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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 ...
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 ...
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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### 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 ...
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,...
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 ...
### 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: ...