lacunary - Mathnotes

Definition: Singularity \@{singularity}

A point z0 is called a singularity of a complex function f if f is not analytic at z0, but every neighborhood of z0 contains at least one point at which f is analytic.

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Definition: Point Singularity (also: Isolated Singularity) \@{point-singularity}

A singularity z0 of a complex function f is said to be a point singularity or isolated singularity if there exists a neighborhood of z0 in which z0 is the only singularity of f.

Alternatively, a point singularity of a function f is a complex number z0 such that f is defined in a neighborhood of z0 but not at the point z0 itself.

Classification of Singularities

Given a @Laurent-series expansion of a function with an isolated singularity at z0, we have the possibility of the singularity being removable, a pole, or an essential singularity.

Definition: Removable Singularity (also: removable) \@{removable-singularity}

A removable singularity occurs if all negative powers in the Laurent series are zero and the function can be redefined at the singularity to be analytic.

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Example \@{example-4}

Consider f(z)=z defined in the @punctured-plane. Then, the @origin is an isolated singularity, because f(z) is not defined there, but is defined everywhere else in a neighborhood of the origin. We can easily define f(0)=0, and this will make f(z) analytic at 0, and so we say this singularity is removable.

Definition: Pole \@{pole}

A pole is present if the Laurent series has a finite number of negative power terms. The largest negative exponent (in absolute value) indicates the order of the pole.

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Example \@{example-6}

Consider f(z)=1/z defined in the @punctured-plane. Then, the @origin is an isolated singularity, but in this case, we can't simply define it in a continuous fashion, because f(z) approaches infinity as z→0. Thus, we call this singularity a pole.

Definition: Essential Singularity \@{essential-singularity}

An essential singularity occurs if there are infinitely many negative powers in the Laurent series.

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Example \@{example-8}

Consider f(z)=e1/z, on the @punctured-plane. As z approach 0 on the positive real axis, f(z) goes to infinity, and as z approaches 0 on the negative real axis, f(z) approaches 0. On the imaginary axis, f(z) oscillates wildly, but remains bounded, as z approaches 0. This is neither a removable singularity nor a pole, and we call it an essential singularity.

Theorem \@{theorem-9}

If f(1/z) has a singularity at 0, then f(z) has a singularity at the @point-at-infinity.