Define Removable singularity and give an example
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removable singularity is a singular point of a function for which it is possible to assign a complex number in such a way that becomes analytic. ... Forexample, the point is a removable singularity in the sinc function , since this function satisfies .
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removable singularity is a singular point of a function for which it is possible to assign a complex number in such a way that becomes analytic. ... Forexample, the point is a removable singularity in the sinc function , since this function satisfies .
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In complex analysis, a removable singularityof a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
For instance, the (unnormalized) sinc function
{\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}}
has a singularity at z = 0. This singularity can be removed by defining f(0) := 1, which is the limit of f as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by f being given an indeterminate form. Taking a power series expansion for {\displaystyle {\frac {\sin(z)}{z}}} around the singular point shows that
{\displaystyle {\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .}
Formally, if {\displaystyle U\subset \mathbb {C} } is an open subset of the complex plane {\displaystyle \mathbb {C} }, {\displaystyle a\in U} a point of {\displaystyle U}, and {\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} } is a holomorphic function, then {\displaystyle a} is called a removable singularity for {\displaystyle f}if there exists a holomorphic function {\displaystyle g:U\rightarrow \mathbb {C} } which coincides with {\displaystyle f} on {\displaystyle U\setminus \{a\}}. We say {\displaystyle f} is holomorphically extendable over {\displaystyle U} if such a {\displaystyle g} exists.
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For instance, the (unnormalized) sinc function
{\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}}
has a singularity at z = 0. This singularity can be removed by defining f(0) := 1, which is the limit of f as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by f being given an indeterminate form. Taking a power series expansion for {\displaystyle {\frac {\sin(z)}{z}}} around the singular point shows that
{\displaystyle {\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .}
Formally, if {\displaystyle U\subset \mathbb {C} } is an open subset of the complex plane {\displaystyle \mathbb {C} }, {\displaystyle a\in U} a point of {\displaystyle U}, and {\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} } is a holomorphic function, then {\displaystyle a} is called a removable singularity for {\displaystyle f}if there exists a holomorphic function {\displaystyle g:U\rightarrow \mathbb {C} } which coincides with {\displaystyle f} on {\displaystyle U\setminus \{a\}}. We say {\displaystyle f} is holomorphically extendable over {\displaystyle U} if such a {\displaystyle g} exists.
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