values at q=0

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1: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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2: 28.4 Fourier Series

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§28.4(iv) Case $q=0$

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3: 28.2 Definitions and Basic Properties

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${\mathrm{se}}_n(z,0)=\sin \left(nz\right)$ , $n=1,2,3,\mathrm{\dots }$.

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4: 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

5: 28.7 Analytic Continuation of Eigenvalues

… ►The only singularities are algebraic branch points, with $a_n\left(q\right)$ and $b_n\left(q\right)$ finite at these points. …All real values of $q$ are normal values. To 4D the first branch points between $a_0\left(q\right)$ and $a_2\left(q\right)$ are at $q_0=\pm \mathrm{i}1.4688$ with $a_0\left(q_0\right)=a_2\left(q_0\right)=2.0886$, and between $b_2\left(q\right)$ and $b_4\left(q\right)$ they are at $q_1=\pm \mathrm{i}6.9289$ with $b_2\left(q_1\right)=b_4\left(q_1\right)=11.1904$. For real $q$ with , $a_0\left(\mathrm{i}q\right)$ and $a_2\left(\mathrm{i}q\right)$ are real-valued, whereas for real $q$ with $|q|>|q_0|$, $a_0\left(\mathrm{i}q\right)$ and $a_2\left(\mathrm{i}q\right)$ are complex conjugates. … ►Analogous statements hold for $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$, and $b_{2n+2}\left(q\right)$, also for $n=0,1,2,\mathrm{\dots }$. …

6: 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 . …

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: 16.2 Definition and Analytic Properties

… ►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty }$. … ►On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\mathrm{\Re }{\gamma }_q>0$, convergent except at $z=1$ if , and divergent if $\mathrm{\Re }{\gamma }_q\le -1$, where … ►(However, except where indicated otherwise in the DLMF we assume that when $p>q+1$ at least one of the $a_k$ is a nonpositive integer.) …

9: 31.9 Orthogonality

… ►Here $\zeta$ is an arbitrary point in the interval $(0,1)$. The integration path begins at $z=\zeta$, encircles $z=1$ once in the positive sense, followed by $z=0$ once in the positive sense, and so on, returning finally to $z=\zeta$. …The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … ►The right-hand side may be evaluated at any convenient value, or limiting value, of $\zeta$ in $(0,1)$ since it is independent of $\zeta$. … ►and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

10: 2.4 Contour Integrals

… ►Except that $\lambda$ is now permitted to be complex, with $\mathrm{\Re }\lambda >0$, we assume the same conditions on $q(t)$ and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of $\mathrm{\Re }z$. … ►

In a neighborhood of $a$

2.4.11

$$p(t)=p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)=\sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

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Symbols:

$\lambda$: constant, $a$: left endpoint, $p(t)$: analytic function, $q(t)$: analytic function, $p_s$: coefficients, $q_s$: coefficients and $\mu$: positive constant

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E11

Encodings:

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See also:

Annotations for §2.4(iii), §2.4 and Ch.2

with $\mathrm{\Re }\lambda >0$, $\mu >0$, $p_0\ne 0$, and the branches of ${(t-a)}^{\lambda }$ and ${(t-a)}^{\mu }$ continuous and constructed with $\mathrm{ph}\left(t-a\right)\to \omega$ as $t\to a$ along $𝒫$.

… ►and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of $p(t)$ and $q(t)$ at $t=t_0$. … ►Cases in which $p^{\prime }(t_0)\ne 0$ are usually handled by deforming the integration path in such a way that the minimum of $\mathrm{\Re }\left(zp(t)\right)$ is attained at a saddle point or at an endpoint. … ►with $p,q$ and their derivatives evaluated at $t_0$. …