Parametric definition for a complex integral

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SUMMARY

The discussion focuses on evaluating a complex integral defined along a piecewise smooth curve γ from -1 to 1. The integrals A and B are defined as A=∫γ(a(x²-y²) + 2bxy) dz and B=∫γ(2axy - b(x²-y²)) dz, with the goal of proving that A + Bi = (2/3)(a - bi). The conversation highlights the importance of understanding "exact" integrands, which yield the same result regardless of the path taken between endpoints, and suggests that familiarity with complex arithmetic is essential for tackling such problems.

PREREQUISITES
  • Understanding of complex functions and integrals
  • Familiarity with piecewise smooth curves in complex analysis
  • Knowledge of exact differentials and their properties
  • Proficiency in complex arithmetic, including operations with complex numbers
NEXT STEPS
  • Study the concept of exact differentials in complex analysis
  • Learn how to parametrize curves in complex integrals
  • Practice complex arithmetic, focusing on operations like multiplication of complex numbers
  • Explore the implications of path independence in complex integrals
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Mathematicians, students of complex analysis, and anyone involved in advanced calculus or mathematical physics who seeks to deepen their understanding of complex integrals and their properties.

thayin
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I have been working in complex functions and this is a new animal I came across.

Let γ be a piecewise smooth curve from -1 to 1, and let
A=∫γa(x2-y2) + 2bxy dz
B=∫γ2axy - b(x2-y2) dz

Prove A + Bi = (2/3)(a-bi)

In the past anything like this defined γ and I would find a parametric definition of the function γ and integrate. This seems like a completely different animal.
Does anyone have any ideas as to how to bet this one rolling?
 
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there are certain integrands called "exact" that give the same answer over every path joining the two endpoints. These are the ones which equal df for some function f. maybe this is one. do you know the test for exactness

or maybe you can just guess f. have you ever multiplied out z^2 = (x+iy)^2?

If not you need to practice many more basic examples of complex arithmetic.
 

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