Suppose we evaluate for a value of very close to one of the roots. In the example of the polynomial , let be the roots in the order in which they were re-turned by Yacas. Then can be found by evaluating at :
P(x):=x^4-5*x^3-25*x^2 +65*x+84 N(1/P(3.000001)) -8928.5702094768
We know that for very close to 3, the expression
will be dominated by the term, so
By the same method we can find the other four constants:
dx:=.000001 N(1/P(7+dx),30)*dx 0.2840908276e-2 N(1/P(-4+dx),30)*dx -0.4329006192e-2 N(1/P(-1+dx),30)*dx 0.1041666664e-1
construct is to tell Yacas to do a numerical calculation rather than an exact symbolic one, and to use 30 digits of precision, in order to avoid problems with rounding errors.) Thus,
The desired integral is
As in the simpler example started off with, where was second or- der and we got , in this example we expect that , for otherwise the large- behavior of the partial-fraction form would be rather than . This is a useful way of checking the result:
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