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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