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21: 31.4 Solutions Analytic at Two Singularities: Heun Functions

§31.4 Solutions Analytic at Two Singularities: Heun Functions

►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . To emphasize this property this set of functions is denoted by ►

31.4.1 $$(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.6

Permalink:

http://dlmf.nist.gov/31.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

… ►with $(s_1,s_2)\in \{0,1,a,\mathrm{\infty }\}$, denotes a set of solutions of (31.2.1), each of which is analytic at $s_1$ and $s_2$. …

22: 4.15 Graphics

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Figure 4.15.4: $\arctan x$ and $\mathrm{arccot}x$. …$\mathrm{arccot}x$ is discontinuous at $x=0$. Magnify

… ►Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$, where $w=\sin z$ and $z=\arcsin w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto ellipses in the $w$-plane with foci at $w=\pm 1$, and lines parallel to the imaginary axis in the $z$-plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty })$. … ►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …

23: 22.5 Special Values

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§22.5(i) Special Values of $z$

►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $\mathrm{i}{K}^{\prime }$. For example, at $z=K+\mathrm{i}{K}^{\prime }$, $\mathrm{sn}(z,k)=1/k$, $d\mathrm{sn}(z,k)/dz=0$. … ►Table 22.5.2 gives $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ for other special values of $z$. … ►

§22.5(ii) Limiting Values of $k$

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24: 7.2 Definitions

… ► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. ►

Values at Infinity

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$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erfc}z=0$ , .

… ► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. ►

Values at Infinity

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25: 3.7 Ordinary Differential Equations

… ►If the solution $w(z)$ that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along $𝒫$ from $a$ to $b$, then $w(z)$ and $w^{\prime }(z)$ may be computed in a stable manner for $z=z_0,z_1,\mathrm{\dots },z_P$ by successive application of (3.7.5) for $j=0,1,\mathrm{\dots },P-1$, beginning with initial values $w(a)$ and $w^{\prime }(a)$. …

26: 8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by …However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ take their principal values; compare §4.2(i). Except where indicated otherwise in the DLMF these principal values are assumed. … ►When $z\ne 0$, $\mathrm{\Gamma }(a,z)$ is an entire function of $a$, and $\gamma (a,z)$ is meromorphic with simple poles at $a=-n$, $n=0,1,2,\mathrm{\dots }$, with residue ${(-1)}^n/n!$. … ►If $w={\mathrm{e}}^z{z}^{1-a}\mathrm{\Gamma }(a,z)$, then …

27: 15.2 Definitions and Analytical Properties

… ►again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. … ►The same is true of other branches, provided that $z=0$, $1$, and $\mathrm{\infty }$ are excluded. As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. … ►(Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does $𝐅(a,b;c;z)$, which is analytic at $c=0,-1,-2,\mathrm{\dots }$.) …

28: 8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ is a multivalued function of $z$ with branch point at $z=0$. Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. … ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). Elsewhere in the sector $|\mathrm{ph}z|\le \pi$ the principal values are defined by analytic continuation from $\mathrm{ph}z=0$; compare §4.2(i). ►From here on it is assumed that unless indicated otherwise the functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ have their principal values. …

29: 6.2 Definitions and Interrelations

… ►The principal value of the exponential integral $E_1\left(z\right)$ is defined by …As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\mathrm{\infty },0]$ and the principal value is two-valued on $(-\mathrm{\infty },0)$. ►Unless indicated otherwise, it is assumed throughout the DLMF that $E_1\left(z\right)$ assumes its principal value. … ► … ►

Values at Infinity

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30: 10.75 Tables

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Abramowitz and Stegun (1964, p. 373) tabulates the three smallest zeros of $Y_0\left(z\right)$, $Y_1\left(z\right)$, $Y_1^{\prime }\left(z\right)$ in the sector , together with the corresponding values of $Y_1\left(z\right)$, $Y_0\left(z\right)$, $Y_1\left(z\right)$, respectively, to 9D. (There is an error in the value of $Y_0\left(z\right)$ at the 3rd zero of $Y_1\left(z\right)$: the last four digits should be 2533; see Amos (1985).)

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