What is Laurent expansion: Definition and 22 Discussions
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}(zc)^{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)}{(zc)^{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 zc=\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
Homework Statement
Find a Laurent series of ##f(z)=ze^{1/z}## in powers of ##z1##. Is there an easier way to go about this as this is not a typical expansion I see on textbooks. It seems that my incomplete solution is too complicated. Please help, exam is in two days and I am working on past...
Homework Statement
Use an appropriate Laurent series to find the indicated residue for ##f(z)=\frac{4z6}{z(2z)}## ; ##\operatorname{Res}(f(z),0)##
Homework Equations
n/a
The Attempt at a Solution
Computations are done such that ##0 \lt \vert z\vert \lt 2##...
Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...
Homework Statement
Find Laurent series of $$z^2sin(\frac{1}{1z})$$ at $$0<\lvert z1 \rvert<\infty$$
Homework Equations
sine series expansion.
The Attempt at a Solution
At first, it seems pretty elementary since you can set
w=\frac{1}{z1} and expand at infinity in z, which is 0 in w...
<Moderator's note: moved from a technical forum, so homework template missing>
Hi. I have solved the others but I am really struggling on 22c. I need it to converge for z>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
So, I was doing a question on Laurent series. Part of it asked me to work out the pole of the function:
$$ exp \bigg[\frac{1}{z1}\bigg]$$
The answer is ##1##  since, we can write out a Maclaurin expansion:
(1) $$ exp\bigg[\frac{1}{z1}\bigg] = 1+\frac{1}{z1}+\frac{1}{2!}\frac{1}{(z1)^{2}}...
Homework Statement
Find the residue at z=2 for
$$g(z) = \frac{\psi(z)}{(z+1)(z+2)^3}$$
Homework Equations
$$\psi(z)$$ represents the digamma function, $$\zeta(z)$$ represents the RiemannZetaFunction.
The Attempt at a Solution
I know that:
$$\psi(z+1) = \gamma  \sum_{k=1}^{\infty}...
Homework Statement
the Laurent expansion of f(z)=e1/sin(z) at the isolated singularity z=π
Homework EquationsThe Attempt at a Solution
I tried rewriting 1/sin(z) into exponential form, but it seems have no help for the expansion. Would someone give me some inspirations?
Homework Statement
Hey guys,
So I need a bit of help with this question:
Find three Laurent expansions around the origin, valid in three regions you should specify, for the function
f(z)=\frac{30}{(1+z)(z2)(3+z)}
Homework Equations
None that I know of...just binomial expansion...
Homework Statement
Expand f(z)=\frac{1}{z4} in a laurent series valid for (a) z<4 and (b) z>4
Homework Equations
The formula for laurent expansion...
\sum_{n=∞}^{+∞}a_{n}(zz_{0})^{n}
where
a_{n}=\frac{1}{2\pi i}\oint_c \frac{f(z)}{(zz_{0})^{n+1}}dz
The Attempt at a...
Hi guys, i need your help to go about his question,
Question:
$$\text{Show that the coefficient }C_n \text{in the Laurent expansion of }$$
$$f(z)=(z+\frac{1}{z}) \text{ about z=0 is given by}$$
$$C_n=\frac{1}{2\pi}\int^{2\pi}_0 \cos(2cos(\theta))cos(n\theta)\, d\theta ,n\in\mathbb{z}$$
Here is the question:
Here is a link to the question:
Find Laurent series, please help?  Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Here is the question:
Here is a link to the question:
Find Laurent series for cosz/z centered at z=0?  Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Homework Statement
I'm supposed to find the Laurent expansion of sin z/(z1) at z=1.
The Attempt at a Solution
I thought about expanding the sine as a power series of (z1) but I'm not so sure if that would be correct since the sine is a function of z and not z1.
Homework Statement
Find the Laurent expansion of f(z)= 1/(z^3  6z^2 + 9z) in the annulus z3>3.
Homework Equations
none
The Attempt at a Solution
I've been spending way too long on this problem.. I can't seem to think of a way to manipulate f to use the geometric series, other...
Homework Statement
I need to calculate the residue of a function at infinity. My teacher does this by expanding the function in a laurent expansion and deduces the value from that. That seems much harder than it needs to be. For example, in the notes he calculates the residue at infinity...
Homework Statement
Find all Laurent expansion of the function f(z) = 1/(z(8(z^3)1)) with centre z = 0.
The Attempt at a Solution
I tried to find all the singularities and came up with z = 0, z = 1/2, z = (1/2)exp((n*pi*i)/3)
where n = +2,+4,+6... . But according to the solution n can only...
Homework Statement
Find all possible Laurent expansions centered at 0 for
(z  1) / (z + 1)
Find the Laurent Expansion centerd at z = 1 that converages at z = 1/2 and determine the largest opens et on which
(z  1) / (z + 1) converges
Homework Equations
The Attempt at a...
Homework Statement
Laurent expansion for 2/z+4 1/z+2
I can derive the laurent expansion, but i would like a better understanding, so if anyone could tell me if my understanding is right or wrong,
Homework Equations
I know that the region we want is an annulus.
But i am trying to...
How do I find the Laurent expansion of a function containing the principal branch cut of the nth root?
Example:
f(z)=iz\cdot\sqrt[4]{1\frac{1}{z^{4}}}
Say you have f(z)=\frac{1}{(z+i)^2(zi)^2}
a past exam question asked me to find and classify the residues of this.
i had to factorise it into this form and then i just said there was a double pole at z=+i,z=i
now for 5 marks, this doesn't seem like very much work.
is it possible to...
Homework Statement
Find the Laurent expansion of f(z) = \sin(1\frac{1}{z}) about z = 0, and state the annulus of convergence.Homework Equations
The Attempt at a Solution
I tried doing the regular expansion of sin(z), then applying the binomial expansion on the (11/z)^n terms, but I can't help...