# Questions tagged [inverse-series]

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### Is the real and imaginary part of the Dirichlet eta function invertible when viewed as single variable function?

If we examine $\Re(\eta(\alpha + \beta i))$ as a function of $\alpha$ or only $\beta$ is $\eta$ invertible? That is, if we define that map $J:\mathbb{R}\rightarrow \mathbb{R}$ as
\begin{equation}\...

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### What is the impact of individual estimate on each other in matrix inversion?

I am looking to understand the impact of each estimate on each other in matrix inversion.
Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...

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### Property of an integer sequence related to series reversion

Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here:
Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative ...

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### Functional inverse of $z=1+w+\cdots+w^{n-1}$

Migrated from the MSE.
I am interested in the functional inverse of
$$
z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1.
$$
This function is strictly increasing on $w\geq0$ and thus admits an inverse.
By ...

3
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1
answer

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### "Lagrange inversion" around an extremum

Cross-posted from Math Stackexchange.
In an older question to which I provided an answer it was asked how to compute a particular limit involving the roots of a transcedental function around its ...

3
votes

1
answer

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### Inverse of the incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind $E(\varphi \, | \,k)$ is defined as follows:
$$E(\varphi \, | \,k) = \int_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $$
Where $0<k^...

2
votes

1
answer

468
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### Correction terms in the asymptotic expansion of hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below),
$$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\frac{1}...

2
votes

1
answer

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### Is $\sigma(n)$ has explicit compositional inverse formula since it has series representation?

let $\sigma_x(n)$ be a power of sum divisor function such that : $\sigma_x(n)=\sum_{d|n} d^x$ . My question is: Does $\sigma$ has explicit compositional inverse formula since it has series ...

1
vote

1
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### Can I apply Lagrange inversion theorem? [closed]

I want to invert the equation
$$\eta = g(x)\sqrt{1+g'(x)^2}$$
to get $x$ as a function of $\eta$. Assume $g(0)=0$, $g'(0)=0$ and $g'(x)>0$ for $x>0$ (Think $g(x) = x^p$ for $p\geq 2$ integer).
...

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### Positivity of coefficients of the inverse of a certain power series

Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation
$$
g(z)-g(z)^8+g(z)^{15}=z,
$$
that is the inverse of
$$
z-z^8+z^{15}
$$
in the group of formal ...