You should not think in terms or *area*. You really are doing a *line* integral. So it's not a question of whether the area is the same, it's the question of whether the contribution *along* the upper and lower contours is the same. What you must ensure is that the contribution along the infinite semi-circle vanishes. As Tide pointed out, this is determined by the sign of the exponent in omega t. For example, if you have [itex] e^{-i \omega t} [/itex], if you close with a contour above the real axis, as t goes to [itex] + i \infty [/itex], the exponent will go to [itex] e^{ + \infty} [/itex] which means that the contribution along that contour will blow up! So you *must* close below in that case.Go37Pi said:I was doing a Fourier Transform Integral, and was wondering if it would be legitimate for me to choose a semicircle CR on the lower half-plane below the real axis rather than choosing a semicircle CR on the upper half-plane above the real axis. I would expect it to be valid because the contour integral should have equal areas regardless of whether it is oriented above or below the Real axis. Is this right to expect?
The Cauchy Principal Value (Cauchy P.V.) of an improper integral is a way of assigning a finite value to an integral that would otherwise be undefined due to a singularity or infinite discontinuity in the integrand. It is denoted by the symbol P.V. and is calculated using the limit of the integral as the singularity approaches a specific point.
The Cauchy P.V. differs from the regular value of an integral because it takes into account the behavior of the integrand near a singularity. The regular value of an integral is calculated by integrating over a specific interval, while the Cauchy P.V. involves taking the limit of the integral as the singularity approaches a certain point.
The Cauchy P.V. is commonly used in mathematical applications involving singularities, such as in the study of complex analysis and Fourier transforms. It is also frequently used in physics, particularly in the calculation of electric and magnetic fields around point charges or moving particles.
The Cauchy P.V. is significant in mathematics because it allows for the evaluation of integrals that would otherwise be undefined due to singularities. This makes it a valuable tool for solving complex mathematical problems and allows for a better understanding of the behavior of functions near singular points.
While the Cauchy P.V. is a useful method for calculating integrals, it does have some limitations. It can only be used for integrals with finite limits and cannot be applied to certain types of singularities, such as essential singularities. Additionally, it may not always provide a unique solution for the integral, as there can be different paths to approach the singularity that result in different Cauchy P.V. values.