Complex Integration: Integrating x^0.5/(1+x^2)

In summary, the conversation discusses a method for integrating x^0.5/(1+x^2) using complex integration, specifically the residue theorem. The suggested approach involves using a half-disc contour in the upper half-plane with an indentation around the branch-point at the origin, with the integral over the two legs on the real axis being equal to 2pi i times the residue at i.
  • #1
hedipaldi
210
0

Homework Statement


integrate x^0.5/(1+x^2) by using complex integration


Homework Equations



residue theorem

The Attempt at a Solution


my attempt at a solution is attached.i need help in finding where am i mistaken.
thank's
Hedi
 

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  • #2
. . . that's too messy to read. For me at least. Try and learn the math formatting language Latex to nicely format your math. Is this what you want to integrate:

[tex]\int_0^{\infty} \frac{\sqrt{x}}{1+x^2}dx[/tex]

If so, how about taking a half-disc contour in the upper half-plane, indented around the branch-point at the origin. For now, just suppose the small and large half-circle contours go to zero and you're left with the two legs on the real axis. Then the integral over those two legs is equal to 2pi i times the residue at i right.
 
  • #3
Thank you, that works.
 

1. What is complex integration?

Complex integration is a mathematical process of finding the area under a complex function in the complex plane. It is similar to the concept of integration in real numbers, but instead of integrating along a straight line, it involves integrating along a curve in the complex plane.

2. What is the formula for integrating x^0.5/(1+x^2)?

The formula for integrating x^0.5/(1+x^2) is ∫ x^0.5/(1+x^2) dx = (2/3)arctan(x) + C, where C is the constant of integration.

3. Can complex integration be solved using traditional integration techniques?

No, complex integration requires the use of special techniques such as Cauchy's integral theorem, residue theorem, and the method of contour integration.

4. How is the constant of integration determined in complex integration?

The constant of integration in complex integration is determined by evaluating the function at a specific point on the complex plane. This point is chosen in such a way that the value of the function at that point is known.

5. What are some real-world applications of complex integration?

Complex integration has many applications in physics, engineering, and other fields. It is used to solve problems involving electric fields, fluid dynamics, and signal processing, among others. It is also used in the study of complex functions and their properties.

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