Complex Analysis Residue Problem

In summary, The conversation discusses finding the integral from 0 to infinity of 1/(x^3 + 1), using specific contours and residues. The singular point is (1+sqrt(3)*i)/2 and the solution involves factoring (x^3 + 1) into (x + 1)(x - (1+sqrt(3)*i)/2)(x - (1-sqrt(3)*i)/2). The correctness of the factoring is confirmed by multiplying all factors back together.
  • #1
tylerc1991
166
0

Homework Statement



The question asks me to find the integral from 0 to infinity of 1/(x^3 + 1), where I have to use the specific contours that they specify. Now I know that I need to use residues (in fact just one here) and the singular point is (1+sqrt(3)*i)/2. Once I can factor the (x^3 + 1) part then I can take the problem from there. Can someone check me to see if I factored this right?

The Attempt at a Solution



(x^3 + 1) has zeros at x = -1, (1+sqrt(3)*i)/2, and (1-sqrt(3)*i)/2, so when I factor this it becomes (x + 1)(x - (1+sqrt(3)*i)/2)(x - (1-sqrt(3)*i)/2). Then I can continue with the residue and solve the problem, but I need to have factored this correctly. Thank you anyone for your help!
 
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  • #2
Yes, you factored correctly. You could check by multiplying all the factors back together.
 

What is a residue in complex analysis?

A residue in complex analysis is the value of a function at a singular point, which is the coefficient of the term with a negative power in the Laurent series expansion of the function around that point.

What is the importance of residues in complex analysis?

Residues play a crucial role in calculating complex integrals, as they provide a way to evaluate these integrals without having to actually integrate. They also help in finding the poles and zeros of a function, which are important in understanding the behavior of the function.

How do you find the residues of a function?

To find the residue of a function at a singular point, you can use the formula Res(f, z0) = limz→z0[(z-z0)f(z)], where z0 is the singular point and f(z) is the function in consideration. You can also use the Cauchy integral formula or the Laurent series expansion to calculate residues.

What are the applications of complex analysis residue problem?

Complex analysis residue problem has a wide range of applications in fields such as physics, engineering, and finance. It is used to solve problems involving complex integrals, such as in the evaluation of Fourier and Laplace transforms. It also plays a crucial role in the study of differential equations and in signal processing.

Can residues have negative or fractional values?

Yes, residues can have negative or fractional values depending on the function and the singular point at which they are being evaluated. In some cases, the residue may be zero if the function is analytic at the singular point. However, it is important to note that residues are always complex numbers.

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