https://en.formulasearchengine.com/api.php?action=feedcontributions&feedformat=atom&user=213.37.229.63 2026-05-02T15:53:56Z User contributions MediaWiki 1.43.0-wmf.28 https://en.formulasearchengine.com/index.php?title=Multidisciplinary_design_optimization&diff=5506 2014-01-08T20:15:55Z

<p>213.37.229.63: /* Recent MDO methods */</p> <hr /> <div>In [[complex analysis]], a '''zero''' (sometimes called a '''root''') of a [[holomorphic function]] ''f'' is a [[complex number]] ''a'' such that ''f''(''a'') = 0.<br /> <br /> ==Multiplicity of a zero==<br /> A complex number ''a'' is a '''simple zero''' of ''f'', or a '''zero of multiplicity 1''' of ''f'', if ''f'' can be written as<br /> <br /> :&lt;math&gt;f(z)=(z-a)g(z)\,&lt;/math&gt;<br /> <br /> where ''g'' is a [[holomorphic function]] ''g'' such that ''g''(''a'') is not zero.<br /> <br /> Generally, the '''[[multiplicity (mathematics)|multiplicity]]''' of the zero of ''f'' at ''a'' is the positive integer ''n'' for which there is a holomorphic function ''g'' such that<br /> <br /> :&lt;math&gt;f(z)=(z-a)^ng(z)\ \mbox{and}\ g(a)\neq 0.\,&lt;/math&gt;<br /> <br /> The multiplicity of a zero ''a'' is also known as the '''order of vanishing''' of the function at ''a''.<br /> <br /> ==Existence of zeros==<br /> The [[fundamental theorem of algebra]] says that every nonconstant [[polynomial]] with complex coefficients has at least one zero in the [[complex plane]]. This is in contrast to the situation with [[real number|real]] zeros: some polynomial functions with real coefficients have no real zeros. An example is ''f''(''x'') = ''x''&lt;sup&gt;2&lt;/sup&gt; + 1.<br /> <br /> ==Properties==<br /> An important property of the set of zeros of a holomorphic function of one variable (that is not identically zero) is that the zeros are isolated. In other words, for any zero of a holomorphic function there is a small disc around the zero which contains no other zeros.<br /> There are also some theorems in complex analysis which show the connections between the zeros of a holomorphic (or meromorphic) function and other properties of the function. In particular [[Jensen's formula]] and [[Weierstrass factorization theorem]] are results for complex functions which have no counterpart for functions of a real variable.<br /> <br /> ==See also==<br /> * [[Root of a function]]<br /> * [[Pole (complex analysis)]]<br /> * [[Hurwitz's theorem (complex analysis)]]<br /> * [[Rouché's theorem]]<br /> * [[Filter design]]<br /> * [[Nyquist stability criterion]] in [[control theory]]<br /> * [[Marden's theorem]]<br /> * [[Sendov's conjecture]]<br /> * [[Gauss–Lucas theorem]]<br /> <br /> ==References==<br /> *{{cite book |last=Conway |first=John |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=0-387-90328-3}}<br /> *{{cite book |last=Conway |first=John |title=Functions of One Complex Variable II |year=1995 |publisher=Springer |isbn=0-387-94460-5}}<br /> <br /> == External links ==<br /> * {{MathWorld | urlname= Root | title= Root}}<br /> * [http://math.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html Module for Zeros and Poles by John H. Mathews]<br /> * [http://directory.fsf.org/wiki/ZerSol Free C++ solver for zeros of an analytic function (with C and Fortran bindings) by Ivan B. Ivanov]<br /> [[Category:Complex analysis]]<br /> [[Category:Zero]]</div>

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