Complex analysis - area inside a simple closed curve

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SUMMARY

The area enclosed by a simple closed curve C can be calculated using the formula 1/2i * ∫(conjugate of z) dz over the curve C. To extend this proof from a circle to any arbitrary closed curve, one must express the area as a double integral. Applying Green's Theorem is essential for transforming this double integral into a single integral over the curve C, facilitating the calculation of the area for any closed curve.

PREREQUISITES
  • Understanding of complex analysis concepts, specifically contour integration.
  • Familiarity with Green's Theorem and its applications in vector calculus.
  • Knowledge of polar coordinates and their use in integration.
  • Proficiency in manipulating complex functions and their conjugates.
NEXT STEPS
  • Study the application of Green's Theorem in complex analysis.
  • Learn how to express areas enclosed by curves as double integrals.
  • Explore the use of polar coordinates in contour integration.
  • Investigate the properties of complex conjugates in integration.
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in advanced calculus techniques for calculating areas enclosed by curves.

sweetvirgogirl
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Let C be a simple closed curve. Show that the area enclosed by C is given by 1/2i * integral of conjugate of z over the curve C with respect to z.

the hint says: use polar coordinates

i can prove it for a circle, but i am not sure how to extend it to prove it for any given closed curve
 
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How would you express the enclosed by an arbitrary closed curve as a double integral? Can you apply Green's theorem to turn this into a single integral over C?
 

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