Math 2.0

definition

Essential singularity

Definition

Let cc be an isolated singularity of ff. The singularity cc is essential if it is neither a removable singularity nor a pole of ff.

Remarks

The singularity cc is essential if and only if the principal part of ff at cc has infinitely many nonzero terms. Equivalently, f(z)f(z) has no limit in C\mathbb{C} as z→cz \to c, and ∣f(z)∣|f(z)| does not tend to ∞\infty. The standard example is exp(1/z) at 0. The function sin⁡(1/z)\sin(1/z) also has an essential singularity at 00. Near an essential singularity, ff comes close to each complex value, by the Casorati–Weierstrass theorem.

Explanations

L1 human

An essential singularity is a bad point that is neither a removable hole nor a point where the function simply goes to infinity. Near such a point, the function behaves wildly: along different paths it can go to 00, to infinity, or to any other value.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionIsolated singularity, definition, no evidence levelIsolated singularitydefinitionPole, definition, no evidence levelPoledefinitionRemovable singularity, definition, no evidence levelRemovable singularitydefinitionEssential singularity, definition, no evidence levelEssential singularitydefinition
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
  5. Supremum definition
  6. Completeness axiom statement
  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Modulus of a complex number definition
  10. Complex differentiability definition
  11. Holomorphic function definition
  12. Isolated singularity definition
  13. Pole definition
  14. Removable singularity definition
  15. Essential singularity definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Essential_singularity, Formal description
  2. Web https://en.wikipedia.org/wiki/Essential_singularity, Alternative descriptions

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