The argument principle states that if is meromorphic
in a region
enclosed by a contour, and and are respectively the numbers of complexroots and poles of in , with each zero and pole counted according to its multiplicity
or order, then
Duren, P.; Hengartner, W.; and Laugessen, R. S. "The Argument Principle for Harmonic Functions." Amer. Math. Monthly103,
411-415, 1996.Knopp, K. Theory
of Functions, Parts I and II. New York: Dover, pp. 132-134, 1996.Krantz,
S. G. "The Argument Principle." Ch. 5 in Handbook
of Complex Variables. Boston, MA: Birkhäuser, pp. 69-78, 1999.
is meromorphic
in a region 
enclosed by a contour 
, and 
and 
are respectively the numbers of complex roots and poles of 
in 
, with each zero and pole counted according to its multiplicity
or order, then
and 
gives
