What is an integral with a circle through it?

In summary, an integral with a circle through it is a symbol used in mathematics to represent a contour integral, which integrates a complex-valued function along a given curve in the complex plane. It differs from a regular integral in that it integrates over a path or curve in the complex plane rather than over a range of real numbers. The circle in the symbol represents the path or contour over which the integration is performed, and it is evaluated using techniques from complex analysis. Contour integrals have applications in various fields such as physics, engineering, and mathematics.
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Luongo
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[tex]\oint[/tex] what's the difference?
 
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This is a symbol usually used to denote a line integral over a closed curve (a curve homeomorphic to the circle, hence the circle symbol), but it can also be used for an integral over a closed manifold in more general circumstances.
 

What is an integral with a circle through it?

An integral with a circle through it is a symbol used in mathematics to represent a specific type of integral called a contour integral. It is used to compute the integral of a complex-valued function along a given curve in the complex plane.

How is a contour integral different from a regular integral?

A contour integral differs from a regular integral in that it integrates over a path or curve in the complex plane rather than over a range of real numbers. It is a useful tool for solving problems involving complex functions and is often used in physics and engineering.

What does the circle in the integral symbol represent?

The circle in the integral symbol represents the path or contour over which the integration is to be performed. It is a closed loop in the complex plane that encloses the region of interest.

How is a contour integral evaluated?

A contour integral is evaluated using techniques from complex analysis, such as Cauchy's integral theorem and Cauchy's integral formula. These techniques involve calculating the integral over a simpler path and then using the properties of analytic functions to extend the result to the desired contour.

What are some applications of contour integrals?

Contour integrals have many applications in physics, engineering, and mathematics. They are used to solve problems in fluid dynamics, electromagnetism, and quantum mechanics. They are also used in the study of complex functions, such as the Riemann zeta function and the gamma function.

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