a principal-value integral
/ koh-SHEE for Cauchy /
Some integrals are divergent in the ordinary sense because the integrand blows up at an interior point, yet the blow-up is anti-symmetric — it shoots to plus infinity on one side and minus infinity on the other, and those infinities want to cancel. The Cauchy principal value is a careful way of taking that cancellation seriously, assigning a finite, meaningful number to an integral that would otherwise be undefined.
The definition: suppose f has a single bad point at x = c inside [a, b]. The ordinary integral fails because each one-sided piece diverges. The principal value, written P.V. integral, is the limit as epsilon shrinks to 0 of the sum of the integral from a to c - epsilon and the integral from c + epsilon to b — that is, you cut out a symmetric little interval of half-width epsilon around c and let it close in from both sides at the same rate. The symmetry is the whole point: the infinities on the two sides, being cut off at equal distances, cancel, and a finite limit survives. For the integral from minus infinity to infinity it likewise means the symmetric limit, from -R to R as R grows, which can converge even when the integral is not absolutely convergent.
Principal values are exactly what contour integration produces when a pole sits on the path of integration rather than off it. Indenting the contour with a small semicircle around the real pole splits the contribution into a principal value (the symmetric real part) plus a half-residue term (the indentation), and the residue theorem then expresses the principal value cleanly. The honest caveat: a principal value is not the same as an ordinary convergent integral — it is a particular regularized value that depends on the symmetric cancellation, and you must announce that you are taking it; a different cutoff (non-symmetric) could give a different or no limit.
The integral from -1 to 1 of dx / x diverges ordinarily, but its principal value is 0, because the limit of (integral from -1 to -epsilon plus integral from epsilon to 1) of dx / x cancels by the odd symmetry of 1/x.
Cut out a symmetric epsilon-window around the bad point and let it close in evenly.
A principal value can exist even when the integral diverges in the ordinary or absolute sense, so it is a weaker notion; you must state P.V. explicitly. The symmetric cutoff is essential — an asymmetric one can change the value entirely.