Argument principle

The argument principle relates the change in a meromorphic function’s argument along a closed contour to the number of its zeros minus its poles inside.

Logarithmic derivative: The derivative of a function divided by the function itself. Integrating it around the contour records the function’s zeros and poles through their residues.
Rouché's theorem: A theorem that gives equal zero counts for two holomorphic functions when one dominates their difference on a contour. It uses the argument principle to compare the zeros of perturbed functions.
Holomorphic function: A complex function that is complex differentiable throughout an open domain. The meromorphic setting extends holomorphic functions by allowing isolated poles.
Schwarz lemma: A theorem bounding holomorphic self-maps of the unit disk that fix the origin. Unlike the argument principle, it constrains function values rather than counting enclosed zeros.
Winding number: An integer measuring how many times a closed curve winds around a point, with orientation. The image contour’s winding around zero is the argument change counted by the theorem.
Nyquist stability criterion: A control-theory test that infers closed-loop stability from a frequency-response plot’s encirclements. Its encirclement count is an application of the argument principle to a transfer function.
Contour integral: An integral of a function along an oriented curve in the complex plane. The theorem can be expressed as a contour integral of the logarithmic derivative.
Maximum modulus principle: A theorem stating that a nonconstant holomorphic function cannot attain its maximum modulus inside its domain. It constrains interior values, whereas the argument principle counts zeros and poles from boundary behavior.
Residue theorem: A theorem equating a complex contour integral to the sum of enclosed residues, multiplied by 2πi. Applied to the logarithmic derivative, it produces the zero-minus-pole count.
Fundamental theorem of algebra: The theorem that every nonconstant complex polynomial has a complex root. The argument principle provides a complex-analysis route to counting polynomial roots.

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