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