# small x

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##### 2: 26.6 Other Lattice Path Numbers
For sufficiently small $|x|$ and $|y|$,
26.6.6 $\sum_{n=0}^{\infty}D(n,n)x^{n}=\frac{1}{\sqrt{1-6x+x^{2}}},$
26.6.7 $\sum_{n=0}^{\infty}M(n)x^{n}=\frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}},$
26.6.9 $\sum_{n=0}^{\infty}r(n)x^{n}=\frac{1-x-\sqrt{1-6x+x^{2}}}{2x}.$
##### 3: 11.13 Methods of Computation
Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that $\mathbf{H}_{\nu}\left(x\right)$ and $\mathbf{L}_{\nu}\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution $\mathbf{K}_{\nu}\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\tfrac{1}{2}$. …
##### 5: 36.7 Zeros
Near $z=z_{n}$, and for small $x$ and $y$, the modulus $|\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …
36.11.1 $t_{1}(\mathbf{x})
36.11.2 $\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).$
36.11.4 $\Psi_{3}\left(x,0,0\right)=\frac{\sqrt{2\pi}}{(5|x|^{3})^{1/8}}\begin{cases}% \exp\left(-2\sqrt{2}(\ifrac{x}{5})^{5/4}\right)\left(\cos\left(2\sqrt{2}(% \ifrac{x}{5})^{5/4}-\tfrac{1}{8}\pi\right)+o\left(1\right)\right),&x\to+\infty% ,\\ \cos\left(4(\ifrac{|x|}{5})^{5/4}-\tfrac{1}{4}\pi\right)+o\left(1\right),&x\to% -\infty.\end{cases}$
36.11.5 $\Psi_{3}\left(0,y,0\right)=\overline{\Psi_{3}(0,-y,0)}=\exp\left(\tfrac{1}{4}i% \pi\right)\sqrt{\ifrac{\pi}{y}}\left(1-(i/{\sqrt{3}})\exp\left(\tfrac{3}{2}i(% \ifrac{2y}{5})^{5/3}\right)+o\left(1\right)\right),$ $y\to+\infty$.
36.11.7 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$,
##### 7: 10.75 Tables
• British Association for the Advancement of Science (1937) tabulates $J_{0}\left(x\right)$, $J_{1}\left(x\right)$, $x=0(.001)16(.01)25$, 10D; $Y_{0}\left(x\right)$, $Y_{1}\left(x\right)$, $x=0.01(.01)25$, 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $Y_{0}\left(x\right)$, $Y_{1}\left(x\right)$ for small values of $x$, as well as auxiliary functions to compute all four functions for large values of $x$.

• British Association for the Advancement of Science (1937) tabulates $I_{0}\left(x\right)$, $I_{1}\left(x\right)$, $x=0(.001)5$, 7–8D; $K_{0}\left(x\right)$, $K_{1}\left(x\right)$, $x=0.01(.01)5$, 7–10D; $e^{-x}I_{0}\left(x\right)$, $e^{-x}I_{1}\left(x\right)$, $e^{x}K_{0}\left(x\right)$, $e^{x}K_{1}\left(x\right)$, $x=5(.01)10(.1)20$, 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $K_{0}\left(x\right)$, $K_{1}\left(x\right)$ for small values of $x$.

• Young and Kirk (1964) tabulates $\operatorname{ber}_{n}x$, $\operatorname{bei}_{n}x$, $\operatorname{ker}_{n}x$, $\operatorname{kei}_{n}x$, $n=0,1$, $x=0(.1)10$, 15D; $\operatorname{ber}_{n}x$, $\operatorname{bei}_{n}x$, $\operatorname{ker}_{n}x$, $\operatorname{kei}_{n}x$, modulus and phase functions $M_{n}\left(x\right)$, $\theta_{n}\left(x\right)$, $N_{n}\left(x\right)$, $\phi_{n}\left(x\right)$, $n=0,1,2$, $x=0(.01)2.5$, 8S, and $n=0(1)10$, $x=0(.1)10$, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$. (Concerning the phase functions see §10.68(iv).)

• ##### 8: 10.45 Functions of Imaginary Order
In consequence of (10.45.5)–(10.45.7), $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either $\widetilde{I}_{\nu}\left(x\right)$ and $(1/\pi)\sinh\left(\pi\nu\right)\widetilde{K}_{\nu}\left(x\right)$, or $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu\neq 0$ or $\nu=0$. …
##### 9: 10.24 Functions of Imaginary Order
Also, in consequence of (10.24.7)–(10.24.9), when $x$ is small either $\widetilde{J}_{\nu}\left(x\right)$ and $\tanh\left(\tfrac{1}{2}\pi\nu\right)\widetilde{Y}_{\nu}\left(x\right)$ or $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu\neq 0$ or $\nu=0$. …
##### 10: 8.7 Series Expansions
For an expansion for $\gamma\left(a,ix\right)$ in series of Bessel functions $J_{n}\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\geq 0$) is small or moderate in magnitude see Barakat (1961).