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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.^{ [1] }

- Convergent Laurent series
- Uniqueness
- Laurent polynomials
- Principal part
- Multiplication and sum
- See also
- References
- External links

The Laurent series for a complex function *f*(*z*) about a point *c* is given by

where *a*_{n} and *c* are constants, with *a*_{n} defined by a line integral that generalizes Cauchy's integral formula:

The path of integration is counterclockwise around a Jordan curve enclosing *c* and lying in an annulus *A* in which is holomorphic (analytic). The expansion for will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled . If we take to be a circle , where , this just amounts to computing the complex Fourier coefficients of the restriction of to . The fact that these integrals are unchanged by a deformation of the contour is an immediate consequence of Green's theorem.

One may also obtain the Laurent series for a complex function *f*(*z*) at . However, this is the same as when (see the example below).

In practice, the above integral formula may not offer the most practical method for computing the coefficients for a given function ; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is unique whenever it exists, any expression of this form that actually equals the given function in some annulus must actually be the Laurent expansion of .

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 with . As a real function, it is infinitely differentiable everywhere; as a complex function however it is not differentiable at *x* = 0. By replacing *x* with −1/*x*^{2} in the power 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

for *N* = 1, 2, 3, 4, 5, 6, 7 and 50. As *N* → ∞, the approximation becomes exact for all (complex) numbers *x* except at the singularity *x* = 0.

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

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:

- The Laurent series converges on the open annulus
*A*≡ {*z*:*r*< |*z*−*c*| <*R*} . 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*f*(*z*) on the open annulus. - Outside the annulus, the Laurent series diverges. That is, at each point of the exterior of
*A*, the positive degree power series or the negative degree power series diverges. - On the boundary of the annulus, one cannot make a general statement, except to say that there is at least one point on the inner boundary and one point on the outer boundary such that
*f*(*z*) cannot be holomorphically continued to those points.

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:

We take *R* to be infinite when this latter lim sup is zero.

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, consider the following rational function, along with its partial fraction expansion:

This function has singularities at *z* = 1 and *z* = 2*i*, where the denominator of the expression is zero and the expression is therefore undefined. A Taylor series about *z* = 0 (which yields a power series) will only converge in a disc of radius 1, since it "hits" the singularity at 1.

However, there are three possible Laurent expansions about 0, depending on the radius of *z*:

- One series is defined on the inner disc where |
*z*| < 1; it is the same as the Taylor series,This follows from the partial fraction form of the function, along with the formula for the sum of a geometric series, for . - The second series is defined on the middle annulus where 1 < |
*z*| is caught between the two singularities:Here, we use the alternative form of the geometric series summation, for . - The third series is defined on the infinite outer annulus where 2 < |
*z*| < ∞, (which is also the Laurent expansion at )This series can be derived using geometric series as before, or by performing polynomial long division of 1 by (*x*− 1)(*x*− 2i), not stopping with a remainder but continuing into*x*^{−n}terms; indeed, the "outer" Laurent series of a rational function is analogous to the decimal form of a fraction. (The "inner" Taylor series expansion can be obtained similarly, just reversing the term order in the division algorithm.)

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

This function is holomorphic everywhere except at *z* = 0.

To determine the Laurent expansion about *c* = 0, we use our knowledge of the Taylor series of the exponential function:

We find that the residue is 2.

One example for expanding about :

Suppose a function *f*(*z*) holomorphic on the annulus *r* < |*z* − *c*| < *R* has two Laurent series:

Multiply both sides by , where k is an arbitrary integer, and integrate on a path γ inside the annulus,

The series converges uniformly on , where *ε* is a positive number small enough for *γ* to be contained in the constricted closed annulus, so the integration and summation can be interchanged. Substituting the identity

into the summation yields

Hence the Laurent series is unique.

A **Laurent polynomial** is a Laurent series in which only finitely many coefficients are non-zero. Laurent polynomials differ from ordinary polynomials in that they may have terms of negative degree.

The **principal part** of a Laurent series is the series of terms with negative degree, that is

If the principal part of *f* is a finite sum, then *f* has a pole at *c* of order equal to (negative) the degree of the highest term; on the other hand, if *f* has an essential singularity at *c*, the principal part is an infinite sum (meaning it has infinitely many non-zero terms).

If the inner radius of convergence of the Laurent series for *f* is 0, then *f* has an essential singularity at *c* if and only if the principal part is an infinite sum, and has a pole otherwise.

If the inner radius of convergence is positive, *f* may have infinitely many negative terms but still be regular at *c*, as in the example above, in which case it is represented by a *different* Laurent series in a disk about *c*.

Laurent series with only finitely many negative terms are well-behaved—they are a power series divided by , and can be analyzed similarly—while Laurent series with infinitely many negative terms have complicated behavior on the inner circle of convergence.

Laurent series cannot in general be multiplied. Algebraically, the expression for the terms of the product may involve infinite sums which need not converge (one cannot take the convolution of integer sequences). Geometrically, the two Laurent series may have non-overlapping annuli of convergence.

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.

Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.

Also, for a field , by the sum and multiplication defined above, formal Laurent series would form a field which is also the field of fractions of the ring of formal power series.

- Puiseux series
- Mittag-Leffler's theorem
- Formal Laurent series – Laurent series considered
*formally*, with coefficients from an arbitrary commutative ring, without regard for convergence, and with only*finitely*many negative terms, so that multiplication is always defined. - Z-transform – the special case where the Laurent series is taken about zero has much use in time-series analysis.
- Fourier series – the substitution transforms a Laurent series into a Fourier series, or conversely. This is used in the
*q*-series expansion of the*j*-invariant. - Padé approximant – Another technique used when a Taylor series is not viable.

In complex analysis, an **entire function**, also called an **integral function,** is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function *f*(*z*) has a root at *w*, then *f*(*z*)/(*z−w*), taking the limit value at *w*, is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function.

In mathematics, the **gamma function** is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n,

In mathematics, a **series** is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

The **Riemann zeta function** or **Euler–Riemann zeta function**, *ζ*(*s*), is a mathematical function of a complex variable *s*, and can be expressed as:

In mathematics, a **power series** is an infinite series of the form

In mathematics, an **analytic function** is a function that is locally given by a convergent power series. There exist both **real analytic functions** and **complex analytic functions**. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about *x*_{0} converges to the function in some neighborhood for every *x*_{0} in its domain.

In mathematics, more specifically complex analysis, the **residue** is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

In complex analysis, a branch of mathematics, **analytic continuation** is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.

The **Euler–Mascheroni constant** is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

In mathematics, a **modular form** is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

In mathematics, the **digamma function** is defined as the logarithmic derivative of the gamma function:

In complex analysis, a branch of mathematics, **Morera's theorem**, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

In mathematics, the **upper** and **lower incomplete gamma functions** are types of special functions which arise as solutions to various mathematical problems such as certain integrals.

In complex analysis a complex-valued function *ƒ* of a complex variable *z*:

In mathematics, a **divergent series** is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

In mathematics, **infinite-dimensional holomorphy** is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces, typically of infinite dimension. It is one aspect of nonlinear functional analysis.

In mathematics, and particularly in the field of complex analysis, the **Weierstrass factorization theorem** asserts that every entire function can be represented as a product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root.

In mathematics, **holomorphic functional calculus** is functional calculus with holomorphic functions. That is to say, given a holomorphic function *f* of a complex argument *z* and an operator *T*, the aim is to construct an operator, *f*(*T*), which naturally extends the function *f* from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of *T* to the bounded operators.

In mathematics, **Maass forms** or **Maass wave forms** are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are Eigenforms of the hyperbolic Laplace Operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.

In mathematics, a **bilateral hypergeometric series** is a series Σ*a*_{n} summed over *all* integers *n*, and such that the ratio

- ↑ Rodriguez, Rubi; Kra, Irwin; Gilman, Jane P. (2012),
*Complex Analysis: In the Spirit of Lipman Bers*, Graduate Texts in Mathematics,**245**, Springer, p. 12, ISBN 9781441973238 .

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