|
Regular Part
In mathematics, the regular part of a Laurent series consists of the series of terms with positive powers.. That is, if :f(z) = \sum_^ a_n (z - c)^n, then the regular part of this Laurent series is :\sum_^ a_n (z - c)^n. In contrast, the series of terms with negative powers is the principal part In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function. Laurent series definition The principal part at z=a of a function : f(z) = \sum_^\infty a_k .... References Complex analysis {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) 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. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.. Definition The Laurent series for a complex function f(z) about a point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where a_n and c are constants, with a_n defined by a line integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic (analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Part
In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function. Laurent series definition The principal part at z=a of a function : f(z) = \sum_^\infty a_k (z-a)^k is the portion of the Laurent series consisting of terms with negative degree. That is, : \sum_^\infty a_ (z-a)^ is the principal part of f at a . If the Laurent series has an inner radius of convergence of 0 , then f(z) has an essential singularity at a, if and only if the principal part is an infinite sum. If the inner radius of convergence is not 0, then f(z) may be regular at a despite the Laurent series having an infinite principal part. Other definitions Calculus Consider the difference between the function differential and the actual increment: :\frac=f'(x)+\varepsilon : \Delta y=f'(x)\Delta x +\varepsilon \Delta x = dy+\varepsilon \Delta x The differential ''dy'' is sometimes called the principal (linear) part of the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
