Numerical Integration of Function That Has a Pole

It is common to need to integrate numerically
functions that diverge somewhere outside the 
range of integration.  Even if the divergence occurs quite
far away, integration formulas like Simpson's, 
that depend on fitting a polynomial, usually will be
inaccurate: near a pole they will be very bad.  
A method is described that gives formulas that will integrate
functions of this kind accurately if the 
orders and positions of the poles are known.  Explicit
formulas are given that are easy to use on an 
automatic computer.  It is shown that they can be used
for some other singularities as well as poles. 
 If the integral converges, integration can be carried
to the singularity.  The accuracy of the integration 
with a pole of second order is discussed, and, as an example,
the new formula is compared with Simpson's. 
 The new formulas are useful even far from the pole,
while near the pole their advantage is overwhelming.

CACM April, 1967

Eisner, E.

CA670407 JB February 28, 1978  11:20 AM

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