For a more sophisticated application of complex integration from advanced physics (quantum mechanics), see this video: https://www.youtube.com/watch?v=0Won5Vs_65E complex integration starts around time stamp 18:15
Differentiation has similar meaning as derivatives in 2D euclidean plane with the additional constraint between the slope along x-axis and slope along y-axis prescribed by the CR-conditions. For significance of integration, look up the wikipedia page on "circulation (Fluid Dynamics)": think about a vortex whose circulation intensity you want to find, so you need to compute the circulation integral given in the wiki page, but a vortex has a hole at the center (if you imagine the picture of a vortex), a hole is nothing but a singularity or a point of non - analyticity of the velocity field, so we will need to use Cauchy integral formulas and defamation of contour concepts. Likewise there are many applications of complex integration in electrodynamics but you guys are in a pure mathematics program where the syllabus is not designed to discuss physical application problems. Look up your textbook (Ablowitz book) for applications
For a more sophisticated application of complex integration from advanced physics (quantum mechanics), see this video: https://www.youtube.com/watch?v=0Won5Vs_65E complex integration starts around time stamp 18:15
Differentiation has similar meaning as derivatives in 2D euclidean plane with the additional constraint between the slope along x-axis and slope along y-axis prescribed by the CR-conditions. For significance of integration, look up the wikipedia page on "circulation (Fluid Dynamics)": think about a vortex whose circulation intensity you want to find, so you need to compute the circulation integral given in the wiki page, but a vortex has a hole at the center (if you imagine the picture of a vortex), a hole is nothing but a singularity or a point of non - analyticity of the velocity field, so we will need to use Cauchy integral formulas and defamation of contour concepts. Likewise there are many applications of complex integration in electrodynamics but you guys are in a pure mathematics program where the syllabus is not designed to discuss physical application problems. Look up your textbook (Ablowitz book) for applications