Originally posted by DeepThought
A Cauchy principle value is a way of assigning a finite result to the problem of assigning a number to an integral which is not absolutely convergent.
The integral isn't difficult. With a change of parameters, the integral is:
∫[–∞, ∞] x / ( b^2 + (x – a)^2 ) dx
where I've ignored the factor of |h - u| as it is just a constant multiplier and ...[text shortened]... /List_of_integrals_of_rational_functions#Integrands_of_the_form_xm_.2F_.28a_x2_.2B_b_x_.2B_c.29n
Arr; I have just spotted an edit error I have been repeatedly making which could help explain at least some of my confusion!
I have edited the equation in wrong here because I kept leaving out pi !
The integral for the mean average, just as I correctly written it in my computer program but had incorrectly edited in the OP, should
have been written;
∫[–∞, ∞] x * |h – u| / ( pi
((h – u)^2 + (x – u)^2) ) dx
and this is for a probability distribution with probability density defined by;
probability_density(x) = |h – u| / ( pi
((h – u)^2 + (x – u)^2 )) )
OK, NOW when I enter that integral into wolframAlpha, although it still says it doesn't converge, it then says using catchy principle value that it has the required value of u just as I would expect if
mean average is defined.
So, given that the catchy principle value says the integral for the mean (IF defined) is u, does that mean, despite
the integral for that mean average being divergent, the mean average for this distribution IS defined or not?
My intuition says it is defined but need to be absolutely sure before writing that as a fact my book. I don't want any falsehoods asserted in my book!