# odd part

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## 1—10 of 17 matching pages

##### 1: 26.21 Tables
Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\not\equiv\pm 2\pmod{5}$, partitions into parts $\not\equiv\pm 1\pmod{5}$, and unrestricted plane partitions up to 100. …
##### 2: 1.12 Continued Fractions
If $C^{\prime}_{n}=C_{2n+1}$, $n=0,1,2,\dots$, then $C^{\prime}$ is called the odd part of $C$. The odd part of $C$ exists iff $b_{2k+1}\not=0$, $k=0,1,2,\dots$, and up to equivalence is given by … and the even and odd parts of the continued fraction converge to finite values. …
##### 3: 26.10 Integer Partitions: Other Restrictions
$p\left(\mathcal{O},n\right)$ denotes the number of partitions of $n$ into odd parts. …
##### 4: 28.23 Expansions in Series of Bessel Functions
When $j=2,3,4$ the series in the even-numbered equations converge for $\Re z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\Re z>0$ and $|\sinh z|>1$. …
##### 5: 18.17 Integrals
18.17.41 $\int_{0}^{\infty}{\mathrm{e}}^{-ax}\mathit{He}_{n}\left(x\right)x^{z-1}\,% \mathrm{d}x=\Gamma\left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-\tfrac{1}{2}n,-% \tfrac{1}{2}n+\tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-% \tfrac{1}{2}n+\tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),$ $\Re a>0$. Also, $\Re z>0$, $n$ even; $\Re z>-1$, $n$ odd.
##### 6: 9.13 Generalized Airy Functions
9.13.10 $A_{n}\left(-z\right)=\begin{cases}2\sqrt{p/\pi}\cos\left(\tfrac{1}{2}p\pi% \right)z^{-n/4}\left(\cos\left(\zeta-\tfrac{1}{4}\pi\right)+e^{|\Im\zeta|}O% \left(\zeta^{-1}\right)\right),&\text{|\operatorname{ph}z|\leq 2p\pi-\delta,% n odd},\\ \sqrt{p/\pi}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}\right)\right),&\text{|% \operatorname{ph}z|\leq p\pi-\delta, n even},\end{cases}$
9.13.12 $B_{n}\left(-z\right)=\begin{cases}-(\ifrac{2}{\sqrt{\pi}})\sin\left(\tfrac{1}{% 2}p\pi\right)z^{-n/4}\left(\sin\left(\zeta-\tfrac{1}{4}\pi\right)+e^{\left|\Im% \zeta\right|}O\left(\zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|% \leq 2p\pi-\delta,n\text{ odd},\\ (\ifrac{1}{\sqrt{\pi}})\sin\left(p\pi\right)z^{-n/4}e^{-\zeta}\left(1+O\left(% \zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|\leq 3p\pi-\delta,n% \text{ even}.\end{cases}$
##### 7: 23.15 Definitions
In §§23.1523.19, $k$ and $k^{\prime}$ $(\in\mathbb{C})$ denote the Jacobi modulus and complementary modulus, respectively, and $q=e^{i\pi\tau}$ ($\Im\tau>0$) denotes the nome; compare §§20.1 and 22.1. … A modular function $f(\tau)$ is a function of $\tau$ that is meromorphic in the half-plane $\Im\tau>0$, and has the property that for all $\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})$, or for all $\mathcal{A}$ belonging to a subgroup of SL$(2,\mathbb{Z})$,
23.15.5 $f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),$ $\Im\tau>0$,
where $c_{\mathcal{A}}$ is a constant depending only on $\mathcal{A}$, and $\ell$ (the level) is an integer or half an odd integer. …
##### 8: 20.2 Definitions and Periodic Properties
For fixed $\tau$, each $\theta_{j}\left(z\middle|\tau\right)$ is an entire function of $z$ with period $2\pi$; $\theta_{1}\left(z\middle|\tau\right)$ is odd in $z$ and the others are even. For fixed $z$, each of $\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}$, $\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}$, $\theta_{3}\left(z\middle|\tau\right)$, and $\theta_{4}\left(z\middle|\tau\right)$ is an analytic function of $\tau$ for $\Im\tau>0$, with a natural boundary $\Im\tau=0$, and correspondingly, an analytic function of $q$ for $\left|q\right|<1$ with a natural boundary $\left|q\right|=1$. …
##### 9: 25.16 Mathematical Applications
$H\left(s\right)$ is analytic for $\Re s>1$, and can be extended meromorphically into the half-plane $\Re s>-2k$ for every positive integer $k$ by use of the relations … $H\left(s\right)$ has a simple pole with residue $\zeta\left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\dots$. …
##### 10: 31.7 Relations to Other Functions
The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …