# products

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## 1—10 of 175 matching pages

##### 1: 30.10 Series and Integrals

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►For product formulas and convolutions see Connett et al. (1993).
…For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).

##### 2: 13.12 Products

###### §13.12 Products

… ►For integral representations, integrals, and series containing products of $M(a,b,z)$ and $U(a,b,z)$ see Erdélyi et al. (1953a, §6.15.3).##### 3: 13.25 Products

###### §13.25 Products

… ►For integral representations, integrals, and series containing products of ${M}_{\kappa ,\mu}\left(z\right)$ and ${W}_{\kappa ,\mu}\left(z\right)$ see Erdélyi et al. (1953a, §6.15.3).##### 4: 27.4 Euler Products and Dirichlet Series

###### §27.4 Euler Products and Dirichlet Series

►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the*Euler product*of the series. If $f(n)$ is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes … ►Euler products are used to find series that generate many functions of multiplicative number theory. …

##### 5: 4.22 Infinite Products and Partial Fractions

###### §4.22 Infinite Products and Partial Fractions

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4.22.1
$$\mathrm{sin}z=z\prod _{n=1}^{\mathrm{\infty}}\left(1-\frac{{z}^{2}}{{n}^{2}{\pi}^{2}}\right),$$

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4.22.2
$$\mathrm{cos}z=\prod _{n=1}^{\mathrm{\infty}}\left(1-\frac{4{z}^{2}}{{(2n-1)}^{2}{\pi}^{2}}\right).$$

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##### 6: 4.36 Infinite Products and Partial Fractions

###### §4.36 Infinite Products and Partial Fractions

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4.36.1
$$\mathrm{sinh}z=z\prod _{n=1}^{\mathrm{\infty}}\left(1+\frac{{z}^{2}}{{n}^{2}{\pi}^{2}}\right),$$

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4.36.2
$$\mathrm{cosh}z=\prod _{n=1}^{\mathrm{\infty}}\left(1+\frac{4{z}^{2}}{{(2n-1)}^{2}{\pi}^{2}}\right).$$

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##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►A complex linear vector space
${V}$
is called an

*inner product space*if an*inner product*${\u27e8}{u}{,}{v}{\u27e9}{\in}{\u2102}$ is defined for all ${u}{,}{v}{\in}{V}$ with the properties: (i) ${\u27e8}{u}{,}{v}{\u27e9}$ is complex linear in ${u}$ ; (ii) ${\u27e8}{u}{,}{v}{\u27e9}{=}\overline{{\u27e8}{v}{,}{u}{\u27e9}}$ ; (iii) ${\u27e8}{v}{,}{v}{\u27e9}{\ge}{0}$ ; (iv) if ${\u27e8}{v}{,}{v}{\u27e9}{=}{0}$ then ${v}{=}{0}$ . With*norm*defined by … ►The inner product of ${v}$ and ${w}{=}{(}{{d}}_{{0}}{,}{{d}}_{{1}}{,}{{d}}_{{2}}{,}{\mathrm{\dots}}{)}$ is … ►Functions ${f}{,}{g}{\in}{{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ for which ${\u27e8}{f}{-}{g}{,}{f}{-}{g}{\u27e9}{=}{0}$ are identified with each other. The space ${{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ becomes a separable Hilbert space with inner product … ►and functions ${f}{}{(}{x}{)}{,}{g}{}{(}{x}{)}{\in}{{C}}^{{2}}{}{(}{a}{,}{b}{)}$ , assumed real for the moment. The adjoint ${{T}}^{{*}}$ of ${T}$ does satisfy ${\u27e8}{T}{}{f}{,}{g}{\u27e9}{=}{\u27e8}{f}{,}{{T}}^{{*}}{}{g}{\u27e9}$ where ${\u27e8}{f}{,}{g}{\u27e9}{=}{{\int}}_{{a}}^{{b}}{f}{}{(}{x}{)}{}{g}{}{(}{x}{)}{}{d}{x}$ . We integrate by parts twice giving: …