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Questions tagged [lagrange-inversion]

Use of the Lagrange–Bürmann formula, which gives the Taylor series expansion of the inverse function of an analytic function.

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Is the solution to the functional equation $\widehat{F}(z) = z\widehat{G}(\widehat{F}(z))$ unique?

I am reading Martin Aigner's A Course in Enumeration, and in $\S$3.3 The Exponential Formula, the author states and proves the following theorem (on page 117): Theorem 3.8. Suppose $F(z) = zG(F(z))$...
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Finding generating function and coefficient with symbolic method and Lagrange

I would appreciate any help on the following two problem: Given the symbolic equation $\mathcal{T}=\mathcal{Q}^{[\ast]}$ and $\mathcal{Q}=Δ∗\mathcal{T}∗\mathcal{T}$, I am trying to build the ...
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111 views

What is the compositional inverse of Riemann zeta function near $s=0$?

This question is related to my question here, I have used The Riemann-Siegle theta function particulary its taylor series arround $t=0$ to check wether Riemann zeta function has an explicite inversion ...
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77 views

How many ways to split a convex polygon to squares?

If $a_0 = 0$ and $a_1 = 1$, and $a_n$ stands for the number of ways to split a convex polygon with $n+1$ angles to squares, is given by $$a_n = \sum_{k+l+m = n }a_ka_la_m$$ Now, using generating ...
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63 views

Proving $\int_0^{\infty}f(x) $ converges using Lagrange

Let $f:[0,∞) \to \Bbb R$ be a differentiable such that $f(x) > 0$ for every $x \in [0, \infty)$ and a positive function. Assume there exits an $0 < L < \infty$ such that $$\lim_{x\to \infty}\,...
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76 views

Unexpected (incorrect) solution to Lagrange Inversion solution to $x^4 - x^3 - x^2 - x - 1 = 0$ about the solution near $x = 2$

I am developing generalized hypergeometric solutions for a set of such polynomials. With this example we can write $x^4 - x^3 - x^2 - x - 1 = \frac{x^5 - 2 x^4 + 1}{x - 1}$. Lagrange Inversion ...
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Inverting a function from asymptotic expansion

Can I invert the following functions to obtain $r(\rho)$? $\rho=r+a+b r^{q}$, where $q<0$ $\rho=cr+dr^{q}$, where $q>0$ $\rho=r+\frac{b_{0}}{2}\left(-1+\ln[2]-\ln[b_{0}/2]-\ln[1/r]\right)$ ...
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63 views

Lagrange inversion formula example unclear

The following example is from De Bruijn's Asymptotic methods in analysis (page 24). The considered equation is $x^t = e^{-x}$ The author wants to transform the equation into the form: $w=z/f(z)$, ...
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Does anyone know of a good expression for this Maclaurin series?

I'm looking for a usable (something I can program) closed form or recursive formula for the coefficients $a_n$ of this series for an even-symmetry function of a real variable: $$ \sum\limits_{n=0}^{\...
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1answer
462 views

Lagrange Inversion Theorem Proof

Note: throughout this question, I'll be using the following notation convention: $$f^{(n)}(x)=\frac{d^nf}{dx^n}(x)$$ I was browsing through Wikipedia and even MSE's related questions searching for a ...
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1answer
238 views

Requesting basic explanation of the Lagrange inversion theorem

I'm interested in the Lagrange inversion theorem (https://en.wikipedia.org/wiki/Lagrange_inversion_theorem#Theorem_Statement), despite several attempts on different pdfs, on this site and on wikipedia,...
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1answer
68 views

Asymptotics of the inverse of an analytic equation

What is an asymptotics for the solution of the analytic (transcendental) equation $$e^{-x^2}P\Big(\frac1x \Big)=\frac1y$$ where $P$ is a polynomial, as $y\rightarrow\infty$? Would the Lagrange ...
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139 views

Functions satsify $f'=f^{-1}$ with $f^{-1}$ is compostional inverse of$ f$

let $f$ be a function such that :$f:\mathbb{C}\to \mathbb{C}$ and $f^{-1}$ is the compositional inverse of $f$, I seek for the analyticity of $f$ at $0$, then my question here is : Question: ...
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49 views

Isabelle — Strange proof under Formal_Power_Series

While working towards the Lagrange Inversion theorem in Isabelle to do some formal combinatorics I am following: http://users.math.msu.edu/users/magyar/Math880/Lagrange.pdf I get to Lemma 1, ii . $\...
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103 views

Invert an odd-symmetry function of one variable (with a parameter) using two terms and Lagrange Inversion Theorem.

First of all, I don't have much rep to spend over here, but I gotta lot more at the DSP SE. So if you wanna earn some rep at the DSP SE, please go to my question over there because tomorrow I am ...
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Inversion of so-called probability-generating functional

In this paper here, the authors defined the probability-generating functional for a counting process $N_t$ as \begin{align*} G[u(t)] = \mathbb{E}\left[\exp\left\{\int \log u(t) dN_t\right\}\right] \...
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178 views

Lagrange Inversion - Analytic to Formal Power Series

Suppose one has a proof of the Lagrange Inversion formula in the case for power series with some nonzero radius of convergence (that is, power series which actually describe analytic functions in some ...
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Extending an Analytic Proof for Lagrange Inversion of Formal Power Series

In Appendix A.6 of Sedgewick and Flajolet's Analytic Combinatorics, the authors present a proof of the assertion that if we have formal power series $y(z)$ and $\phi(z)$ (where the constant term of ...
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281 views

Solving $x^p-x-1=0$ with Lagrange Inversion Formula

I am working through the proof that one can solve quintic equations first by reducing the polynomial to one of the form $x^5-x-t$, and then solving $x^5-x-t=0$ using the Lagrange Inversion Formula on ...
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Formula for implicit functions

The Lagrange inversion theorem tells us about the coefficients of the inverse of a function. Is there any other formula for $y$ in terms of $x$ satisfying $G(x,y)=0$?
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Numerical inverse of a function

How to approximate the inverse of the function below? $$f(x) = \frac34 x - \frac 12\sin(2x) + \frac 1{16} \sin(4x)$$ The goal is to get $x$ values (range $[0, \pi]$) from values of $f$. The function ...
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How to find the power series of the inverse of a function?

Question: I wish to find the inverse of the following function: $f(x)=\frac{1}{2}\left(\arctan(x) + \ln\left(\sqrt{\frac{1+x}{1-x}}\right)\right)$ This is the equation for a radial null geodesic in a ...
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204 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function (...
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1answer
2k views

How to deal with this double summation?

I'm stuck with the proof of this result: $$2^n = \sum_{t=-\frac{n-1}{2}}^{\frac{n-1}{2}} \binom{n+1}{\frac{n+1}{2} + t} \sum_{k=\vert t \vert}^{\frac{n-1}{2}} \binom{\frac{n-1}{2}+k}{k} \binom{2k}{k+t}...
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98 views

Lagrange Expansion

I have to show that $$\frac{2}{1+(1+x)^{1/2}} = 1 - \frac{k}1\cdot\frac{x}4 + \frac{k(k+3)}{2!}\cdot\left(\frac{x}4\right)^2-\frac{k(k+4)(k+5)}{3!}\cdot\left(\frac{x}4\right)^3+\ldots$$ I need to ...
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399 views

Generating function of $\binom{3n}{n}$ [duplicate]

Wolfram alpha tells me the ordinary generating function of the sequence $\{\binom{3n}{n}\}$ is given by $$\sum_{n} \binom{3n}{n} x^n = \frac{2\cos[\frac{1}{3}\sin^{-1}(\frac{3\sqrt{3}\sqrt{x}}{2})]}{\...
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1answer
246 views

Proof of lagrange inversion of taylor series

is there a proof for the lagrange inversion of taylor series? The formula is given in http://en.wikipedia.org/wiki/Lagrange_inversion_theorem#Theorem_statement The proof cannot be found in the ...
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563 views

What does the $q$-Catalan Numbers count?

I had completed a paper describing the $q$-Catalan numbers, which is the $q$-analog of the Catalan numbers. The $n$-th Catalan numbers can be represented by: $$C_n=\frac{1}{n+1}{2n \choose n}$$ and ...