Laurent series Definition and 162 Threads

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




f
(
z
)
=



n
=








a

n


(
z

c

)

n


,


{\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n},}
where an and c are constants, with an defined by a line integral that generalizes Cauchy's integral formula:





a

n


=


1

2
π
i






γ





f
(
z
)


(
z

c

)

n
+
1






d
z
.


{\displaystyle a_{n}={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-c)^{n+1}}}\,dz.}
The path of integration



γ


{\displaystyle \gamma }
is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which



f
(
z
)


{\displaystyle f(z)}
is holomorphic (analytic). The expansion for



f
(
z
)


{\displaystyle f(z)}
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



γ


{\displaystyle \gamma }
. If we take



γ


{\displaystyle \gamma }
to be a circle




|

z

c

|

=
ϱ


{\displaystyle |z-c|=\varrho }
, where



r
<
ϱ
<
R


{\displaystyle r<\varrho <R}
, this just amounts
to computing the complex Fourier coefficients of the restriction of



f


{\displaystyle f}
to



γ


{\displaystyle \gamma }
. The fact that these
integrals are unchanged by a deformation of the contour



γ


{\displaystyle \gamma }
is an immediate consequence of Green's theorem.
One may also obtain the Laurent series for a complex function f(z) at



z
=



{\displaystyle z=\infty }
. However, this is the same as when



R




{\displaystyle R\rightarrow \infty }
(see the example below).
In practice, the above integral formula may not offer the most practical method for computing the coefficients





a

n




{\displaystyle a_{n}}
for a given function



f
(
z
)


{\displaystyle f(z)}
; 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




f
(
z
)


{\displaystyle f(z)}
in some annulus must actually be the Laurent expansion of



f
(
z
)


{\displaystyle f(z)}
.

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  1. L

    Whats a Laurent series? And how do I use one to represent a function?

    Homework Statement Write TWO laurent series in powers of z that represent the function f(z)= \frac{1}{z(1+z^2)} In certain domains, and specify the domains Homework Equations Well that's my prob, not sure what the terms in the Laurent series are The formula I'm looking at is...
  2. N

    How to find this Laurent series?

    I understand perfectly well how to do Taylor series, but I am foggy on these Laurent series. Say, we have something like, f\left( z \right)\; =\; \frac{1}{z^{2}\cdot \sin \left( z \right)} I think I need to use the taylor series expressions for sin(z) but otherwise, I am not sure what to...
  3. N

    Why are Laurent Series manipulated differently for different regions?

    Say we have the function: \frac{1}{\left( z-1 \right)\left( z+2 \right)} Using partial fractions, \frac{1}{\left( z-1 \right)\left( z+2 \right)}\; =\; \frac{1}{z-2}\; -\; \frac{1}{z\; -\; 1} My question comes in on why and how these equations are manipulted for different regions. Now for a)...
  4. M

    Laurent Series and Singularity Proofs.

    Homework Statement Let D be a subset of C and D is open. Suppose a is in D and f:D\{a} -> C is analytic and injective. Prove the following statements: a) f has in a, a non-essential singularity. b) If f has a pole in a, then it is a pole of order 1. c) If f has a removable singularity...
  5. mattmns

    What is the region of convergence for the Laurent series?

    This is more of a general question than a specific question. ---------- Find the annulus of convergence for the Laurent series \sum_{n=-\infty}^{-1} \left( \frac{z}{2} \right)^n + \sum_{n=0}^\infty \frac{z^n}{n!} -------- I know what to do for the second series, but I am not sure about the...
  6. P

    Integrating e^x /x using Laurent series

    How does one integrate \int_{}^{} \frac{e^x}{x}dx I could expand it using a Laurent series and than integrating term by term but are there more elementary methods?
  7. A

    Algebraic Closure of Laurent Series

    Define: \mathbb{C}((t)) = \{t^{-n_0}\sum_{i=0}^{\infty}a_it^i\ :\ n_0 \in \mathbb{N}, a_i \in \mathbb{C}\} What is its algebraic closure? My notes say that it is "close" to: \bigcup _{m \in \mathbb{N}}\mathbb{C}((t))(t^{1/m}) where \mathbb{C}((t))(t^{1/m}) is the extention of the...
  8. W

    Calculating the Laurent Series of $\frac{1}{e^z-1}$

    Just wondering where to go with this one.. calculate the laurent series of \frac{1}{e^z-1} don't even know where to start on it I know e^z={{\sum^{\infty}}_{j=0}}\frac{z^j}{j!} but not much else...
  9. H

    Laurent series of the function f(z)=1/z^2 for |z-a|>|a|

    Hi all,, I have an Exam tommorow and this question is irritating me...Pls help Laurent series of the function f(z)=1/z^2 for |z-a|>|a| .a is not equal to zero... I am waiting for yours responses...I will be highly thankful to you.
  10. K

    Get Expert Help with Laurent Series for Convergence of e^z/(z-z^2)"

    I need help with a problem from Complex Analysis. The directions say find the Laurent series that converges for 0<|z|<R and determine the precise region of convergence. The expression is : e^z/(z-z^2). I understand how to do the other 7 problems in this section but not this one. Can someone...
  11. S

    Coefficients of a complex laurent series

    My question is about the coefficients of a complex laurent series. As far as I know, there are three kinds of series: those which converge in a finite circular region around the expansion point z0,(aka taylor series), those that converge in a ring shaped region between two circles centered at...
  12. R

    Laurent Series Expansion for Complex Functions

    Does anyone know of any examples of the explicit calculation of the Laurent series of a complex function? Any information would be appreciated.
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