# values at q=0

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##### 3: 28.2 Definitions and Basic Properties
$\operatorname{se}_{n}\left(z,0\right)=\sin\left(nz\right)$ , $n=1,2,3,\dots$.
##### 4: 20.4 Values at $z$ = 0
20.4.12 $\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}=\frac{\theta% _{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}+\frac{\theta_{3}''\left(0,% q\right)}{\theta_{3}\left(0,q\right)}+\frac{\theta_{4}''\left(0,q\right)}{% \theta_{4}\left(0,q\right)}.$
##### 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 $|q|<|q_{0}|$, $a_{0}\left(\mathrm{i}q\right)$ 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 $n=0,1,2,\dots$. …
##### 6: 31.4 Solutions Analytic at Two Singularities: Heun Functions
For an infinite set of discrete values $q_{m}$, $m=0,1,2,\dots$, of the accessory parameter $q$, the function $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic at $z=1$, and hence also throughout the disk $|z|. …
##### 7: 31.3 Basic Solutions
$\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ 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 $p\leq q$ 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 $+\infty$ on the real axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of ${{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$; 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 $\infty$. … On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\Re\gamma_{q}>0$, convergent except at $z=1$ if $-1<\Re\gamma_{q}\leq 0$, and divergent if $\Re\gamma_{q}\leq-1$, 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 $\mathcal{L}_{1}$, $\mathcal{L}_{2}$ 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 $\Re\lambda>0$, 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 $\Re z$. …
• (a)

In a neighborhood of $a$

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

• 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 $p^{\prime}(t_{0})\neq 0$ are usually handled by deforming the integration path in such a way that the minimum of $\Re\left(zp(t)\right)$ is attained at a saddle point or at an endpoint. … with $p,q$ and their derivatives evaluated at $t_{0}$. …