essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essential singularity is...

singularity is removable). The point a {\displaystyle a} is an essential singularity of f {\displaystyle f} if it is neither a removable singularity nor...

The technological singularity—or simply the singularity—is a hypothetical point in time at which technological growth becomes completely alien to humans...

Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity at a point w {\textstyle w} , then on any punctured neighborhood...

function, then a {\displaystyle a} is an isolated singularity of f {\displaystyle f} . Every singularity of a meromorphic function on an open subset U ⊂...

highest term; on the other hand, if f {\displaystyle f} has an essential singularity at c {\displaystyle c} , the principal part is an infinite sum (meaning...

on U ∖ { z 0 } {\displaystyle U\setminus \{z_{0}\}} , but has an essential singularity at z 0 {\displaystyle z_{0}}  . The Casorati–Weierstrass theorem...

then f instead has an essential singularity at c. If the limit is 0, then f is either analytic at c or has a removable singularity there. If the limit is...

certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function...

{\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}} has a singularity at z = 0. This singularity can be removed by defining sinc ( 0 ) := 1 , {\displaystyle...

singularity. The function f ( z ) = sin ⁡ 1 z {\displaystyle f(z)=\sin {\frac {1}{z}}} is not meromorphic either, as it has an essential singularity at...

then f instead has an essential singularity at c. If the limit is 0, then f is either analytic at c or has a removable singularity there. If the limit is...

functions near essential singularities is described by Picard's theorem. Functions that have only poles but no essential singularities are called meromorphic...

which a multiple-valued function has nontrivial monodromy and an essential singularity. In geometric function theory, unqualified use of the term branch...

formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued...

grows in size, refining the Picard theorem on behaviour close to an essential singularity. The theory exists for analytic functions (and meromorphic functions)...

complex infinity on the Riemann sphere the zeta function has an essential singularity. For sums involving the zeta function at integer and half-integer...

or discontinuity of the second kind. (This is distinct from an essential singularity, which is often used when studying functions of complex variables)...

formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued...

mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable...

{\displaystyle \mathbb {C} \cup \{\infty \}} . Viewed this way, the only possible singularity for entire functions, defined on C ⊂ C ∪ { ∞ } {\displaystyle \mathbb...

center a {\displaystyle a} to the nearest non-removable singularity; if there are no singularities (i.e., if f {\displaystyle f} is an entire function),...

of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0. Complex numbers have applications in...

f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e1/z), then ( f , − f ) {\displaystyle (f,-f)}...

formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued...

z-z_{Z}}+{g'(z) \over g(z)}.} Since g(zZ) ≠ 0, it follows that g' (z)/g(z) has no singularities at zZ, and thus is analytic at zZ, which implies that the residue of...

often used to try to make models amenable to extension beyond curvature singularities, for example to permit description of the universe even before the Big...

formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued...

harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution...

century, leading to a singularity. Kurzweil elaborates on his views in his books The Age of Spiritual Machines and The Singularity Is Near. In the natural...