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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 $M(a,b,z)$ 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

►

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

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

►

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

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

…

6: 4.36 Infinite Products and Partial Fractions

§4.36 Infinite Products and Partial Fractions

►

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

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.68

Permalink:

http://dlmf.nist.gov/4.36.E1

Encodings:

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See also:

Annotations for §4.36 and Ch.4

►

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

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.69

Permalink:

http://dlmf.nist.gov/4.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

…

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 $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$ ; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$ ; (iii) $⟨v,v⟩\ge 0$ ; (iv) if $⟨v,v⟩=0$ 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 $⟨f-g,f-g⟩=0$ 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 $⟨Tf,g⟩=⟨f,T^{*}g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$ . We integrate by parts twice giving: …

8: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

►

§20.5(i) Single Products

… ►

Jacobi’s Triple Product

… ►

§20.5(iii) Double Products

… ►

9: 10.28 Wronskians and Cross-Products

§10.28 Wronskians and Cross-Products

…

10: 10.63 Recurrence Relations and Derivatives

… ►

§10.63(ii) Cross-Products

… ►

10.63.7 $$p_{\nu }{s}_{\nu }=r_{\nu }^2+q_{\nu }^2.$$

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Symbols:

$\nu$: complex parameter, $p_{\nu }$: cross-product, $q_{\nu }$: cross-product, $r_{\nu }$: cross-product and $s_{\nu }$: cross-product

A&S Ref:

9.9.20

Referenced by:

§10.63(ii)

Permalink:

http://dlmf.nist.gov/10.63.E7

Encodings:

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See also:

Annotations for §10.63(ii), §10.63 and Ch.10

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