Last revised: 2008Apr01
Integrals with Singularities or Discontinuous Derivatives
Numerical integration algorithms such as Simpson’s rule are designed to work with
“smooth” integrands, because they assume that the function is locally a polynomial.
If
the integrand has discontinuous derivatives or poles or a branch point in the integration
region, such integration rules will generally do poorly. However, one can convert the original
integral into a mathematically equivalent one that
is
smooth in this sense. Here are some
strategies (check
Numerical Recipes
and the HjorthJensen notes for others):
•
If there is a discontinuous derivative somewhere in the integrand, then split the integral
into two integrals at the discontinuity. For example, if there is an absolute value, then
Z
1

1

x

f
(
x
)
dx
=
Z
0

1
(

x
)
f
(
x
)
dx
+
Z
1
0
x f
(
x
)
dx ,
(1)
if
f
(
x
) is smooth. (One can combine the integrals after taking
x
→ 
x
in the first.)
•
If there is a branch point, such as
x
1
/n
with
n
an integer greater than one, a simple
variable change can usually be found to convert the integrand to a smooth one. For
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 Winter '11
 Furnstahl
 Physics, Derivative, Work, dx, Principal value

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