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e −1/x 2 and its Laurent approximations (labeled) with the negative degree rising. The neighborhood around the zero singularity can never be approximated. e −1/x 2 and its Laurent approximations. As the negative degree of the Laurent series rises, it approaches the correct function.
Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(z) = 1/2. In a neighbourhood of a point , a nonzero meromorphic function f is the sum of a Laurent series with at most finite principal part (the terms with negative index values):
The partial fraction expansion for a function can also be used to find a Laurent series for it by simply replacing the rational functions in the sum with their Laurent series, which are often not difficult to write in closed form. This can also lead to interesting identities if a Laurent series is already known. Recall that
Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a −1 of (z − c) −1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to ...
Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a −1 of (z − c) −1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to ...
The principal part at = of a function = = ()is the portion of the Laurent series consisting of terms with negative degree. [1] That is, = is the principal part of at .If the Laurent series has an inner radius of convergence of , then () has an essential singularity at if and only if the principal part is an infinite sum.
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 z 0 is an isolated singularity of a function f if there exists an open disk D centered at z 0 such that f is holomorphic on D \ {z 0}, that is, on the set obtained from D by taking z 0 out.
A Laurent polynomial over may be viewed as a Laurent series in which only finitely many coefficients are non-zero. The ring of Laurent polynomials R [ X , X − 1 ] {\displaystyle R\left[X,X^{-1}\right]} is an extension of the polynomial ring R [ X ] {\displaystyle R[X]} obtained by "inverting X {\displaystyle X} ".