value at z=0

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1: 20.4 Values at $z$ = 0

§20.4 Values at $z$ = 0

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20.4.12 $$\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{{\theta }_1^{\prime }(0,q)}=\frac{{\theta }_2^{\prime \prime }(0,q)}{{\theta }_2(0,q)}+\frac{{\theta }_3^{\prime \prime }(0,q)}{{\theta }_3(0,q)}+\frac{{\theta }_4^{\prime \prime }(0,q)}{{\theta }_4(0,q)}.$$

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Symbols:

${\theta }_j(z,q)$: theta function and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

2: 12.2 Differential Equations

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§12.2(ii) Values at $z=0$

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3: 18.6 Symmetry, Special Values, and Limits to Monomials

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Laguerre

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4: 12.14 The Function $W(a,x)$

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§12.14(ii) Values at $z=0$ and Wronskian

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5: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. …

6: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. …

7: 31.3 Basic Solutions

… ► $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. …

8: 10.25 Definitions

… ►In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►It has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. ►Both $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ are real when $\nu$ is real and $\mathrm{ph}z=0$. ►For fixed $z$ $(\ne 0)$ each branch of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ is entire in $\nu$. …

9: 10.2 Definitions

… ►This differential equation has a regular singularity at $z=0$ with indices $\pm \nu$, and an irregular singularity at $z=\mathrm{\infty }$ of rank $1$; compare §§2.7(i) and 2.7(ii). … ►This solution of (10.2.1) is an analytic function of $z\in ℂ$, except for a branch point at $z=0$ when $\nu$ is not an integer. … ►Whether or not $\nu$ is an integer $Y_{\nu }\left(z\right)$ has a branch point at $z=0$. … ►Each solution has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. …

10: 13.14 Definitions and Basic Properties

… ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. …