Hermite–Biehler theorem

E502194

mathematical theorem theorem in control theory

The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.

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Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in control theory ⓘ
appliesTo	Hurwitz stable polynomials ⓘ univariate complex polynomials ⓘ
assumes	polynomial has no zeros on the real axis in standard form ⓘ
characterizes	when a complex polynomial has all zeros in the open upper half-plane ⓘ
concerns	distribution of zeros of entire functions in some extensions ⓘ relationship between real and imaginary parts of a polynomial on the real axis ⓘ
conditionOnCoefficients	complex polynomial has real coefficients in many standard formulations ⓘ
conditionOnZeros	associated real polynomials must have only real zeros ⓘ zeros of the two associated real polynomials must interlace on the real axis ⓘ
equivalentTo	a real-axis interlacing condition on two real polynomials ⓘ
field	complex analysis ⓘ control theory ⓘ polynomial theory ⓘ
givesCriterionFor	all zeros of a polynomial lying in the open left half-plane via change of variables ⓘ all zeros of a polynomial lying in the open upper half-plane ⓘ
hasGeneralization	Hermite–Biehler class of entire functions NERFINISHED ⓘ
historicalPeriod	late 19th to early 20th century mathematics ⓘ
implies	sign alternation properties of associated real polynomials on intervals between zeros ⓘ
involves	decomposition of a complex polynomial into real and imaginary parts ⓘ two associated real polynomials ⓘ
mathematicalDomain	complex function theory ⓘ real algebraic geometry ⓘ
namedAfter	Charles Hermite NERFINISHED ⓘ Ludwig Bieberbach Biehler ⓘ
relatedConcept	entire functions of bounded type in a half-plane ⓘ interlacing of real roots ⓘ real-rooted polynomials ⓘ self-inversive polynomials ⓘ
relatedTo	Hurwitz stability criterion NERFINISHED ⓘ Routh–Hurwitz theorem NERFINISHED ⓘ stability of linear time-invariant systems ⓘ
subject	complex polynomials ⓘ interlacing of zeros ⓘ location of zeros ⓘ real polynomials ⓘ upper half-plane ⓘ
typeOfCriterion	frequency-domain stability criterion ⓘ
usedFor	analyzing root loci of polynomials ⓘ characterizing stable polynomials ⓘ designing stable feedback systems ⓘ
usedIn	control theory ⓘ filter design ⓘ signal processing ⓘ stability analysis of dynamical systems ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Charles Hermite → hasConceptNamedAfter → Hermite–Biehler theorem ⓘ