# Lagrange inversion theorem

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##### 3: 22.15 Inverse Functions
###### §22.15(i) Definitions
Each of these inverse functions is multivalued. The principal values satisfy …
##### 4: 3.3 Interpolation
###### §3.3(i) Lagrange Interpolation
With an error term the Lagrange interpolation formula for $f$ is given by …
##### 6: 18.40 Methods of Computation
###### Histogram Approach
In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: … Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …
##### 7: 2.2 Transcendental Equations
where $F_{0}=f_{0}$ and $sF_{s}$ ($s\geq 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of $(f(x))^{s}$ (Lagrange’s formula for the reversion of series). …
4.38.19 $\operatorname{Arctanh}u\pm\operatorname{Arccoth}v=\operatorname{Arctanh}\left(% \frac{uv\pm 1}{v\pm u}\right)=\operatorname{Arccoth}\left(\frac{v\pm u}{uv\pm 1% }\right).$