# Laurent series

The Laurent series (after Pierre Alphonse Laurent ) is an infinite series similar to a power series , but with additional negative exponents . In general, a Laurent series with a development point has this shape: ${\ textstyle x}$${\ textstyle c}$

${\ displaystyle f (x) = \ sum _ {n = - \ infty} ^ {\ infty} a_ {n} (xc) ^ {n}}$

Here are the and most complex numbers , but there are other options in the section Formal Laurent series are described below. For complex Laurent series one usually uses the variable instead . ${\ textstyle a_ {n}}$${\ textstyle c}$${\ textstyle z}$${\ textstyle x}$

Summands in which is are usually not written down, so not every Laurent series has to reach infinitely in both directions. Just like it is handled with power series and similar to the representation of breaking decimal fractions , in which formally an infinite number of zeros are behind the last digit. ${\ textstyle a_ {n} = 0}$

The series of terms with negative exponents is called the main part of the Laurent series, the series of terms with nonnegative exponents is called the secondary part or the regular part .

A Laurent series with a vanishing main part is a power series ; if it also has a finite number of terms, then it is a polynomial . If a Laurent series only has a finite number of terms (with negative or positive exponent), then it is called a Laurent polynomial .

The Laurent series was introduced in 1843 by the French mathematician Pierre Alphonse Laurent . However, records in the estate of the German mathematician Karl Weierstrass indicate that he had discovered it as early as 1841.

## Laurent decomposition

The principle of developing a holomorphic function into a Laurent series is based on the Laurent decomposition. To do this, consider a circular ring area . Now define two holomorphic functions and : ${\ displaystyle {\ mathcal {R}} = \ {z \ in \ mathbb {C} \; | \; r <| z | ${\ displaystyle g}$${\ displaystyle h}$

${\ displaystyle g \ colon U_ {R} (0) \ rightarrow \ mathbb {C}}$
${\ displaystyle h \ colon U _ {\ frac {1} {r}} (0) \ rightarrow \ mathbb {C}}$.

That is, the functions and are holomorphic on a circular disk of radius or around the center point. Since the argument of the function must lie within the thus defined circle area, you can quickly see that the function is defined for values . This is also the sum of the two functions ${\ displaystyle g}$${\ displaystyle h}$${\ displaystyle R}$${\ displaystyle 1 / r}$${\ displaystyle h}$${\ displaystyle h (1 / z)}$${\ displaystyle | z |> r}$

${\ displaystyle f (z) = g (z) + h \ left ({\ frac {1} {z}} \ right)}$

analytical on the circular ring . It can be shown that every holomorphic function on a ring area can be decomposed in this way. If one also presupposes, the decomposition is clear. ${\ displaystyle {\ mathcal {R}}}$${\ displaystyle h (0) = 0}$

If you develop this function in the form of power series, the following representation results:

${\ displaystyle f (z) = g (z) + h \ left ({\ frac {1} {z}} \ right) = \ sum _ {n = 0} ^ {\ infty} a_ {n} z ^ {n} + \ sum _ {n = 1} ^ {\ infty} b_ {n} z ^ {- n} = \ sum _ {n = - \ infty} ^ {\ infty} a_ {n} z ^ { n}}$.

It was defined. It also follows from the condition . If one extends these considerations to the development by a point , and not as in the case of the origin, the definition of the Laurent series for a holomorphic function around the development point, as given at the beginning, results : ${\ displaystyle b_ {n} \ equiv a _ {- n}}$${\ displaystyle b_ {0} = 0}$${\ displaystyle h (0) = 0}$${\ displaystyle c}$${\ displaystyle f}$${\ displaystyle c}$

${\ displaystyle f (z) = \ sum _ {n = - \ infty} ^ {\ infty} a_ {n} (zc) ^ {n}}$

## example

In the following, either the real or complex numbers are designated . ${\ textstyle \ mathbb {K}}$

${\ displaystyle f \ colon \ mathbb {K} \ to \ mathbb {K} \ colon x \ mapsto {\ begin {cases} \ exp \ left (- {\ frac {1} {x ^ {2}}} \ right), & x \ neq 0 \\ 0, & {\ text {otherwise}} \ end {cases}}}$.

The function is infinitely often real differentiable , but it is not complex differentiable at this point and even has an essential singularity there . ${\ displaystyle x = 0}$

By inserting into the power series expansion of the exponential function , one obtains the Laurent series of with expansion point : ${\ textstyle - {\ frac {1} {x ^ {2}}}}$${\ textstyle f}$${\ textstyle 0}$

Approach of the Laurentreihen for different to the function .${\ displaystyle n}$${\ displaystyle f}$
${\ displaystyle f (x) = \ sum _ {j = 0} ^ {\ infty} (- 1) ^ {j} {\ frac {x ^ {- 2j}} {j!}} = \ sum _ { j = - \ infty} ^ {- 1} (- 1) ^ {j} {\ frac {x ^ {2j}} {(- j)!}} + 1}$

It converges for every complex number . ${\ textstyle x \ neq 0}$

The picture on the right shows how the partial sum sequence

${\ displaystyle f_ {n} (x) = \ sum _ {j = 0} ^ {n} (- 1) ^ {j} {\ frac {x ^ {- 2j}} {j!}}}$

approximates the function.

## Convergence of Laurent Series

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

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

Be

${\ displaystyle \ sum _ {n = - \ infty} ^ {\ infty} a_ {n} (zc) ^ {n}}$

a Laurent series in with complex coefficients and expansion point . Then there are two uniquely determined numbers and such that: ${\ displaystyle z}$${\ displaystyle a_ {n}}$${\ displaystyle c}$${\ displaystyle r}$${\ displaystyle R}$

• The Laurent series converges normally on the open annulus , i.e. especially absolutely and locally uniformly . This means that the main and secondary parts converge normally. Locally uniform convergence implies uniform convergence on every compact subset of , i.e. in particular on the images of curves in . The Laurent series defines a holomorphic function .${\ displaystyle A: = \ {z: r <\ vert zc \ vert ${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle f}$
• Outside the annulus the Laurent series diverges. This means that for each point in the exterior of , , the series diverges the terms positive or terms with negative exponents.${\ displaystyle A}$${\ displaystyle \ {z: r> \ vert zc \ vert \ vee \ vert zc \ vert> R \}}$
• No general statements can be made on the edge of the annulus, except that there is at least one point on the outer boundary and at least one point on the inner boundary, which cannot be continued holomorphically.${\ displaystyle f}$

It is possible that and is, but it can also be that is. The two radii can be calculated as follows using Cauchy-Hadamard's formula: ${\ displaystyle r = 0}$${\ displaystyle R = \ infty}$${\ displaystyle r = R}$

${\ displaystyle r = \ limsup _ {n \ to \ infty} \ vert a _ {- n} \ vert ^ {1 / n}}$
${\ displaystyle R = {\ frac {1} {\ limsup _ {n \ to \ infty} \ vert a_ {n} \ vert ^ {1 / n}}}}$

One puts and in the second formula. ${\ displaystyle {\ frac {1} {0}} = \ infty}$${\ displaystyle {\ frac {1} {\ infty}} = 0}$

Conversely, one can start with a circular ring and a holomorphic function . Then there always exists a clearly determined Laurent series with a development point , which (at least) converges on and corresponds there with . The following applies to the coefficients ${\ displaystyle A: = \ {z: r <\ vert zc \ vert ${\ displaystyle A}$${\ displaystyle f}$${\ displaystyle c}$${\ displaystyle A}$${\ displaystyle f}$

${\ displaystyle a_ {n} = {\ frac {1} {2 \ pi \ mathrm {i}}} \ oint _ {\ partial U _ {\ varrho} (c)} {\ frac {f (\ zeta)} {\ left (\ zeta -c \ right) ^ {n + 1}}} \ mathrm {d} \ zeta}$

for all and one . Because of Cauchy's integral theorem, the choice of is irrelevant. ${\ displaystyle n \ in \ mathbb {Z}}$${\ displaystyle \ varrho \ in (r, R)}$${\ displaystyle \ varrho}$

The case , i.e. that of a holomorphic function on a perforated circular disk , is particularly important. The coefficient of the Laurent series expansion of is called the residual of in the isolated singularity , it plays a major role in the residual theorem . ${\ displaystyle r = 0}$${\ displaystyle f}$${\ displaystyle c}$${\ displaystyle a _ {- 1}}$${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle c}$

## Formal Laurent range

Formal Laurent series are Laurent series in an indefinite state , whose convergence behavior at a point of development (like the one in the introduction) is (at least initially) not interested. The coefficients can then come from any commutative ring with one element. Usually formal Laurent series are considered with only a finite number of negative exponents, i.e. with a so-called finite main part . ${\ displaystyle X}$${\ displaystyle c}$${\ displaystyle a_ {k}}$ ${\ displaystyle R}$

The formal Laurent series thus correspond to infinite sequences in which only a finite number of coefficients with a negative index differ from zero. The indefinite corresponds to the sequence ${\ displaystyle R ^ {\ mathbb {Z}}}$${\ displaystyle X}$

${\ displaystyle X = \ left (c_ {k} \ right) _ {k \ in \ mathbb {Z}}}$with and for ,${\ displaystyle c_ {1} = 1}$${\ displaystyle c_ {k} = 0}$${\ displaystyle k \ in \ mathbb {Z} \! \ setminus \! \ {1 \}}$

so

 ${\ displaystyle X = (\ dotsc, \, 0, \, 0,}$ ${\ displaystyle 0,}$ ${\ displaystyle 1, \, 0, \, 0, \ dotsc)}$ ${\ displaystyle \ uparrow}$ ${\ displaystyle \ uparrow}$ index 0 1

By definition, two formal Laurent series are equal if and only if they agree in all coefficients. Two Laurent series are added by adding the coefficients with the same index ( i.e. component-wise ) and, because they only have finitely many terms with negative exponents, they can be multiplied by convolution of their coefficient sequences, as is done with power series. With these links, the set of all Laurent series becomes a commutative ring, which is denoted by. ${\ displaystyle R ((X))}$

Is a body , then form the formal power series in the indeterminate over a integral domain , which with is called. Its quotient body is isomorphic to the body of the Laurent series above . ${\ displaystyle K}$${\ displaystyle X}$${\ displaystyle K}$${\ displaystyle K [[X]]}$${\ displaystyle K ((X))}$${\ displaystyle K}$