evaluating the integrals. There are discussions about the nature of singularities and analyticity of the function at specific points.

Discussion Status

There are varying interpretations of the integral's value, with some participants suggesting different approaches, including the use of residues. The conversation reflects an active exploration of the problem, with no clear consensus on the correct method or outcome yet.

Contextual Notes

Participants question the implications of singularities and analyticity, particularly regarding the behavior of f(z) at z = 0. There is also mention of limits of integration and the nature of the function around the specified contours.

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Homework Statement


We know sin(z) has zeros at integral multiples of pi. Let f(z) = z2/sin2(z)
How do I find the integral of f(z) dz around C1 (C1 is the circle |z| = 1 orientated anti-clockwise) and how do I find the integral of f(z) dz around C2 (C2 is the circle |z - pi| = 1 orientated anti-clockwise).

Homework Equations





The Attempt at a Solution


Do I use the Cauchy Integral formula for these integrals.
If not, how would I go about doing these.
 
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how about thinking residues?
 
I got 1 as my integral.
 
No. 1 is my value of f(z) (using l'hospital's rule and the fact that f(z) has a removable singularity at z = 0 so this function is analytic).
My limits of integration are 0 and 2pi so my integral is 2pi.
 
No again. We use the residue theorem
integral = 2 pi i (sum of the residues)
= 2 pi i (1)
= 2 pi i
and for the next integral I got 4 pi^2 i
 
Last edited:
If your function is analytic how can it have a residue at 0?
 

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