# products

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##### 1: 30.10 Series and Integrals
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 $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$ 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. …
##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
A complex linear vector space $V$ is called an inner product space if an inner product $\left\langle u,v\right\rangle\in\mathbb{C}$ is defined for all $u,v\in V$ with the properties: (i) $\left\langle u,v\right\rangle$ is complex linear in $u$ ; (ii) $\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$ ; (iii) $\left\langle v,v\right\rangle\geq 0$ ; (iv) if $\left\langle v,v\right\rangle=0$ then $v=0$ . With norm defined byThe inner product of $v$ and $w=(d_{0},d_{1},d_{2},\ldots)$ isFunctions $f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)$ for which $\left\langle f-g,f-g\right\rangle=0$ are identified with each other. The space $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ becomes a separable Hilbert space with inner productand functions $f(x),g(x)\in C^{2}(a,b)$ , assumed real for the moment. The adjoint ${T}^{*}$ of $T$ does satisfy $\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle$ where $\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x$ . We integrate by parts twice giving: