# divided differences

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##### 1: 3.3 Interpolation
###### §3.3(iii) DividedDifferences
$\left[z_{0}\right]f=f_{0},$
$\left[z_{0},z_{1}\right]f=(\left[z_{1}\right]f-\left[z_{0}\right]f)/(z_{1}-z_{% 0}),$
$\left[z_{0},z_{1},z_{2}\right]f=(\left[z_{1},z_{2}\right]f-\left[z_{0},z_{1}% \right]f)/(z_{2}-z_{0}),$
3.3.39 $x(f)=\left[f_{0}\right]x+(f-f_{0})\left[f_{0},f_{1}\right]x+(f-f_{0})(f-f_{1})% \left[f_{0},f_{1},f_{2}\right]x;$
##### 2: 18.26 Wilson Class: Continued
18.26.14 $\ifrac{\delta_{y}\left(W_{n}\left(y^{2};a,b,c,d\right)\right)}{\delta_{y}(y^{2% })}=-n(n+a+b+c+d-1)\*W_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac% {1}{2},d+\tfrac{1}{2}\right).$
18.26.15 $\ifrac{\delta_{y}\left(S_{n}\left(y^{2};a,b,c\right)\right)}{\delta_{y}(y^{2})% }=-nS_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac{1}{2}\right).$
##### 3: 21.7 Riemann Surfaces
21.7.15 $4\boldsymbol{{\eta}}^{1}(T)\cdot\boldsymbol{{\eta}}^{2}(T)=\tfrac{1}{2}\left(|% T\ominus U|-g-1\right)\pmod{2},$
##### 4: 20.2 Definitions and Periodic Properties
20.2.7 $\theta_{2}\left(z+(m+n\tau)\pi\middle|\tau\right)=(-1)^{m}q^{-n^{2}}e^{-2inz}% \theta_{2}\left(z\middle|\tau\right),$
20.2.8 $\theta_{3}\left(z+(m+n\tau)\pi\middle|\tau\right)=q^{-n^{2}}e^{-2inz}\theta_{3% }\left(z\middle|\tau\right),$
20.2.12 $\theta_{2}\left(z\middle|\tau\right)=\theta_{1}\left(z+\tfrac{1}{2}\pi\middle|% \tau\right)=M\theta_{3}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=M\theta_% {4}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$
20.2.13 $\theta_{3}\left(z\middle|\tau\right)=\theta_{4}\left(z+\tfrac{1}{2}\pi\middle|% \tau\right)=M\theta_{2}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=M\theta_% {1}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$
##### 5: 18.1 Notation
$\delta_{x}\left(f(x)\right)=\left(f(x+\tfrac{1}{2}\mathrm{i})-f(x-\tfrac{1}{2}% \mathrm{i})\right)/\mathrm{i},$
##### 6: 24.10 Arithmetic Properties
Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. …
##### 7: 19.36 Methods of Computation
The cases $k_{c}^{2}/2\leq p<\infty$ and $-\infty require different treatment for numerical purposes, and again precautions are needed to avoid cancellations. …
##### 8: Errata
• Equation (3.3.34)

In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

• Equation (10.22.72)
10.22.72 $\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)J_{\nu}\left(ct% \right)t^{1-\mu}\,\mathrm{d}t=\frac{(bc)^{\mu-1}\sin\left((\mu-\nu)\pi\right)(% \sinh\chi)^{\mu-\frac{1}{2}}}{(\frac{1}{2}{\pi}^{3})^{\frac{1}{2}}a^{\mu}}{% \mathrm{e}}^{(\mu-\frac{1}{2})\mathrm{i}\pi}Q^{\frac{1}{2}-\mu}_{\nu-\frac{1}{% 2}}\left(\cosh\chi\right),$ $\Re\mu>-\tfrac{1}{2},\Re\nu>-1,a>b+c,\cosh\chi=(a^{2}-b^{2}-c^{2})/(2bc)$

Originally, the factor on the right-hand side was written as $\frac{(bc)^{\mu-1}\cos\left(\nu\pi\right)(\sinh\chi)^{\mu-\frac{1}{2}}}{(\frac% {1}{2}{\pi}^{3})^{\frac{1}{2}}a^{\mu}}$, which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associated Legendre function of the second kind $Q^{\mu}_{\nu}$. Watson’s $Q_{\nu}^{\mu}$ equals $\frac{\sin\left((\nu+\mu)\pi\right)}{\sin\left(\nu\pi\right)}{\mathrm{e}}^{-% \mu\pi\mathrm{i}}Q^{\mu}_{\nu}$ in the DLMF.

Reported by Arun Ravishankar on 2018-10-22

• ##### 9: 4.24 Inverse Trigonometric Functions: Further Properties
4.24.2 $\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3% \cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right),$ $|1-z|\leq 2$.
##### 10: 7.7 Integral Representations
7.7.7 $\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x)\right)+e^{-2ab}\operatorname% {erfc}\left(ax-(b/x)\right)\right),$ $x>0$, $|\operatorname{ph}a|<\tfrac{1}{4}\pi$.
7.7.8 $\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 2a}e^{-2ab},$ $|\operatorname{ph}a|<\tfrac{1}{4}\pi$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$.