Search results
Results from the Health.Zone Content Network
In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
Formal power series. In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms ...
Complex analysis. In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points { ak } k, even if ...
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re (s) > 1 and a ≠ 0, −1, −2, … by. This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ ...
Laurent series definition. 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.
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} ".
Puiseux series were first introduced by Isaac Newton in 1676 [1] and rediscovered by Victor Puiseux in 1850. [2] The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator n, a Puiseux series becomes a Laurent series in an n th root of the indeterminate.
Pierre Alphonse Laurent. Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an expansion of a function into an infinite power series, generalizing the Taylor series expansion. He was born in Paris, France.