# as x→±1

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##### 2: 18.31 Bernstein–Szegő Polynomials
Let $\rho(x)$ be a polynomial of degree $\ell$ and positive when $-1\leq x\leq 1$. The Bernstein–Szegő polynomials $\{p_{n}(x)\}$, $n=0,1,\dots$, are orthogonal on $(-1,1)$ with respect to three types of weight function: $(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}$, $(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}$, $(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}$. …
##### 3: 4.5 Inequalities
4.5.1 $\frac{x}{1+x}<\ln\left(1+x\right) $x>-1$, $x\neq 0$,
4.5.2 $x<-\ln\left(1-x\right)<\frac{x}{1-x},$ $x<1$, $x\neq 0$,
4.5.7 $e^{-x/(1-x)}<1-x $x<1$,
4.5.11 $x $x<1$,
4.5.12 $e^{x/(1+x)}<1+x,$ $x>-1$,
##### 5: 14.10 Recurrence Relations and Derivatives
14.10.1 ${\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,$
14.10.2 ${\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)% \mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}% \left(x\right)=0,$
14.10.4 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x% \mathsf{P}^{\mu}_{\nu}\left(x\right)},$
14.10.6 ${P^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}P^{\mu+1}_{% \nu}\left(x\right)}-(\nu-\mu)(\nu+\mu+1)P^{\mu}_{\nu}\left(x\right)=0,$
14.10.7 ${\left(x^{2}-1\right)^{1/2}P^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)P^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)xP^{\mu}_{\nu}\left(x\right)=0.$
##### 6: 12.6 Continued Fraction
For a continued-fraction expansion of the ratio $\ifrac{U\left(a,x\right)}{U\left(a-1,x\right)}$ see Cuyt et al. (2008, pp. 340–341).
##### 7: 4.12 Generalized Logarithms and Exponentials
4.12.1 $\phi(x+1)=e^{\phi(x)},$ $-1,
and is strictly increasing when $0\leq x\leq 1$. …It, too, is strictly increasing when $0\leq x\leq 1$, and …
4.12.5 $\phi(x)=\psi(x)=x,$ $0\leq x\leq 1$.
4.12.6 $\phi(x)=\ln\left(x+1\right),$ $-1,
##### 8: 27.7 Lambert Series as Generating Functions
27.7.1 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}.$
If $|x|<1$, then the quotient $x^{n}/(1-x^{n})$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: … Again with $|x|<1$, special cases of (27.7.2) include:
27.7.3 $\sum_{n=1}^{\infty}\mu\left(n\right)\frac{x^{n}}{1-x^{n}}=x,$
27.7.4 $\sum_{n=1}^{\infty}\phi\left(n\right)\frac{x^{n}}{1-x^{n}}=\frac{x}{(1-x)^{2}},$
##### 9: 19.32 Conformal Map onto a Rectangle
with $x_{1},x_{2},x_{3}$ real constants, has differential …
19.32.3 $x_{1}>x_{2}>x_{3},$
$z(x_{1})=R_{F}\left(0,x_{1}-x_{2},x_{1}-x_{3}\right)\quad\text{(>0)},$
$z(x_{2})=z(x_{1})+z(x_{3}),$
$z(x_{3})=R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-iR_{F}\left(0,x_{1}-x_{3% },x_{2}-x_{3}\right).$