The inverse tag has no usage guidance.

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### Convergence of generalized inverses

During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation to post more links), I've got stuck, trying to understand more deeply one of ...

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367 views

### Equation between the two branches of the lambert w function

My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$?
For example the two square roots $r_1(y)$ and $r_2(y)$ of ...

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### Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $...

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### What is the inverse kernel of this integral transform?

I am looking for the associated inverse kernel to the integral transform $T$ defined by
$(Tf)(u) = \int_{-\infty}^{+\infty} K(u,t)f(t) \ dt,\ \ u \in \mathbb{R^+}$
whose kernel is $K(u,t) = \frac{...

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265 views

### How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\
G(x,y) &=& c_k\...

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### radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke:
Teorem. Let $f : U ⊂ \mathbb{R}^n ...

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### Inverses of two-argument functions with respect to one argument

I asked a shorter version of this question at math.stackexchange.com four days ago but it hasn't gotten any answers or comments.
Consider a function $f : A \times B \to C$ and two inverses, each with ...

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132 views

### Inverse problems for an asymptotic series which depends on a parameter?

I have the series
$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,
where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An ...

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255 views

### Non-linear 1st order difference equation

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$
I have tried various substitutions, simplifications ...

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579 views

### How to prove Lambert's W function is not elementary?

Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois thory extends that to Bessel functions, say. But what tools exist for ...