How Can the Residue Theorem Be Applied to Prove This Integral?

In summary, the conversation discusses using the Cauchy integral theorem to show that the integral of a complex function over a circle centered at the origin is equal to 2πi times the limit of the function as z approaches infinity. This can be proven by considering the function on a Riemann sphere and using the Residue Theorem.
  • #1
de1irious
20
0
Hi, so suppose f(z) is a complex function analytic on {z|1<|z|} (outside the unit circle). Also, we know that limit as z-->infinity of zf(z) = A. Now I need to show that for any circle [tex]C_{R}[/tex] centered at origin with radius R>1 and counterclockwise orientation, that

[tex]\oint f(z)dz = 2\pi iA[/tex]

Any ideas? I'm trying to use Cauchy integral theorem somehow but it's not working.
 
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  • #2
Are you familiar with the Riemann sphere?

In this picture, every closed curve in C can be considered to go around infinity (which is just one point on the sphere). If you choose a circle P with radius larger than R then your function is analytic in the connected component of C\P which contains infinity.

The residue of your function at infinity is A so the Residue Theorem implies your assertion at once.
 
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FAQ: How Can the Residue Theorem Be Applied to Prove This Integral?

What is Cauchy integral theorem?

Cauchy integral theorem, also known as Cauchy's integral formula, is a fundamental theorem in complex analysis that states that if a function is holomorphic on a simply connected domain, then the value of its contour integral around any closed curve in that domain is equal to the sum of its values at all points inside the curve.

Who discovered Cauchy integral theorem?

Augustin-Louis Cauchy, a French mathematician, is credited with discovering Cauchy integral theorem in the early 19th century. However, other mathematicians such as Leonhard Euler and Joseph-Louis Lagrange had made similar discoveries before Cauchy.

What is the significance of Cauchy integral theorem?

Cauchy integral theorem is significant because it allows us to evaluate complex integrals using simple methods, as well as make connections between complex analysis and real analysis. It also lays the foundation for other important theorems in complex analysis, such as Cauchy's integral formula and the Cauchy-Riemann equations.

What are some applications of Cauchy integral theorem?

Cauchy integral theorem has numerous applications in physics, engineering, and other fields. It is used to solve problems in fluid dynamics, electromagnetism, and quantum mechanics. It is also used in the study of conformal mappings and the behavior of analytic functions.

Is Cauchy integral theorem limited to complex functions?

Yes, Cauchy integral theorem only applies to complex functions, which are functions of a complex variable. It cannot be applied to real-valued functions. However, it has been extended to higher dimensions in the form of Cauchy's integral theorem for polytopes.

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