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  2. Essential singularity - Wikipedia

    en.wikipedia.org/wiki/Essential_singularity

    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 ...

  3. Zeros and poles - Wikipedia

    en.wikipedia.org/wiki/Zeros_and_poles

    t. e. In complex analysis (a branch of mathematics), a pole is a 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 (see essential singularity). Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is ...

  4. Isolated singularity - Wikipedia

    en.wikipedia.org/wiki/Isolated_singularity

    In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z0 is an isolated singularity of a function f if there exists an open disk D centered at z0 such that f is holomorphic on D \ {z 0}, that is, on the set obtained from D by taking z0 out.

  5. Removable singularity - Wikipedia

    en.wikipedia.org/wiki/Removable_singularity

    Removable singularity. In complex analysis, a removable singularity of 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, as defined by.

  6. Singularity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Singularity_(mathematics)

    Singularity (mathematics) In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. [1][2][3] For example, the reciprocal function has a singularity at , where the ...

  7. Casorati–Weierstrass theorem - Wikipedia

    en.wikipedia.org/wiki/Casorati–Weierstrass_theorem

    A short proof of the theorem is as follows: Take as given that function f is meromorphic on some punctured neighborhood V \ {z 0}, and that z 0 is an essential singularity. . Assume by way of contradiction that some value b exists that the function can never get close to; that is: assume that there is some complex value b and some ε > 0 such that ‖ f(z) − b ‖ ≥ ε for all z in V at ...

  8. Branch point - Wikipedia

    en.wikipedia.org/wiki/Branch_point

    Suppose that g is a global analytic function defined on a punctured disc around z 0.Then g has a transcendental branch point if z 0 is an essential singularity of g such that analytic continuation of a function element once around some simple closed curve surrounding the point z 0 produces a different function element.

  9. Picard theorem - Wikipedia

    en.wikipedia.org/wiki/Picard_theorem

    Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P1 (C) = C ∪ {∞} denotes the Riemann sphere and f : M \ {w} → P1 (C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f (z) attains all but at most two points of P1 (C) infinitely often.

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