# values at infinity

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## 21—30 of 119 matching pages

##### 21: 13.2 Definitions and Basic Properties
This equation has a regular singularity at the origin with indices $0$ and $1-b$, and an irregular singularity at infinity of rank one. …In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\infty$ coalesce into an irregular singularity at $\infty$. … In general, $U\left(a,b,z\right)$ has a branch point at $z=0$. The principal branch corresponds to the principal value of $z^{-a}$ in (13.2.6), and has a cut in the $z$-plane along the interval $(-\infty,0]$; compare §4.2(i). …
##### 22: 1.10 Functions of a Complex Variable
The singularities of $f(z)$ at infinity are classified in the same way as the singularities of $f(1/z)$ at $z=0$. … A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. … In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $\mathbb{C}$ with at most one exception. … Functions which have more than one value at a given point $z$ are called multivalued (or many-valued) functions. … Then the value of $F(z)$ at any other point is obtained by analytic continuation. …
##### 23: 28.5 Second Solutions $\mathrm{fe}_{n}$, $\mathrm{ge}_{n}$
As a consequence of the factor $z$ on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as $z\to\pm\infty$ on $\mathbb{R}$. …
##### 24: 13.4 Integral Representations
At the point where the contour crosses the interval $(1,\infty)$, $t^{-b}$ and the ${{}_{2}{\mathbf{F}}_{1}}$ function assume their principal values; compare §§15.1 and 15.2(i). …
##### 25: 32.11 Asymptotic Approximations for Real Variables
Next, for given initial conditions $w(0)=0$ and $w^{\prime}(0)=k$, with $k$ real, $w(x)$ has at least one pole on the real axis. There are two special values of $k$, $k_{1}$ and $k_{2}$, with the properties $-0.45142\;8, $1.85185\;3, and such that: … Conversely, for any nonzero real $k$, there is a unique solution $w_{k}(x)$ of (32.11.4) that is asymptotic to $k\mathrm{Ai}\left(x\right)$ as $x\to+\infty$. … If $|k|>1$, then $w_{k}(x)$ has a pole at a finite point $x=c_{0}$, dependent on $k$, and … Now suppose $x\to-\infty$. …
##### 26: 22.11 Fourier and Hyperbolic Series
22.11.1 $\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.2 $\operatorname{cn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}},$
22.11.3 $\operatorname{dn}\left(z,k\right)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{% \infty}\frac{q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}.$
In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. …
22.11.14 $k^{2}{\operatorname{sn}}^{2}\left(z,k\right)=\frac{{E^{\prime}}}{{K^{\prime}}}% -\left(\frac{\pi}{2{K^{\prime}}}\right)^{2}\sum_{n=-\infty}^{\infty}\left({% \operatorname{sech}}^{2}\left(\frac{\pi}{2{K^{\prime}}}(z-2nK)\right)\right),$
##### 27: 2.10 Sums and Sequences
In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\geq\frac{1}{2}(\alpha+1)$. For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a). … For an extension to integrals with Cauchy principal values see Elliott (1998). … The singularities of $f(z)$ on the unit circle are branch points at $z=e^{\pm i\alpha}$. To match the limiting behavior of $f(z)$ at these points we set …
##### 28: 9.19 Approximations
• Martín et al. (1992) provides two simple formulas for approximating $\mathrm{Ai}\left(x\right)$ to graphical accuracy, one for $-\infty, the other for $0\leq x<\infty$.

• Moshier (1989, §6.14) provides minimax rational approximations for calculating $\mathrm{Ai}\left(x\right)$, $\mathrm{Ai}'\left(x\right)$, $\mathrm{Bi}\left(x\right)$, $\mathrm{Bi}'\left(x\right)$. They are in terms of the variable $\zeta$, where $\zeta=\tfrac{2}{3}x^{3/2}$ when $x$ is positive, $\zeta=\tfrac{2}{3}(-x)^{3/2}$ when $x$ is negative, and $\zeta=0$ when $x=0$. The approximations apply when $2\leq\zeta<\infty$, that is, when $3^{2/3}\leq x<\infty$ or $-\infty. The precision in the coefficients is 21S.

• These expansions are for real arguments $x$ and are supplied in sets of four for each function, corresponding to intervals $-\infty, $a\leq x\leq 0$, $0\leq x\leq b$, $b\leq x<\infty$. …
• Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and $\mathrm{Ai}'\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, $\mathrm{Bi}'\left(z\right)$.

• MacLeod (1994) supplies Chebyshev-series expansions to cover $\mathrm{Gi}\left(x\right)$ for $0\leq x<\infty$ and $\mathrm{Hi}\left(x\right)$ for $-\infty. The Chebyshev coefficients are given to 20D.

• ##### 29: 14.16 Zeros
For uniform asymptotic approximations for the zeros of $\mathsf{P}^{-m}_{n}\left(x\right)$ in the interval $-1 when $n\to\infty$ with $m$ $(\geq 0)$ fixed, see Olver (1997b, p. 469).
###### §14.16(iii) Interval $1
$P^{\mu}_{\nu}\left(x\right)$ has exactly one zero in the interval $(1,\infty)$ if either of the following sets of conditions holds: … For all other values of $\mu$ and $\nu$ (with $\nu\geq-\frac{1}{2}$) $P^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$. $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$ when $\nu>-1$, and at most one zero in the interval $(1,\infty)$ when $\nu<-1$.
##### 30: 4.2 Definitions
This is a multivalued function of $z$ with branch point at $z=0$. … $\ln z$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,0]$ and real-valued when $z$ ranges over the positive real numbers. … The principal value is …This is an analytic function of $z$ on $\mathbb{C}\setminus(-\infty,0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in\mathbb{Z}$. …