Questions tagged [inverse-series]
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5
questions
3
votes
1answer
237 views
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
1answer
237 views
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),
$$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\frac{1}...
3
votes
1answer
220 views
Is $\sigma(n)$ has explicit compositional inverse formula since it has series representation?
let $\sigma_x(n)$ be a power of sum divisor function such that : $\sigma_x(n)=\sum_{d|n} d^x$ . My question is: Does $\sigma$ has explicit compositional inverse formula since it has series ...
1
vote
1answer
136 views
Can I apply Lagrange inversion theorem? [closed]
I want to invert the equation
$$\eta = g(x)\sqrt{1+g'(x)^2}$$
to get $x$ as a function of $\eta$. Assume $g(0)=0$, $g'(0)=0$ and $g'(x)>0$ for $x>0$ (Think $g(x) = x^p$ for $p\geq 2$ integer).
...
18
votes
1answer
440 views
Positivity of coefficients of the inverse of a certain power series
Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation
$$
g(z)-g(z)^8+g(z)^{15}=z,
$$
that is the inverse of
$$
z-z^8+z^{15}
$$
in the group of formal ...
