99 research outputs found
Non-integer valued winding numbers and a generalized Residue Theorem
We define a generalization of the winding number of a piecewise cycle
in the complex plane which has a geometric meaning also for points which lie on
the cycle. The computation of this winding number relies on the Cauchy
principal value, but is also possible in a real version via an integral with
bounded integrand. The new winding number allows to establish a generalized
residue theorem which covers also the situation where singularities lie on the
cycle. This residue theorem can be used to calculate the value of improper
integrals for which the standard technique with the classical residue theorem
does not apply.Comment: Final version, 19 pages, 7 figure
An integral that counts the zeros of a function
Given a real function on an interval satisfying mild regularity
conditions, we determine the number of zeros of by evaluating a certain
integral. The integrand depends on and . In particular, by
approximating the integral with the trapezoidal rule on a fine enough grid, we
can compute the number of zeros of by evaluating finitely many values of
and . A variant of the integral even allows to determine the number
of the zeros broken down by their multiplicity.Comment: 20 pages, 1 figure, final versio
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