Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Removable singularity
open-in-new
<h2 id="riemanns-theorem">Riemann's theorem</h2> <p><a href="/facts/Bernhard_Riemann/oV2fL0k8">Riemann's</a> theorem on removable singularities is as follows: </p> <p>Theorem— Let D ⊂ C {\displaystyle D\subset \mathbb {C} } be an open subset of the complex plane, a ∈ D {\displaystyle a\in D} a point of D {\displaystyle D} and f {\displaystyle f} a holomorphic function defined on the set D ∖ { a } {\displaystyle D\setminus \{a\}} . The following are equivalent: </p> <ol><li> f {\displaystyle f} is holomorphically extendable over a {\displaystyle a} .</li> <li> f {\displaystyle f} is continuously extendable over a {\displaystyle a} .</li> <li>There exists a <a href="/facts/Neighborhood_(topology)/XHKgrpkp">neighborhood</a> of a {\displaystyle a} on which f {\displaystyle f} is <a href="/facts/Bounded_function/whVjbtuk">bounded</a>.</li> <li> lim z → a ( z − a ) f ( z ) = 0 {\displaystyle \lim _{z\to a}(z-a)f(z)=0} .</li></ol> <p>The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at a {\displaystyle a} is equivalent to it being analytic at a {\displaystyle a} (<a href="/facts/Proof_that_holomorphic_functions_are_analytic/4zLtJPBv">proof</a>), i.e. having a power series representation. Define </p> h ( z ) = { ( z − a ) 2 f ( z ) z ≠ a , 0 z = a . {\displaystyle h(z)={\begin{cases}(z-a)^{2}f(z)&z\neq a,\\0&z=a.\end{cases}}} <p>Clearly, <i>h</i> is holomorphic on D ∖ { a } {\displaystyle D\setminus \{a\}} , and there exists </p> h ′ ( a ) = lim z → a ( z − a ) 2 f ( z ) − 0 z − a = lim z → a ( z − a ) f ( z ) = 0 {\displaystyle h'(a)=\lim _{z\to a}{\frac {(z-a)^{2}f(z)-0}{z-a}}=\lim _{z\to a}(z-a)f(z)=0} <p>by 4, hence <i>h</i> is holomorphic on <i>D</i> and has a <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> about <i>a</i>: </p> h ( z ) = c 0 + c 1 ( z − a ) + c 2 ( z − a ) 2 + c 3 ( z − a ) 3 + ⋯ . {\displaystyle h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots \,.} <p>We have <i>c</i>0 = <i>h</i>(<i>a</i>) = 0 and <i>c</i>1 = <i>h'</i>(<i>a</i>) = 0; therefore </p> h ( z ) = c 2 ( z − a ) 2 + c 3 ( z − a ) 3 + ⋯ . {\displaystyle h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots \,.} <p>Hence, where z ≠ a {\displaystyle z\neq a} , we have: </p> f ( z ) = h ( z ) ( z − a ) 2 = c 2 + c 3 ( z − a ) + ⋯ . {\displaystyle f(z)={\frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+\cdots \,.} <p>However, </p> g ( z ) = c 2 + c 3 ( z − a ) + ⋯ . {\displaystyle g(z)=c_{2}+c_{3}(z-a)+\cdots \,.} <p>is holomorphic on <i>D</i>, thus an extension of f {\displaystyle f} . </p> <h2 id="other-kinds-of-singularities">Other kinds of singularities</h2> <p>Unlike functions of a real variable, holomorphic functions are sufficiently rigid that their isolated singularities can be completely classified. A holomorphic function's singularity is either not really a singularity at all, i.e. a removable singularity, or one of the following two types: </p> <ol><li>In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number m {\displaystyle m} such that lim z → a ( z − a ) m + 1 f ( z ) = 0 {\displaystyle \lim _{z\rightarrow a}(z-a)^{m+1}f(z)=0} . If so, a {\displaystyle a} is called a <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">pole</a> of f {\displaystyle f} and the smallest such m {\displaystyle m} is the order of a {\displaystyle a} . So removable singularities are precisely the <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">poles</a> of order 0. A holomorphic function blows up uniformly near its other poles.</li> <li>If an isolated singularity a {\displaystyle a} of f {\displaystyle f} is neither removable nor a pole, it is called an <a href="/facts/Essential_singularity/KMb1R292">essential singularity</a>. The <a href="/facts/Picard_Theorem/0cVTD10S">Great Picard Theorem</a> shows that such an f {\displaystyle f} maps every punctured open neighborhood U ∖ { a } {\displaystyle U\setminus \{a\}} to the entire complex plane, with the possible exception of at most one point.</li></ol> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Analytic_capacity/CWAoWnlt">Analytic capacity</a></li> <li><a href="/facts/Removable_discontinuity/TGLMsQom">Removable discontinuity</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php/Removable_singular_point">Removable singular point</a> at <a href="http://www.encyclopediaofmath.org/">Encyclopedia of Mathematics</a></li></ul>