Parametric definition for a complex integral

In summary, the conversation discusses a new type of integration problem involving a piecewise smooth curve and two integrands, A and B. The goal is to prove that A + Bi equals (2/3)(a-bi). The person is unsure of how to approach this problem since it is different from past integration problems. They mention the concept of "exact" integrands and suggest finding a parametric definition or guessing the function f. The conversation ends with a suggestion to practice more basic examples of complex arithmetic.
  • #1
thayin
4
0
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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  • #2
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.
 

1. What is a parametric definition for a complex integral?

A parametric definition for a complex integral is a mathematical tool that allows for the calculation of integrals using parametric equations. This method involves reparameterizing a complex curve into a simpler form, making it easier to integrate.

2. How does a parametric definition differ from a traditional definition of an integral?

A traditional definition of an integral involves finding the area under a curve by summing infinitesimally small rectangles. A parametric definition, on the other hand, involves reparameterizing the curve into simpler forms and then integrating the resulting equations.

3. What are the benefits of using a parametric definition for a complex integral?

Parametric definitions for complex integrals can be used to solve a wider range of integrals compared to traditional methods. They are also more efficient and can be applied to complex curves that are difficult to integrate using traditional methods.

4. Are there any limitations to using a parametric definition for a complex integral?

Parametric definitions may not be applicable to all integrals, especially those with highly irregular curves. It is also important to note that parametric equations need to be carefully chosen to ensure accurate results.

5. How can I apply a parametric definition to solve real-world problems?

Parametric definitions for complex integrals are useful in a variety of scientific fields, such as physics, engineering, and economics, for solving real-world problems. They can be used to calculate areas, volumes, and other quantities that involve integration of complex curves.

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