Consider the the following [Improper integrals](Improper%20integrals.md) of a real valued function with limits $-\infty$ to $\infty$  If this integral [converges,](Improper%20integrals.md#Convergence%20of%20improper%20integrals) the value to which it integral converges is referred to as the _Cauchy principle value,_ which is expressed with the following notation. $\mbox{P.V.}\int_{-\infty}^{\infty} dx f(x) = \lim_{R \rightarrow \infty} \int_{-R}^{R} dx f(x)$ However, improper integrals that _do not converge_ may also have a Cauchy principle value. # Finding the Cauchy principal value ## As Residues along complex contours One way of finding the [cauchy principal value](Cauchy%20principal%20value.md) is [by evaluating the improper integral on a contour along the real axis](Real%20integral%20on%20the%20complex%20plane.md#Integrals%20over%20all%20mathbb%20R) of the complex plane. This may be done if there are [simple poles](Pole.md) along that contour. In this case the principle value will be the sum of [[Residues]]s at points $x_i$ corresponding to a simple poles on the real axis by the [residue theorem](Residue%20theorem.md) while the entire contour extends across either the entire upper lower lower half plane (shown below)  <font size="2"> A closed contour on the upper half complex plane with a simple pole at $z=x_0$. Image adapted from Arken G. B., _Mathematical methods for Physicists_</font> As such poles will lie right on the edge of the contour we need to infinitessimally deform the contour int semicircles around these poles and evaluate the integral using the [the residue theorem](Residue%20theorem.md#The%20Residue%20Theorem%20for%20infinitessimal%20circular%20arcs) as it applies along an infinitessimal semicircular arc on the real axis. Depending when whether the contour is clockwise our counterclockwise as shown below the integral is either positive or (counterclockwise) or negative (clockwise) around the simple poles, following from the residue theorem.  <font size="2"> Infinitessimal semi-circles around poles along the real axis. Image adapted from _Arken G. B., _Mathematical methods for Physicists_ </font> ### Applications Identifying the [Cauchy principal value](Cauchy%20principal%20value.md) is particularly useful when dealing with integrals that have singularities along the contour itself, such as when singularities exist at $R=\infty$ since it lets us extract a number corresponding corresponding to an integral that otherwise diverges by integrating _around_ singularity. --- # Proofs and Examples #MathematicalFoundations/Analysis/Integrals #MathematicalFoundations/Analysis/ComplexAnalysis/Integrals