Note that terms where a_{k} = 0 are generally not written, so a specific Laurent series may not appear to go infinitely far to the left or the right. (This is much the same phenomenon as a terminating decimal expansion of a real number, which can actually be thought of as having infinitely many 0 digits to the right.)
A Laurent series with no nonzero terms of negative degree is just a power series. A Laurent series with only a finite number of nonzero terms is called a Laurent polynomial. Finally, a Laurent series with no terms of negative degree and with only a finite number of nonzero terms is a polynomial.
Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.
Consider for instance the function f(x) = e^{1/x²} with f(0)=0. As a real function, it is infintitely often differentiable everywhere; as a complex function however it is not differentiable at x=0. By plugging 1/x² into the series for the exponential function, we obtain its Laurent series which converges and is equal to f(x) for all complex numbers x except at the singularity x=0. The graph opposite shows e^{1/x²} in black and its Laurent approximations
More generally, Laurent series can be used to express holomorphic functions defined on an annulus[?], much as power series are used to express holomorphic functions defined on a disc.
Suppose ∑_{−∞<n<∞} a_{n}(z − c)^{n} is a given Laurent series with complex coefficients a_{n} and a complex center c. Then there exists a unique inner radius r and outer radius R such that:
It is possible that r may be zero or R may be infinite; at the other extreme, it's not necessarily true that r is less than R. These radii can be computed as follows:
Conversely, if we start with an annulus of the form A = {z : r < z − c < R} and a holomorphic function f(z) defined on A, then there always exists a unique Laurent series with center c which converges (at least) on A and represents the function f(z).
As an example, let
However, there are three possible Laurent expansions about z = 0:
The coefficients of the Laurent series can be determined with an integral formula which generalizes Cauchy's integral formula: Pick any rectifiable path[?] γ in A which is closed (has the same beginning and ending points), does not have any selfintersections, and moves around the annulus counterclockwise. Then for n in Z, the coefficient a_{n} of the Laurent series is given by the path integral
The case r = 0, i.e. a holomorphic function f(z) which may be undefined at a single point c, is especially important. The coefficient a_{1} of the Laurent expansion of such a function is called the residue of f(z) at the singularity c; it plays a prominent role in the residue theorem.
For an example of this, consider
Formal Laurent series are Laurent series that are used without regard for their convergence. The coefficients a_{k} may then be taken from any commutative ring K. In this context, one only considers Laurent series where all but finitely many of the negativedegree coefficients are zero. Furthermore, the center c is taken to be zero.
Two such formal Laurent series are equal if and only if their coefficient sequences are equal. The set of all formal Laurent series in the variable x over the coefficent ring K is denoted by K((x)). Two such formal Laurent series may be added by adding the coefficients, and because of the finiteness of the negativedegree coefficients, they may also be multiplied using convolution of the coefficient sequences. With these two operations, K((x)) becomes a commutative ring.
If K is a field, then the formal power series over K form an integral domain K[[x]]. The field of quotients of this integral domain can be identified with K((x)).
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