The Cauchy principal value allows one to make sense of the definite integral of a real function of a real variable over a portion of the real-axis where the function has isolated first-order singularities. For example, the simple integral of f(x)=1/x over the reals is undefined, but given the fact that f(x) is odd we might intuitively wish the integral to be 0.

The Cauchy principal value integral (from -b to b), denoted by a P in front of the normal integral sign, of a function f(x)/(x-a), where f(x) is differentiable, is defined as the limit as δ goes to the 0 of the integral (from -b to a-δ) + integral (from a+δ to b) of f(x)/(x-a).

Using the definition of the principal value integral, we can find the integral of 1/x between -b and b (and letting b go to infinity over the whole real axis). The principal value integral of 1/x = limit as δ goes to 0 of integral (from -b to -δ) + integral (from δ to b) of 1/x = 0.

It can be shown that if a complex function f(z) is analytic over the upper half of the complex plane and if |f(z)| goes to 0 as |z| goes to infinity, then the limit as b goes to infinity of the principal value integral of f(x)/(x-a) over the reals is given by iπf(a). This is the basis for the Hilbert Transform.

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