Questions tagged [lagrange-inversion]
Use of the Lagrange–Bürmann formula, which gives the Taylor series expansion of the inverse function of an analytic function.
65
questions
0
votes
0
answers
30
views
How to prove/disprove these two formulations of Lagrange inversion formula are equivalent
I have once stumbled at comprehending the original form of Lagrange inversion formula:
If $z=f(w)$, then $$
f^{-1}(z)=a+\sum_{n=1}^{\infty}\left(\lim_{w\to a}\frac{d^{n-1}}{dw^{n-1}}\left[\left(\frac{...
- 256
0
votes
0
answers
26
views
Limits coming from Lagrange Inversion Theorem
$y=e^x\implies x=\ln y=\ln(1+(y-1))\implies$
$$x=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}(y-1)^n\tag1$$
On the other hand, by
Lagrange inversion theorem,
with $y=f(x)=e^x$, $a=0$, we have
$$x=g(y)=\sum_{...
- 11.9k
2
votes
1
answer
79
views
Residue of inverse function using Lagrange inversion
Let $v(z)$ be a infinite power series of terms $z^k$ where $k>0$ with coefficients $v_k$ and a analytic function except at z=0 and $V(z)$ be the inverse of $v(z)$,then show that $Res_0(V(z)^{-k})=k\...
- 53
1
vote
1
answer
51
views
Question on Lagrange series inversion proof
I am stuck in understanding a passage of a proof I have found for Lagrange series inversion formula.
The theorem is proven in the following form:
Consider the equation $x=y+\epsilon f(x)$ where $f$ ...
- 121
3
votes
0
answers
42
views
Applications of q-Lagrange inversion
I was reading a text on q,t-Catalan numbers and Diagonal Harmonics by Haglund, where they mention the following $q$-analogue of Lagrange Inversion, taken from Page 53:
Let $e_n, h_n$ denote the ...
- 143
4
votes
0
answers
132
views
How helpful is $f^{-1}(z)=\frac1{2\pi i}\oint\ln(1-\frac z{f(w)})dw$, or the method to find it, in deriving integral representations of $f^{-1}(z)$?
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following $\erf^{-1}(z)$ series:
$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via ...
- 12.5k
0
votes
0
answers
74
views
Airy function zeros.
I'm studying the Airy functions, and I got particularly interested in the behavior of the real zeros. I was wondering if there's some formula to express the n-th zero of the Airy functions, since they ...
0
votes
0
answers
61
views
Series reversion using reciprocal sum recursion confusion
This answer
claims to give series reversion/Lagrange inversion coefficients recursively for:
$$d_n = \frac{d^{n-1}}{dy^{n-1}}\Big( \frac{y}{5/2(e^{-y}-1)+e^{-y}(2y+y^2/2)} \Big)^n\Big|_{y=0} = (n-1)! ...
- 12.5k
4
votes
1
answer
149
views
Simplified expression for the nth derivative of the n+1th power of a function
$$\frac{d^n}{dx^n}(f(x))^n \text{ and similar expressions}$$
I was trying to work out the Taylor series for the solution to the general cubic using Lagrange Inversion (out of curiosity: I was ...
- 503
1
vote
0
answers
86
views
Multivariate function inversion - Lagrange Inversion
If I have a (bijective) scalar function $f : \mathbb{R}^N \to \mathbb{R}$ and $\vec{x}\in \mathbb{R}^N$. Is it possible to obtain an inverse function $h:\mathbb{R} \to \mathbb{R}^N $ such that $\vec{h}...
- 11
1
vote
0
answers
39
views
Reference for Lagrange inversion formula
I am writing a piece of research where I use the Lagrange inversion formula for solving some equation. More precisely, for $y=f(x)$ where $f$ is analytic at $x=a$ and such that $f'(a)\neq 0$, we have
\...
- 11
2
votes
0
answers
100
views
Evaluating $\frac{d^n}{dt^n}e^{a(t-e^t)}$ as a single series to extend region of convergence for super root function
$\def\srt{\operatorname{srt}}$Introduction:
There is a multiple series expansion for the super root $\srt_n(z)$ valid near $0.7<|z|<1.4$. However, for around $0<|z|<1.3$, there is this ...
- 12.5k
6
votes
2
answers
181
views
Convert $\frac1b\sum_{n=1}^\infty\frac{(b e^a)^n}{n!}B_{n-1}(an)$ to integral using $B_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x(e^t-1)}}{t^{n+1}}dt$
$\def\B{\operatorname B}$
In
How to solve $x^{y^z}=z$
A solution uses Bell polynomials $\B_n(x)$
$$e^{ae^{bz}}=z=1+\frac1b\sum_{n=1}^\infty \frac{(ae^b)^n}{nn!}\B_n(b n)=\frac1b\sum_{n=1}^\infty\...
- 12.5k
5
votes
3
answers
137
views
Inverting $\frac{\xi}{2}(1+\tanh(\xi))=\lambda$ using the Lagrange-Burmann Theorem
For my quantum mechanics homework, I developed the transcendental equation $\frac{\xi}{2}(1+\tanh(\xi))$ for the well-posedness of symmetric potential formed from two delta functions. The professor ...
- 1,222
1
vote
0
answers
77
views
Taylor series expansion in Lagrange-Bürman Theorem
The Lagrange inversion theorem is as follows: Let $f$ be a function which is holomorphic in a neighborhood of a with $f(a) = 0, f'(a) \neq 0$. We know that the functional inverse of f is well-defined ...
- 11