Hoenders and Slump[7]
describe a method for determining number and multiplicities of zeros of a function
based on a numerical quadrature technique, but as they show in their Tables 4-5, this
technique is unstable. It seems unlikely to us that any numerical technique
could decisively solve this problem. Symbolic methods can come to our aid in
some instances. First, if
are rational functions in
must be zero. Hence
A piecewise holomorphic function defined by
consisting of finitely many holomorphic
An expression
Our algorithm is restricted to a finite interval
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Multiple Zeros, Non-Holomorphic Functions, Infinite Intervals
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is a rational function over a suitable computable extension
of
, then square-free factorization is applicable. Second, if
where
is a purely transcendental extension of a computable extension
is a multiple zero, and
where
is a rational function, then
and
identical to zero. Since
is a common root of both
and
, the resultant
and is algebraic over
can be solved symbolically for
and then this solution substituted into
and
to ascertain if
proving
can be decomposed as a composition
,
with multiplicity
,
is a root of
with multiplicity
, then
.
may be treated as
containing non-holomorphic subexpressions can sometimes be rewritten to become holomorphic
or piecewise holomorphic. As an example, suppose
where
and
are holomorphic. Then solve
and replace
by better expressions
.
. In some cases, such as
, this is necessary, for otherwise there would be an infinite number of solutions.
In other cases, such as
, there are only finitely many solutions even on
. The strategy proposed in this case is to use some asymptotic analysis to find
such that
and
leaving only
to contend with.