Laurent series

The Laurent series (after Pierre Alphonse Laurent ) is similar to a power series, but in addition with negative exponents an infinite series. General has a Laurent series in x with development point c this form:

Here, the c and the most complex numbers, but there are also other ways that are described in the section "Formal Laurent Series". For complex Laurent series one usually uses the z variable instead of x.

Summands, where an = 0, are usually not written, so not every Laurent series go in both directions to infinity. ( How it is handled even with power series, and similarly canceled, to represent decimal fractions that actually have an infinite number of zeros after the last digit. )

The number of terms with negative exponents is called the principal part of the Laurent series, the series of terms with non-negative exponents is called the secondary part.

A Laurent series with vanishing main part is a power series, it also has only finitely many terms with non-negative exponents, then it is a polynomial. Has a Laurent series a total of only finitely many terms ( negative or positive exponent), then they are called a Laurent polynomial.

Example

Be,

For is infinitely differentiable, however, for it is differentiable in not complex, it has there even an essential singularity.

Now by substituting in the power series expansion of the exponential function, we obtain the Laurent series of f with expansion point 0:

It converges for every complex number x except for (where the summands are not already defined).

The picture on the right shows how the partial sums

To the function approaches (the curve for f itself).

Convergence of Laurent series

Laurent series are important tools of the theory of functions, especially for the study of functions with isolated singularities.

Laurent series describe complex functions that are holomorphic on an annulus, as power series describe functions that are holomorphic on a circular disk.

Be

A Laurent series with complex coefficients and Development in point. Then there are two uniquely determined numbers and so the following applies:

• The Laurent series converges on the open annulus normal, so in particular absolutely and locally uniformly. So you would think that major and minor part converge normally. Local uniform convergence implies uniform convergence on every compact subset of, ie in particular the images of curves in. The Laurent series defined on a holomorphic function.
• Outside of the annulus diverges Laurent series. That is, for each point in the diverging exterior of, the number of terms with positive or with negative exponent terms.
• On the edge of the annulus can not make general statements, except that there is at least one point on the outer boundary and at least one point on the inner boundary, in which can not be continued holomorphically.

It is possible that and is, but it may also be that is. The two radii can be calculated using the formula of Cauchy -Hadamard as follows:

It is in the second formula.

Conversely, it is with a circle and a start on holomorphic function. Then there always exists a uniquely determined Laurent series with expansion point, the (at least) to converge and there with matches. For the coefficients

For all and. Because of the integral theorem of Cauchy, it does not depend on the choice of.

The case, which is the function of a holomorphic on a perforated disc to be particularly important. The coefficient of the Laurent series expansion of means of residual in the isolated singularity, it plays an important role in residue theorem.

Formal Laurent series

Formal Laurent series are Laurent series in the indeterminates X, which are used without regard to convergence considerations. The coefficients ak can be derived then for any commutative ring K. In this situation, however, it only makes sense to consider Laurent series with only finitely many negative exponents (ie, a so-called finite principal part ), and to define the development point c to 0.

Two such formal Laurent series are by definition equal if and only if they coincide in all coefficients. Two Laurent series are added by adding their corresponding coefficients, and because they are only a finite number of terms with negative exponents, they can be multiplied by convolution of their coefficient sequences ( multiplying them simply from formal, how to do it with power series ). These links are the set of all Laurent series over a commutative ring K a commutative ring K ( (X)) is called.

If K is a field, then form the formal power series in the indeterminate X over K an integrity ring, which is denoted by K [ [X ] ]. Its quotient field is isomorphic to the field K ( (X )) of Laurent series over K.