Need help with Caucy Integral

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In summary, a Cauchy Integral is a type of contour integral used to evaluate complex functions along a closed path in the complex plane. It is calculated using the Cauchy Integral Formula, and its significance lies in Cauchy's Integral Theorem, which states that the value of the integral is not affected by the contour's shape or size. This makes it a powerful tool in complex analysis and has various real-world applications in fields such as physics, engineering, and signal processing. However, there are limitations to Cauchy Integrals, such as only being able to evaluate closed contours and potential convergence issues with singularities on the contour. Alternative methods may need to be used in these cases.
  • #1

marcusl

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I have a function
[tex]G_0=\frac{1-\alpha/u^2}{1-\alpha u^2} .[/tex]

Since [tex]0<\alpha<1[/tex], [tex]G_0[/tex] has zeroes but no poles inside the unit circle.

I need to evaluate
[tex]\Gamma(\tau)=\frac{1}{2\pi i}\oint{\frac{\ln{G_0 (u)}}{u-\tau}du}[/tex]
where the integral is around the unit circle. How do I evaluate the poles of the integrand so I can evaluate this using residues?

EDIT: Oops, G0 has a second order singularity at u=0, too.
 
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  • #2
maybe power series wouldwork, first the geometric series for G0 then the series for log(1+w)?
 
  • #3


To evaluate the poles of the integrand, we can use the fact that the integrand has a logarithmic singularity at u=0. This means that we can expand the integrand as a Laurent series around u=0:

\ln{G_0(u)} = \ln{\left(\frac{1-\alpha/u^2}{1-\alpha u^2}\right)} = -\ln{(1-\alpha u^2)} - \ln{(1-\alpha/u^2)} = -\sum_{n=1}^{\infty} \frac{\alpha^n}{n} u^{2n} - \sum_{n=1}^{\infty} \frac{\alpha^n}{n} \frac{1}{u^{2n}}

We can see that the first term in the series has a pole of order 2 at u=0, while the second term has a pole of order 2 at u=\infty. This means that the integrand has poles at u=0 and u=\infty, both of order 2.

To evaluate the residue at u=0, we can use the formula for the coefficient of the Laurent series:

Res(f,u_0) = \frac{1}{2\pi i} \oint{\frac{f(u)}{(u-u_0)^{n+1}}du}

In this case, we have n=2 and u_0=0. Plugging in the Laurent series for \ln{G_0(u)}, we get:

Res(\ln{G_0(u)}, 0) = \frac{1}{2\pi i} \oint{\frac{-\sum_{n=1}^{\infty} \frac{\alpha^n}{n} u^{2n}}{u^3}du} = \frac{-\alpha}{2\pi i} \oint{\frac{1}{u}du} = -\frac{\alpha}{2}

Similarly, we can evaluate the residue at u=\infty by using the series expansion for \ln{G_0(u)} around u=\infty:

Res(\ln{G_0(u)}, \infty) = \frac{1}{2\pi i} \oint{\frac{\sum_{n=1}^{\infty} \frac{\alpha^n}{n} \frac{1}{u^{2n}}
 

What is a Cauchy Integral?

A Cauchy Integral is a type of contour integral which is used to evaluate complex functions along a closed path in the complex plane.

How is a Cauchy Integral calculated?

A Cauchy Integral is calculated using the Cauchy Integral Formula, which states that the value of the integral is equal to the sum of the function evaluated at points inside the contour, multiplied by the complex derivative of the function at each point.

What is the significance of Cauchy's Integral Theorem?

Cauchy's Integral Theorem states that the value of a Cauchy Integral is not affected by the shape or size of the contour, as long as the contour does not cross any singularities of the function. This makes it a powerful tool in complex analysis and allows for the evaluation of complex integrals without having to explicitly calculate the integral.

What are some real-world applications of Cauchy Integrals?

Cauchy Integrals have many applications in physics, engineering, and other scientific fields. They are used in the study of fluid dynamics, electromagnetism, and heat transfer, among others. They are also used in the field of signal processing to analyze signals with complex components.

Are there any limitations to Cauchy Integrals?

One limitation of Cauchy Integrals is that they can only be used to evaluate integrals over closed contours. Additionally, they may not always converge if the function being integrated has singularities on the contour. In these cases, alternative methods such as the Cauchy Principal Value may need to be used.

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