The Laurent-Series is a generalisation of the power series which allows negative indices and is essential for investigating the behaviour of functions near poles.

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Laurent Series expansion without geometric series

There are several functions in complex analysis which I have not been able to get the Laurent expansion for, both of which are very different from the examples I see online and in the (4) textbooks I ...
4
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3answers
95 views

If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a polynomial

I'm learning about complex analysis and need some help with this problem: If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a ...
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4answers
58 views

Laurent series of $\dfrac{1}{\sin(\frac{1}{z})}$ [on hold]

What is the Laurent series for the function $f(z)=\dfrac{1}{\sin(\frac{1}{z})}$ at the point $z=\dfrac{1}{\pi}$ and $z=0$?
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2answers
59 views

Laurent series expansion of $f$

Find the Laurent series expansion of $f(z)=\dfrac{1}{2z^2-13z+15}$ about the annulus $\dfrac{3}{2}<|z|<5$. I did like this : $f(z)=\dfrac{2}{7}(\dfrac{3}{3-2z}-\dfrac{1}{z-5})$ Then I took ...
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0answers
17 views

Verification on classification of singularities

In an exercise, I'm asked to classify the singularities of these functions: $\qquad i) f(z)=\frac{1}{(z-1)^2} \qquad ii)f(z)=\frac{1-\cos z}{z^2} \qquad iii) f(z)=\frac{z^2-1}{z-1}$ I don't know why ...
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2answers
50 views

Then find the value of $\frac{a_1}{a_2}$

Let $\sum _{n=-\infty}^\infty a_nz^n $ be the Laurent series in the expansion of $f(z)=\dfrac{1}{2z^2-13z+15}$ in the annulus $\frac{3}{2}<|z|<5$ . Then find the value of ...
1
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1answer
7 views

Laurent series of the complex exponential in some annulus around $z$ not zero

How do I find the Laurent series of the exponent of $-z$ around e.g. $z=-1$, in the annulus $|z+1|>0$? Would it just be the Taylor series of the exponent of $-z$ around $-1$ since it converges for ...
3
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2answers
49 views

Finding Laurent's series of a function

I am trying express the function $$f(z)=\frac{z^3+2}{(z-1)(z-2)}$$ like a Laurent's series in each ring centering in $0$, but I do not now how could I express it, in first I said that ...
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3answers
47 views

How to find Laurent expansion

I have been presented with the function $g(z) = \frac{2z}{z^2 + z^3}$ and asked to find the Laurent expansion around the point $z=0$. I split the function into partial fractions to obtain $g(z) = ...
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1answer
17 views

Multivariable Laurent Series

Is it possible to Laurent Expand over two complex variables? for example $\frac{w+\tilde{w}}{(w\tilde{w})^{3}}$ where $w=i\sqrt{2}z+\hat{d}x+i\hat{e}y$ and $\tilde{w}=i\sqrt{2}z-\hat{d}x-i\hat{e}y$ ...
4
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0answers
46 views

Find the value of $a_{-2}$ in the Laurent series expansion of $\sin{(\frac{z}{z+1})}$

If $\sum_{-\infty}^{\infty}a_n (z+1)^n$ is the Laurent series expansion of $f(z)=\sin{(\frac{z}{z+1})}$, then find the value of $a_{-2}$. My work: So we are asked to find the laurent series ...
3
votes
2answers
64 views

Laurent expansion of $\frac{1}{z^2}$

I need to find a Laurent expansion of $\frac{1}{z^2}$ with centre in $z_0 = 1$ and $P(1, 2014, 2015)$. If it was $\frac{1}{z}$, I'd rewrite the fraction like this: $$ \frac{1}{(z-1) +1 } $$ But ...
0
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1answer
23 views

coefficients of Laurent series of rational function

Let $F(z)$ be a rational function $\frac{P(z)}{Q(z)}$ such that the degree of $P(z)$ is less than the degree of $Q(z)$ and suppose that all the zeros of $Q(z)$ are contained in the open disk $|z| < ...
0
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2answers
38 views

The prove of the Laurent series formula

I am trying to understand why at Laurent series we get that . So, I need to understand why this formula is correct: . Could someone help?
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1answer
59 views

How do I find the laurent series for this function?

The function is: $$f(z)=\frac{1-2z}{z^4-1}$$ With the center of $$z=1$$ How can I find the laurent series?
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2answers
37 views

Laurent Series computation

Need to compute the Laurent Series: $$f(z) = \frac{1}{1+z^2} \quad on \quad 0<|z-i|<2$$ $$f(z) = \frac{1}{2i(z-i)} + \frac{i}{2(z+i)}$$ $$\frac{1}{2i(z-i)} = \frac{1}{2zi}\sum_{k=0}^\infty ...
14
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1answer
457 views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
2
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1answer
45 views

Laurent Series on a square annulus

I am trying to prove the following problem: [Source, page 2 problem 3] Let $S_6$ and $S_7$ be the open squares centered at the origin of side length 6 and 7, respectively. Let $\Gamma_6$ and ...
2
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1answer
37 views

Laurent expansion of $\operatorname{sech}(z)$ centred at $\pi i/2$

I have found that the roots of the $\cosh(z)=0$ occur at $\frac{(2k+1)\pi i}{2}$ where $k \in \mathbb{N}\cup{0}$. But I want to find the order the poles of $\operatorname{sech}(z)$ so I'm trying to ...
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0answers
23 views

Taylor and Laurent equal in analytic domain: $\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}=\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}$?

If I have a taylor series around zero that looks like this: $\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}$ Can I claim that this is equal to the first half of the Laurent series: ...
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1answer
29 views

Laurent series in various regions

I have the question: "Find the Laurent series which represents the function $$ f(z) = (z^2 - 1)/(z + 2)(z + 3)\ $$ in the regions (i) $\mid z\mid < 2\ $ (ii) $ 2 < \mid z\mid < 3\ $ ...
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0answers
29 views

Is the integral of any even complex function equal to $0$ on any contour?

Is it true that: $\oint _{C(5i+1,8\sqrt3)} \frac {z}{sh(z)} dz = \oint _{C(i,\sqrt{10})} \frac {z^2}{(1-cos2z)^4}dz = \oint _{C(\pi + i,4)} \frac {z}{tan(z)} dz = 0$ The problem is that i lost my ...
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0answers
36 views

I found a harmonic function from a convergent Laurent series; is this harmonic function unique?

I am guessing that it is simply, "yes", since the Laurent coefficients are unique. I solved for the coefficients to get the Laurent series, showed that it converges, and then took the real part of ...
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1answer
68 views

Determine poles and residues of contour integral using Laurent series

I want to find the residues of the integral $F = \int_{-\infty}^{\infty} \dfrac{1}{x+(a-ib)} \dfrac{1}{\exp(-x/c)-1} dx$ I know that $x=-(a-ib)$ is a simple pole which contributes a non-zero ...
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1answer
22 views

The $c_0$ term in a Laurent expansion

Is the $c_0$ term part of the principal part of a Laurent series, analytic part (positive powers in $z$) or neither or both? I am working on a problem, where I have to match up some coefficients, ...
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1answer
30 views

The convergence domain of a Laurent Series

If the principal part converges for, say, $|z|>1$, and the analytic part (the positive powers in ($z-z_0$)) converges for $|z|<2$, then does the Laurent series, as a whole, converge in the ...
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1answer
69 views

Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
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1answer
39 views

Finding the order of pole of $f(z)=\frac{\sin z}{z-\pi}$

The problem is Kreyszig 10ed international edition : 16.2 #9. What is the order of the pole at $z=\pi$ of the function $f(z)$ below? $$f(z)=\frac{\sin z}{z-\pi}$$ I thought that it will be a simple ...
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1answer
37 views

complex series expansion for $f(z)=\frac{1}{z-1}$

Expand the function $f(z)=\frac{1}{z-1}$ as as a series around $z_{0}$ in two regions a) $$|z-z_{0}| < |1-z_{0}|$$ b) $$|z-z_{0}| > |1-z_{0}|$$ and find coefficient $a_{n}$ is each case. I ...
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1answer
159 views
+100

An entire function bounded outside a strip which contains the reals is constant

Let $f$ be an entire function, which takes real values on the real axis and has no zeros. Suppose $f$ is bounded for $|\operatorname{Im} z| > a > 0$ where $a>0$. Is $f$ a constant? I would ...
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2answers
28 views

Laurent series expansion of a complex function: $\frac{(z+1)}{z(z-4)^3}$

Find the Laurent series for $\frac{(z+1)}{z(z-4)^3} \in 0 < |z-4| < 4$. I get you have to write the denominator in another way, but what are the intermediate steps?
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1answer
36 views

Laurent Series of $f(z) = \sin {\frac{1}{z}}$ at $z=0$

I am attempting to find the Laurent expansion at $z=0$ for $$f(z) = \sin {\frac{1}{z}}$$ where $z$ is a complex number. I rewrote the function as $$f(z)=\frac{ e^{iz^{-1}} - e^{-iz^{-1}}}{2i}$$ ...
2
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1answer
45 views

Laurent Series for $\csc(z)$

I have to find the Laurent series for $$\csc(z), \qquad |z|>0 $$ but I really don't know how to start. Please, guys.
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1answer
40 views

Find the coefficient of $z$ in the Laureant Series expansion of $\frac{e^z}{z-1}$

Find the coefficient of $z$ in the Laureant Series expansion of $\frac{e^z}{z-1}$ in $\{|z|>1\}$. Ok, so for $|z|>1 \iff |\frac{1}{z}|<1$ I can write $\frac{1}{z-1}=\frac{1}{z}\cdot ...
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0answers
39 views

Calculate $\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh \left ( \pi x \right )}dx$ using contour integration

The contour for the complex integral is the rectangle with vertices at $\left ( R,0 \right ), \left ( R,1 \right ),\left ( -R,1 \right ), \left ( -R,0 \right )$ The closed contour integral is equal to ...
2
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0answers
28 views

Expanding $f\left ( z \right )=\frac{e^{az}}{1+e^{z}}$ about $z= i\pi$

$$f\left ( z \right )=\frac{e^{az}}{1+e^{z}} ,\left ( a\in\left ( 0,1 \right ) \right )$$ The point $z=i\pi$ is one of the nonremovable singularities of this function. In order to expand it about that ...
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1answer
47 views

Is my idea of decomposing a meromorphic function into a sum of Laurent series correct?

We know that complex-analytic functions $f(z)$ agree with their power series representations on their domain of analyticity. If a meromorphic function has simple poles at $z_1, ..., z_m$ and ...
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37 views

Series expansion of $ (1-\frac{1}{x})^x $ at infinity

I'm trying to compute the expansion of $ (1-\frac{1}{x})^x $ at infinity, which is given by WolframAlpha as $$ \frac{1}{e} - \frac{1}{2ex} - \frac{5}{24ex^2} - \frac{5}{48ex^3} - \frac{337}{5760ex^4} ...
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1answer
44 views

Laurent series of $ 1\over (z - i) $

I need help in finding the Laurent series of the following complex function: $$ f(z) = {1\over (z-i)} $$ around $z_0 = i$, over the whole complex plane. The part I'm getting confused with is the ...
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1answer
34 views

Laurent series of a quotient of analytic functions

How would you calculate Laurent series of $$f(z) = \frac{\cosh z - 1}{\sinh z - z} \ ?$$ I am struggling to find its residue actually and I believe there is a simple pole at $z = 0$ (as this is ...
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1answer
26 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
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0answers
11 views

Quick method to find Laurant Series? [duplicate]

Is there a quick way to find the Laurant Series of a function without doing a contour integral. Here is an example of the type of problem I am working on Find the Laurant series of the function ...
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1answer
43 views

Laurent series of $\frac{1}{1+z^2}$ on a given set

I want to calculate the Laurent-series of $f: \mathbb C\setminus \{\pm i \} \to \mathbb C, \, \, f(z) = \frac{1}{1+z^2}$ within the domain $D = \{ z \in \mathbb C : 0 \lt \lvert z-i \rvert \lt 2\}$. ...
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1answer
31 views

Region of convergence of a series of finite terms

Say I want to expand \begin{equation} \frac{z^3}{(z+i)^2} \end{equation} about $z=-i$. The answer is straightforward, I get a few terms \begin{equation} \frac{i}{(z+i)^2} - \frac3{z+i}- 3i +(z+i). ...
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1answer
22 views

constructing a Laurent series

I'm trying to find a Laurent series centered at z=0 for $\dfrac{1}{z^4+z^2}$. I'm not sure how to approach this. A partial fraction decomposition gives $\dfrac{1}{z^2}-\dfrac{1}{z^2+1}$, but then I ...
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0answers
23 views

Mistake in my Laurent Series

Expand the function $$f(z) = \frac{1}{z^2(z-2)}$$ into a Laurent series which converges on a region $0<|z|<R$, and determine the greatest possible value of $R$. We can see that we need a ...
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1answer
40 views

Complex Laurent Series and Contour Integral

Let $f(z) = \sin{(\frac{1}{z})}$, where $z \neq 0$. Find a Laurent Series expansion of $f$ around the annulus $D: 1< |z|<3$. Use the result to find $$\oint \limits_C ...
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0answers
33 views

Find a Laurent Series Expansion for the function $f(z) =\frac{5}{z^2 -z-6} $ in the annulus $2< |z|<3$.

Find a Laurent Series Expansion for the function $$f(z) =\frac{5}{z^2 -z-6} $$ in the annulus $2< |z|<3$. My attempt: Notice that we have to find the Laurent series around the point $z=0$, ...
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1answer
43 views

Generating function of Riemann zeta function

I want to know about the generating function of the Riemann zeta function which is related with the Laurent expansion at $z=0$. $f(z) := \dfrac{d}{dz} \log(\sin\pi z)$ $f(z) = \dfrac{1}{z} -2\sum ...
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1answer
40 views

Field of fractions of formal Laurent series

Let $k$ be a field. Consider the formal power series ring $k[[x, y]]$. Its fraction field is the formal Laurent series $k((x,y)) = \{\sum_{m, n\in \mathbb{Z}} k_{m, n} x^m y^n \mid k_{m, n}\in k​\}$. ...