Do you mean the sign of the semicircle around the pole (whose radius t=you take to go to 0 in the end)? But you also have to consider that in one case the pole is inside the contour (so you must use Residue theorem) and in the other is outside (SO by Cauchy theorem the integral around everything would be 0), right?FactChecker said:The sign will change because one path is clockwise around the pole and the other is counter-clockwise.
You'll get the same result IF you do both integrals properly.Silviu said:[...] does it matter how I make the path around the poles on the real line?
Yes. Sorry. My post was too brief. @strangerep gave a better answer.Silviu said:Do you mean the sign of the semicircle around the pole (whose radius t=you take to go to 0 in the end)? But you also have to consider that in one case the pole is inside the contour (so you must use Residue theorem) and in the other is outside (SO by Cauchy theorem the integral around everything would be 0), right?
The purpose of solving complex integral paths with real line poles is to find the value of an integral along a path that passes through a pole on the real line. This is necessary because the integral becomes undefined at the pole, and by finding a complex path that avoids the pole, we can still calculate the value of the integral.
The complex path for solving integrals with real line poles is determined by using the Cauchy Residue Theorem. This theorem states that if a function has a pole of order n at a point c, the integral of the function along a closed curve can be calculated by summing the residues of the function at each of its poles within the curve.
There are several common techniques for solving complex integrals with real line poles, including using partial fractions, contour integration, and the method of residues. These methods involve breaking down the integral into smaller, more manageable parts and using known techniques to solve them.
Yes, complex integral paths with real line poles can be solved numerically using computer software. This involves approximating the complex path and computing the value of the integral using numerical integration methods such as the trapezoidal rule or Simpson's rule.
Yes, there are many real-world applications of solving complex integral paths with real line poles. Some examples include calculating electric fields in physics, evaluating complex integrals in engineering, and solving differential equations in mathematics. These techniques are also used in various areas of science, including fluid dynamics, quantum mechanics, and signal processing.