Mute said:It depends. If it's a straight line contour running through simple pole, I believe it turns out that the contribution is just half the residue (as you could infinitesimally move the pole inside the contour or go around the pole with an infinitesimal semi-circle).
If you have a non-simple pole, I believe it is more complicated. I once found a post online on a different forum where someone proved that straight-line contour that runs through an odd-degree pole still contributes half the residue, but even degree poles or contours which only touch an odd-degree non-simple pole tangentially result in a divergent integral. I'll try to see if I can find that post again.
Edit: Ah, there was a previous thread on Physicsforums in which someone asked the question, and someone linked to this other forum in which a poster proved the statement that the half residue theorem worked for odd-degree poles.
A singularity on the contour is a point on a curve or surface where the function is not continuous or differentiable. In other words, it is a point where the function "breaks" or behaves unexpectedly, often resulting in an infinite or undefined value.
A singularity on the contour can be caused by a variety of factors, such as division by zero, taking the logarithm of zero, or encountering a discontinuity in the function. It can also occur when the function has a pole or essential singularity at a certain point.
Singularities on the contour can significantly alter the behavior of a function. They can cause the function to have infinite or undefined values, discontinuities, or unexpected oscillations. They can also impact the convergence or divergence of series or integrals involving the function.
There are several techniques used to analyze singularities on the contour, including the Residue Theorem, Laurent Series, and Cauchy's Integral Formula. These techniques involve complex analysis and can help determine the type and location of a singularity, as well as its impact on the function.
Singularities on the contour are relevant in many areas of science and engineering, including fluid dynamics, electromagnetism, and quantum mechanics. They can also be found in natural phenomena such as shock waves, vortices, and turbulent flows. Understanding and analyzing singularities on the contour can help explain and predict these phenomena and improve our understanding of the physical world.