Question

How to find Removable Singularity?

Answer 1

Hint: In the question given above, a removable singularity is a function's singular point for which a complex number can be assigned in such a way that the function becomes analytic. A removable singularity can also be defined as a singularity of a function about which the function is bounded.

Complete step-by-step answer:
The concept "removable singularity" is associated with the field of complex analysis, as well as all forms of holomorphic functions.
In simple terms, removable singularities are points on a function's graph where the holomorphic function is still undefined; as a result, we can always redefine the function in such a way that the function becomes normal around a certain neighborhood of the point that makes the function undefined.

Note: Unlike real-valued functions, holomorphic functions are so rigid that their isolated singularities can be classified absolutely. The singularity of a holomorphic function is either not really a singularity at all, i.e. a reversible singularity, or one of the two forms below:
According to Riemann's theorem, given a non-removable singularity, there exists a natural number m such that \[\mathop {\lim }\limits_{z \to a} {\left( {z - a} \right)^{m + 1}}f\left( z \right) = 0\],then \[a\] is the pole of \[f\]and the smallest such \[m\] is the order of \[a\]. As a result, removable singularities are the order \[0\] poles. Near its other poles, a holomorphic function expands uniformly.

An essential singularity is defined as an isolated singularity \[a\] of \[f\] that is neither removable nor pole. The Great Picard Theorem demonstrates that such an \[f\] maps any punctured open neighbourhood \[U\backslash \left\{ a \right\}\] to the entire complex plane, with at most one exception.