Integration techniques and Cauchy prinicpal value

[Stephen Tashi]

Is there a good reference that summarizes what common integration techniques (e.g. change of variables, integration by parts, interchange of the order of integration) can be used on integrands when one is calculating the Cauchy principal value ( http://en.wikipedia.org/wiki/Cauchy_principal_value) and the ordinary integral does not exist?

It looks like a complicated subject, especially if we ask if the ordinary integration methods (like change of variables) are permitted to introduce more Cauchy principal value integrals.

 

[jackmell]

I believe the Cauchy Principal Value is determined by taking a symmetric limit on either side of the singular point so those techniques will not by themselves help you determine this value although they may help you get the problem in a form where you can take these limits. Take for example:

$$P.V. \int_{-1}^{1} \frac{1}{z}dz=\lim_{\epsilon\to 0}\left(\int_{-1}^{-\epsilon} \frac{1}{z}dz+\int_{\epsilon}^1 \frac{1}{z}dz\right)$$

Now in this case, I can find the antiderivative and write:

$$\begin{align} P.V. \int_{-1}^{1} \frac{1}{z}dz&=\lim_{\epsilon\to 0} \left(\log(-\epsilon)-\log(-1)+\log(1)-\log(\epsilon)\right) \\ &=\lim_{\epsilon\to 0}\ln(\epsilon)+\pi i-\ln(1)-\pi i+0-\ln(\epsilon) \\ &=\lim_{\epsilon\to 0} \left(\ln(\epsilon)-\ln(\epsilon) \right)\\ &=0 \end{align}$$

And in general you could do this (taking the limit of the antiderivative) for more complicated integrals. Now sometimes the Residue Theorem can and often is used to find principal values by allowing the radius of an indentation around a singular point go to zero. But even in that case, we are implicitly taking the limit as $\epsilon\to 0$.

 

[Stephen Tashi]

I agree that the Cauchy principal value is defined as a limit. In a given problem, if it happens to be $lim_{A \rightarrow A_0} F(A)$ then I presume we can use any standard integration technique to find an ordinary integral $F(A) =$ some definite integral (in the ordinary sense of the word) of a function $f(x)$ with the constant $A$ involved in the limit of integration.

However, suppose the evaluation of $F(A)$ requires doing several other Cauchy integrations. How safe are the common integration methods then?

For example, consider a change in the order of integration.If your problem involves several Cauchy principal part integrations, you could end up with an expression invovling nested limits:

$lim_{A \rightarrow A_0} ( lim_{B \rightarrow B_0} ( lim_{C \rightarrow C_0} G(A,B,C) ) )$

And if you changed the order of Cauchy integration, you might get the nested limits in different order.

$lim_{A \rightarrow A_0} ( lim_{C \rightarrow C_0} ( lim_{B \rightarrow B_0} G(A,B,C) ) )$

Changing the order of limits in an expression involving nested limits can change the value of the expression. So is there something about the G(A,B,C) involved in Cauchy integration that avoids this problem?

I wondering if someone has worked out a theory of Cauchy principal value integration that is a nice system of rules and checks like we have with ordinary integration. Or are the possibilities so complicated that each problem has to be analyzed on its own merits.