Are Laurent series unique?

Are Laurent series unique?

Uniqueness. Suppose a function f(z) holomorphic on the annulus r < |z − c| < R has two Laurent series: , where k is an arbitrary integer, and integrate on a path γ inside the annulus, Hence the Laurent series is unique.

What is the difference between Taylor series and Laurent series?

A power series with non-negative power terms is called a Taylor series. In complex variable theory, it is common to work with power series with both positive and negative power terms. This type of power series is called a Laurent series.

What is the principal part of Laurent series?

The principal part at of a function. is the portion of the Laurent series consisting of terms with negative degree. That is, is the principal part of at . If the Laurent series has an inner radius of convergence of 0 , then has an essential singularity at , if and only if the principal part is an infinite sum.

Why do we need Laurent series?

The method of Laurent series expansions is an important tool in complex analysis. Where a Taylor series can only be used to describe the analytic part of a function, Laurent series allows us to work around the singularities of a complex function.

What is Laurent’s Theorem?

Proof of Laurent’s theorem f ( z ) = 1 2 π i ∮ C 2 f ( w ) w − z d w − 1 2 π i ∮ C 1 f ( w ) w − z d w . {\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{C_{2}}{\frac {f(w)}{w-z}}\;dw-{\frac {1}{2\pi i}}\oint _{C_{1}}{\frac {f(w)}{w-z}}\;dw.}

What is the point of a Taylor series?

A Taylor series is an idea used in computer science, calculus, chemistry, physics and other kinds of higher-level mathematics. It is a series that is used to create an estimate (guess) of what a function looks like. There is also a special kind of Taylor series called a Maclaurin series.

What are the Latin principal parts?

In Latin grammar: the two principal parts of a noun are the singular nominative and the singular genitive; the three of an adjective are the masculine, the feminine, and the neuter singular nominatives; and the four of a verb are the first-person singular present active indicative, the 1st-pers. sg. perfect act.

What is a principal part of a verb?

: a series of verb forms from which all the other forms of a verb can be derived including in English the infinitive, the past tense, and the present and past participles.

What is the physical significance of Taylor series?

The Taylor series for a function is often useful in physical situations to approximate the value of the function near the expansion point x0. It may be evaluated term-by-term in terms of the derivatives of the function.

What does it mean when the Laurent series converges?

Convergent Laurent series. To say that the Laurent series converges, we mean that both the positive degree power series and the negative degree power series converge. Furthermore, this convergence will be uniform on compact sets. Finally, the convergent series defines a holomorphic function ƒ ( z) on the open annulus.

Where does the path of integration lie in the Laurent series?

The path of integration must lie in an annulus, indicated here by the red color, inside which f ( z) is holomorphic ( analytic ). 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.

How is the Laurent series of a complex function defined?

The Laurent series for a complex function f ( z) about a point c is given by where an and c are constants, with an defined by a line integral that generalizes Cauchy’s integral formula : is holomorphic (analytic). The expansion for will then be valid anywhere inside the annulus.

Can a Laurent series with finitely many negative terms be multiplied?

Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at c, and inner radius of convergence 0, so they both converge on an overlapping annulus. Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms.