# values at q=0

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## 1—10 of 58 matching pages

##### 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 $q=0$

…##### 3: 28.2 Definitions and Basic Properties

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

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##### 4: 20.4 Values at $z$ = 0

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20.4.12
$$\frac{{\theta}_{1}^{\prime \prime \prime}(0,q)}{{\theta}_{1}^{\prime}(0,q)}=\frac{{\theta}_{2}^{\prime \prime}(0,q)}{{\theta}_{2}(0,q)}+\frac{{\theta}_{3}^{\prime \prime}(0,q)}{{\theta}_{3}(0,q)}+\frac{{\theta}_{4}^{\prime \prime}(0,q)}{{\theta}_{4}(0,q)}.$$

##### 5: 28.7 Analytic Continuation of Eigenvalues

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►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.
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►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 $n=0,1,2,\mathrm{\dots}$.
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##### 6: 31.4 Solutions Analytic at Two Singularities: Heun Functions

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►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 $$.
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##### 7: 31.3 Basic Solutions

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$H\mathrm{\ell}(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$
at
$z=0$ and assumes the value
$1$ there.
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##### 8: 16.2 Definition and Analytic Properties

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►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function.
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►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 ${}_{q+1}{}^{}F_{q}^{}(\mathbf{a};\mathbf{b};z)$; 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

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►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.
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►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 $.
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►and the integration paths ${\mathcal{L}}_{1}$, ${\mathcal{L}}_{2}$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$.
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##### 10: 2.4 Contour Integrals

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►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 $\mathrm{\Re}z$.
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(a)
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►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
$t={t}_{0}$.
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►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.
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►with $p,q$ and their derivatives evaluated at
${t}_{0}$.
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In a neighborhood of $a$

2.4.11

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

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

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 $\mathcal{P}$.