Location around which a function displays irregular behavior
For essential singularities of real valued functions, see Classification of discontinuities.
Plot of the function exp(1/z), centered on the essential singularity at z = 0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).Model illustrating essential singularity of a complex function 6w = exp(1/(6z))
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior.
The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some[who?] include non-isolated singularities too; those do not have a residue.
and 25 Related for: Essential singularity information
essentialsingularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essentialsingularity is...
Look up Singularity or singularity in Wiktionary, the free dictionary. Singularity or singular point may refer to: Mathematical singularity, a point at...
The technological singularity—or simply the singularity—is a hypothetical future point in time at which technological growth becomes uncontrollable and...
Picard's Theorem: If an analytic function f {\textstyle f} has an essentialsingularity 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 ⊂...
{\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 essentialsingularity at...
mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essentialsingularity, removable...
which a multiple-valued function has nontrivial monodromy and an essentialsingularity. In geometric function theory, unqualified use of the term branch...
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...
not exist, there is an essentialsingularity there. If it is 0 then it is either analytic there or there is a removable singularity. If it is equal to infinity...
highest term; on the other hand, if f {\displaystyle f} has an essentialsingularity at c {\displaystyle c} , the principal part is an infinite sum (meaning...
or discontinuity of the second kind. (This is distinct from an essentialsingularity, which is often used when studying functions of complex variables)...
grows in size, refining the Picard theorem on behaviour close to an essentialsingularity. The theory exists for analytic functions (and meromorphic functions)...
of the features of holomorphic functions. Other functions have essentialsingularities, such as sin(1/z) at z = 0. Complex numbers have applications in...
complex infinity on the Riemann sphere the zeta function has an essentialsingularity. For sums involving the zeta function at integer and half-integer...
0 {\displaystyle 0} , then f ( z ) {\displaystyle f(z)} has an essentialsingularity at a {\displaystyle a} if and only if the principal part is an infinite...
Weierstrass–Casorati theorem describes the behavior of holomorphic functions near essentialsingularities The Weierstrass preparation theorem describes the behavior of analytic...
f is any function that is holomorphic everywhere except for an essentialsingularity at 0 (for example, e1/z), then ( f , − f ) {\displaystyle (f,-f)}...
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...
Baker domain: these are "domains on which the iterates tend to an essentialsingularity (not possible for polynomials and rational functions)" one example...
bang Stephen W. Hawking (1942–2018) described singularities in general relativity and developed singularity-free models of the big bang; predicted primordial...
functions near essentialsingularities is described by Picard's theorem. Functions that have only poles but no essentialsingularities are called meromorphic...
puncture to two, via the exponential map (which is entire and has an essentialsingularity at infinity, so not defined at infinity, and misses zero and infinity)...
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