The behavior of holomorphic functions near their essential singularities is described by the Casorati Weierstrass theorem and by the considerably stronger Picard's great theorem.
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This plot shows how approaching the essential singularity from different directions yields different behaviors ( as opposed to a pole, which would be uniformly white ).
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And, would the same method prove that e ^ { 1 / z }, or any such function, has an essential singularity at 0.
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However the lines do usually have well determined directions at essential singularities of the function, and there is sometimes a natural choice of these lines as follows.
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In complex analysis, an "'essential singularity "'of a function is a " severe " singularity near which the function exhibits odd behavior.
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This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity.
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The function f ( z ) can have singularities in the complex plane ( branch point singularities, poles or essential singularities ), which limit the radius of convergence of the series.
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In complex analysis, a point z \ in \ mathbb { C } where a holomorphic function is undefined is called a essential singularities, where no meromorphic extension to z exists.
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Indeed, what we have here is an essential singularity, which I know can be a weird phenomenon, but do not understand it completely myself or have done such experimentation with.
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With this generalization, " Little Picard Theorem " follows from " Great Picard Theorem " because an entire function is either a polynomial or it has an essential singularity at infinity.
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