What is Laurent series: Definition and 162 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
Hi, I'm writting because I sort of had an idea that looks that it should work but, I did not find any paper talking about it. I was thinking about approximating something like algebraic functions. That is to say, a function of a complex variable z,(probably multivalued) that obeys something...
First series
\frac{1}{2}\sum^{\infty}_{n=0}\frac{(1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}\frac{1}{2p^4}+\frac{1}{3p^6}\frac{1}{4p^8}+...)
whereas second one is...
(a)
i tried to decompose the fracion as a sum of fractions of form ##\frac{1}{1g}##
$$f=\frac{z}{(1+z)(2z)}=\frac{a}{1+z}+\frac{b}{2z}$$
$$a=\frac{1}{3}, b=\frac{2}{3}$$
$$f=\frac{1}{6}\frac{1}{1+z}\frac{1}{3}\frac{1}{1\frac{z}{2}}$$
$$f=\frac{1}{6}\sum_{n=0}^\infty...
My homework is on mathematical physics and I want to know the concept behind Laurent series. I want to know clearly know the process behind attaining the series representation for the expansion in sigma notation using the formula that can be found on the attached files. There are three questions...
Dear Everyone,
I am wondering how to use the integral formula for a holomorphic function at all points except a point that does not exist in function's analyticity. For instance, Let $f$ be defined as $$f(z)=\frac{z}{e^zi}$$. $f$ is holomorphic everywhere except for $z_n=i\pi/2+2ni\pi$ for...
Not really a homework problem, just an equation from my textbook that I do not understand. I can't think of any way to even begin manipulating the right hand side to make it equal the left hand side.
Just to confirm equality (thanks to another user for suggestion), I multiplied both sides by of...
Homework Statement
Use an appropriate Laurent series to find the indicated residue for ##f(z)=\frac{4z6}{z(2z)}## ; ##\operatorname{Res}(f(z),0)##
Homework Equations
n/a
The Attempt at a Solution
Computations are done such that ##0 \lt \vert z\vert \lt 2##...
Homework Statement
I am looking at the wikipedia proof of uniqueness of laurent series:
https://en.wikipedia.org/wiki/Laurent_seriesHomework Equations
look above or belowThe Attempt at a Solution
I just don't know what the indentity used before the bottom line is, I've never seen it before...
Homework Statement
Find Laurent series of $$z^2sin(\frac{1}{1z})$$ at $$0<\lvert z1 \rvert<\infty$$
Homework Equations
sine series expansion.
The Attempt at a Solution
At first, it seems pretty elementary since you can set
w=\frac{1}{z1} and expand at infinity in z, which is 0 in w...
Homework Statement
Homework EquationsThe Attempt at a Solution
[/B]
Hi,
I am trying to understand the 2nd equality .
I thought perhaps it is an expansion of ##(1\frac{z}{w})^{2}## (and then the ##1## cancels with the ##1## in ##( (1\frac{z}{w})^{2}) 1 ) ##) in the form ##(1x)^{2}##...
Homework Statement
expand f(z)=\frac{1}{z(z1)} in a laurent series valid for the given annular domain.
z> 3
Homework EquationsThe Attempt at a Solution
first I do partial fractions to get
\frac{1}{3z} +\frac{1}{3(z3)}
then in the second fraction I factor out a z in the denominator...
Homework Statement
Hi
I am trying to understand this http://math.stackexchange.com/questions/341406/howdoiobtainthelaurentseriesforfzfrac1cosz41about0
So the long division yields...
<Moderator's note: moved from a technical forum, so homework template missing>
Hi. I have solved the others but I am really struggling on 22c. I need it to converge for z>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
Homework Statement
consider ##f## a meromorphic function with a finite pole at ##z=a## of order ##m##.
Thus ##f(z)## has a laurent expansion: ##f(z)=\sum\limits_{n=m}^{\infty} a_{n} (za)^{n} ##
I want to show that ##f'(z)'/f(z)= \frac{m}{za} + holomorphic function ##
And so where a...
hi, I try to calculate the integral
$$\int_{0}^{1}log(\Gamma (x))dx$$
and the last step To solve the problem is:
$$1 \frac{\gamma }{2} + \lim_{n\rightarrow \infty } \frac{H_{n}}{2} + n + log(\Gamma (n+1))  (n+1)(log(n+1))$$
and wolfram alpha tells me something about series expansion at...
Question 1:
Find the Laurent series of \cos{\frac{1}{z}} at the singularity z = 0.
The answer is often given as,
\cos\frac{1}{z} = 1  \frac{1}{2z^2} + \frac{1}{24z^4}  ...
Which is the MacLaurin series for \cos{u} with u = \frac{1}{z}. The MacLaurin series is the Taylor series when u_0 = 0...
Homework Statement
Expand the function f(z)=1/z(z2) in a Laurent series valid for the annual region 0<z3<1
Homework Equations
I know 1/z(z+1) = 0.5(1/(z2))  0.5(1/z)
Taylor for 0.5(1/(z2)) is : ∑(((1)k/2) * (z3)k) (k is from 0 to ∞)For the second 0.5(1/z) the answer is a...
Homework Statement
Classify the singularities of
##\frac{1}{z^{1/4}(1+z)}##
Find the Laurent series for
##\frac{1}{z^21}## around z=1 and z=1
Homework EquationsThe Attempt at a Solution
So for the first bit there exists a singularity at ##z=0##, but I'm confused about the order of this...
Homework Statement
Classify the singularities of ##\frac{1}{z^2sinh(z)}## and describe the behaviour as z goes to infinity
Find the Laurent series of the above and find the region of convergence
Homework Equations
N/A
The Attempt at a Solution
I thought these two were essentially the same...
Homework Statement
Cassify the singularities of e^\frac{1}{z} and find the Laurent series
Homework Equations
e^\frac{1}{x} =\sum \frac{(\frac{1}{x})^n}{n!}
The Attempt at a Solution
Theres a singularity at z=0, but I need to find the order of the pole
So using the general expression for the...
Homework Statement
Hey guys, I'm just going through a Laurent series example and I'm having trouble understanding how they switched the index on a summation from n=0 to n=1 and then switched the argument from z^(n1) to z^n as well as changing the upper limit to infinity. If anyone could shed...
Homework Statement
I need to find the Laurent Series of Cos[\frac{1}{z}] at z=0
Homework Equations
None
The Attempt at a Solution
I've gone through a lot of these problems and this is one of the last on the problem set. With all the other trig functions it's been just computing their...
Homework Statement
Find the Laurent expansions of
##f(z) = \frac{z+2}{z^2z2}## in ##1 < z<2## and then in ##2 < z< \infty##
in powers of ##z## and ##1/z##.
Homework Equations
Theorem:
Let ##f## be a rational function all of whose poles ##z_1,\dots , z_N## in the plane have order one and...
Homework Statement
Find four terns of the Laurent series for the given function about ##z_0=0##. Also, give the residue of the function at the point.
a) ##\frac{1}{e^z1}##
b) ##\frac{1}{1\cos z}##
Homework Equations
The residue of the function at ##z_0## is coefficient before the...
Problem A now solved!
Problem B:
I am working with two equations:
The first gives me the coefficients for the Laurent Series expansion of a complex function, which is:
f(z) = \sum_{n=\infty}^\infty a_n(zz_0)^n
This first equation for the coefficients is:
a_n = \frac{1}{2πi} \oint...
Homework Statement
Find the Laurent Series of f(z) = \frac{1}{z(z2)^3} about the singularities z=0 and z=2 (separately).
Verify z=0 is a pole of order 1, and z=2 is a pole of order 3.
Find residue of f(z) at each pole.
Homework Equations
The solution starts by parentheses in the form (1 ...
Please help me with this Laurent series example for $\frac{1}{z(z+2)}$ in the region 1 < z1 < 3
Let w = z1, then $ f(z) = \frac{1}{(w+1)(w+3)}=\frac{1}{2} \left[ \frac{1}{w+1}\frac{1}{w+3} \right]$
I get $ \frac{1}{1(w)} = \sum_{n=0}^{\infty}(1)^n w^n, \:for\: w<1;$
$ = ...
Homework Statement
For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin.
The function is...
1/(z*(z1)(z2)^2)
Homework...
Blundering on, this problem will help me confirm what I think I know ...
Find the Laurent series for $ f(z) = \frac{1}{z(z1)(z2)} = \frac{1}{2z}+\frac{1}{1z} \frac{1}{4}\frac{1}{1\frac{z}{2}} $
I found this definition of the LS: $ f(z) = \sum_{\infty}^{+\infty}{a}_{n}(z{z}_{0})^n =...
Hi  I admit to struggling a little with my 1st exposure to complex analysis and Laurent series in particular, so thought I'd try some exercises; always seem to help my understanding.
A function f(z) expanded in Laurent series exhibits a pole of order m at z=z0. Show that the coefficient of $...
My book is a little confusing sometimes, and googling doesn't always help. Just a couple of queries  and please add any of your own 'tips & tricks'...
1) Laurent series (LS) is defined from $ \infty $, yet all the examples I have seen start from 0  I can't think of an annulus with a...
Homework Statement
Evaluate the integral using any method:
∫C (z10) / (z  (1/2))(z10 + 2), where C : z = 1
Homework Equations
∫C f(z) dz = 2πi*(Σki=1 Resp_i f(z)
The Attempt at a Solution
Rewrote the function as (1/(z(1/2)))*(1/(1+(2/z^10))). Not sure if Laurent series expansion is the...
Homework Statement
Find the Laurent series expansion of f(z) = \log\left(1+\frac{1}{z1}\right) in powers of \left(z1\right).
Homework Equations
The function has a singularity at z = 1, and the nearest other singularity is at z = 0 (where the Log function diverges). So in theory there should...
Homework Statement
Determine Laurent Series around z = 1, z = 2, z=0
Determine pole and residue in each case, and solve series in each separate region of C.
f_1(z) = \frac {z}{(z+1)(z2)} Homework EquationsThe Attempt at a Solution
I've determined my partial fractions as
\frac{1}{3}...
Hello.
Can you check if my answer is correct please?
For the region ${\{z\inℂ\big0<z<1\}}$, expand $\frac{1}{z^3z^4}$ that has a center z=0 into Laurent series.
My solution:
$$\frac{1}{z^3(1z)}=\frac{1}{z^3}\sum_{n=0}^{\infty}z^n=\sum_{n=0}^{\infty}z^{n3}$$
Hello.
I am stuck on this question.
Let {##z\in ℂ0<z+i<2##}, expand ##\frac{1}{z^2+1}## where its center ##z=i## into Laurent series.
This is how I start off:
$$\frac{1}{(z+i)(zi)}$$
And then I don't know what to do next. I guess geometric series could be applied later but I...
Hello.
I am stuck on this question.
Let {$z\in ℂ0<z+i<2$}, expand $\frac{1}{z^2+1}$ where its center $z=i$ into Laurent series.
I have no idea how to start.
I guess geometric series could be applied later but I don't know how to start.
Hello.
I need explanation about this Laurent series.
The question is:
Let {##z\inℂ0<z##}, expand ##\frac{e^{z^2}}{z^3}## where the centre z=0 into Laurent series.
And the solution is...
Hello.
I need explanation about this Laurent series.
The question is:
Let {$z\inℂ0<z$}, expand $\frac{e^{z^2}}{z^3}$ where the centre z=0 into Laurent series.
And the solution is...
expand e^{\frac{z}{z2}} in a Laurent series about z=2
I cannot start this.
my attempt so far has been
e^\frac{z}{z2}=1 + \frac{z}{z2} + \frac{z^2}{(z2)^2 2!} + \frac{z^3}{(z2)^3 3!}
This is unlike the other problems I have worked. Seems I need to manipulate this equation some way...
Hi guys,
well i have the problem below,
$$\int_{\gamma(0;1)}\frac{1}{\exp(iz)1}\mathrm{d}z$$
so it is holormorphic in D'(0,1) as it has a point not holomorphic at z=0.
Taking a Laurent Series in the form $$f(z)=\sum_{n=\infty}^{\infty}C_n(z0)^n$$
But i wil get...
Homework Statement
a)Find a harmonic function ##u## on the annulus ##1< z < 2## taking the value 2 in the circle ##z=2## and the value 1 in the circle ##z=1##.
b)Determine all the isolated singularities of the function ##f(z) = \frac{z+1}{z^3+4z^2+5z+2}## and determine the residue at...
Homework Statement
f = \frac{1}{z(z1)(z2)}
Homework Equations
Partial fraction
The Attempt at a Solution
R1 = 0 < z < 1
R2 = 1 < z < 2
R3 = z > 2
f = \frac{1}{z(z1)(z2)} = \frac{1}{z} * (\frac{A}{z1} + \frac{B}{z2})
Where A = 1 , B = 1.
f = \frac{1}{z} *...
Homework Statement
Let f(z) = \frac{1}{z^21}. Find Laurent Series valid for the following regions.
• 0<z−1<2
• 2<z−1<∞
• 0<z<1
Homework Equations
\frac{1}{1z}=\sum^{\infty}_{n=0}z^n,\: z<1
f(z)=\sum^{\infty}_{n=0}a_n(zz_0)^n+\sum^{\infty}_{n=1}b_n(zz_0)^{n}
The Attempt at a...
Okay so the partial fraction decomposition theorem is that if f(z) is a rational function, f(z)=sum of the principal parts of a laurent expansion of f(z) about each root.
I'm working through an example in my book, I am fine to follow it. (method 1 below)
But instinctively , I would have...
Homework Statement
Expand ##f(z)=\frac{1}{z^2(z1)}## in Laurent series for ##0<z1<1##. Use binomial series.
Homework Equations
The Attempt at a Solution
I am looking at this problem for quite some time now and still I got nothing.
I do however think that this will come in...
Homework Statement
Find and determine the type of singularity points for ##f(z)=\frac{\sin(3z)3z}{z^5}##. Also calculate the regular and main part of Laurent series around those points.Homework Equations
The Attempt at a Solution
I am already having troubles with the first part.
Singularity...
So the question I got the represention for both partial fractions after I broke the functions into two partial fraction one I got as 1/3(z + 1) + 2/3(z  2) and I got laurent series represention for both but I was wondering for z < 1 how can they both converge for z < 1 are we acctually...
Homework Statement
Hi! I need to find the laurent series of ##e^{1/(1z)}## to get the residue at ##z=1##. Can somebody help me?
The Attempt at a Solution
https://scontentaams.xx.fbcdn.net/hphotosfrc3/q71/s720x720/1461607_10201796752217165_1002449331_n.jpg
I tried using the taylor series...