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##### 1: 30.5 Functions of the Second Kind
30.5.1 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right),$ $n=m,m+1,m+2,\dots$.
30.5.2 $\mathsf{Qs}^{m}_{n}\left(-x,\gamma^{2}\right)=(-1)^{n-m+1}\mathsf{Qs}^{m}_{n}% \left(x,\gamma^{2}\right),$
30.5.3 $\mathsf{Qs}^{m}_{n}\left(x,0\right)=\mathsf{Q}^{m}_{n}\left(x\right);$
30.5.4 $\mathscr{W}\left\{\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),\mathsf{Qs}^{m}% _{n}\left(x,\gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})\quad(\neq 0),$
##### 2: 30.1 Special Notation
The main functions treated in this chapter are the eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right)$, $\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right)$, and $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathsf{Ps}$, $\mathsf{Qs}$, $\mathit{Ps}$, $\mathit{Qs}$, respectively. … Flammer (1957) and Abramowitz and Stegun (1964) use $\lambda_{mn}(\gamma)$ for $\lambda^{m}_{n}\left(\gamma^{2}\right)+\gamma^{2}$, $R_{mn}^{(j)}(\gamma,z)$ for $S^{m(j)}_{n}\left(z,\gamma\right)$, and
$S^{(1)}_{mn}(\gamma,x)=d_{mn}(\gamma)\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}% \right),$
$S^{(2)}_{mn}(\gamma,x)=d_{mn}(\gamma)\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}% \right),$
##### 3: 30.4 Functions of the First Kind
The eigenfunctions of (30.2.1) that correspond to the eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ are denoted by $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$, $n=m,m+1,m+2,\dots$. …the sign of $\mathsf{Ps}^{m}_{n}\left(0,\gamma^{2}\right)$ being $(-1)^{(n+m)/2}$ when $n-m$ is even, and the sign of $\ifrac{\mathrm{d}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)}{\mathrm{d}x}|_{% x=0}$ being $(-1)^{(n+m-1)/2}$ when $n-m$ is odd. When $\gamma^{2}>0$ $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ is the prolate angular spheroidal wave function, and when $\gamma^{2}<0$ $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ is the oblate angular spheroidal wave function. If $\gamma=0$, $\mathsf{Ps}^{m}_{n}\left(x,0\right)$ reduces to the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$: … $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ has exactly $n-m$ zeros in the interval $-1. …
##### 4: 30.7 Graphics Figure 30.7.5: Ps n 0 ⁡ ( x , 4 ) , n = 0 , 1 , 2 , 3 , - 1 ≤ x ≤ 1 . Magnify Figure 30.7.6: Ps n 0 ⁡ ( x , - 4 ) , n = 0 , 1 , 2 , 3 , - 1 ≤ x ≤ 1 . Magnify Figure 30.7.7: Ps n 1 ⁡ ( x , 30 ) , n = 1 , 2 , 3 , 4 , - 1 ≤ x ≤ 1 . Magnify Figure 30.7.8: Ps n 1 ⁡ ( x , - 30 ) , n = 1 , 2 , 3 , 4 , - 1 ≤ x ≤ 1 . Magnify Figure 30.7.11: Qs n 0 ⁡ ( x , 4 ) , n = 0 , 1 , 2 , 3 , - 1 < x < 1 . Magnify
##### 5: 14.22 Graphics Figure 14.22.1: P 1 / 2 0 ⁡ ( x + i ⁢ y ) , - 5 ≤ x ≤ 5 , - 5 ≤ y ≤ 5 . There is a cut along the real axis from - ∞ to - 1 . Magnify 3D Help Figure 14.22.2: P 1 / 2 - 1 / 2 ⁡ ( x + i ⁢ y ) , - 5 ≤ x ≤ 5 , - 5 ≤ y ≤ 5 . There is a cut along the real axis from - ∞ to 1 . Magnify 3D Help Figure 14.22.3: P 1 / 2 - 1 ⁡ ( x + i ⁢ y ) , - 5 ≤ x ≤ 5 , - 5 ≤ y ≤ 5 . There is a cut along the real axis from - ∞ to 1 . Magnify 3D Help Figure 14.22.4: Q 0 0 ⁡ ( x + i ⁢ y ) , - 5 ≤ x ≤ 5 , - 5 ≤ y ≤ 5 . There is a cut along the real axis from - 1 to 1 . Magnify 3D Help
##### 6: 30.10 Series and Integrals
Integrals and integral equations for $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …
##### 7: 4.37 Inverse Hyperbolic Functions
These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. …
###### §4.37(iv) Logarithmic Forms
On the cutsOn the part of the cuts from $-1$ to $1$On the cuts
##### 8: 30.6 Functions of Complex Argument
30.6.4 $\mathit{Ps}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)=(\mp\mathrm{i})^{m}% \mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$
30.6.5 $\mathit{Qs}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)={(\mp\mathrm{i})^{m% }\left(\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)\mp\tfrac{1}{2}\mathrm{i}% \pi\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)\right)}.$
##### 9: 4.2 Definitions
Most texts extend the definition of the principal value to include the branch cut …where $k$ is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense. In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by $z=x+\mathrm{i}0$, the other set corresponding to the “lower side” and denoted by $z=x-\mathrm{i}0$. …Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … 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 $a\in\mathbb{Z}$. …
##### 10: 4.39 Continued Fractions
where $z$ is in the open cut plane of Figure 4.37.1(i). …where $z$ is in the open cut plane of Figure 4.37.1(iii). …