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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:

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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:

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

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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:

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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 …Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. … ►

1.18.3 $$c_n=⟨v,v_n⟩.$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $n$: nonnegative integer

Referenced by:

§1.18(i)

Permalink:

http://dlmf.nist.gov/1.18.E3

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Annotations for §1.18(i), §1.18 and Ch.1

… ►thus generalizing the inner product of (1.18.9). …

8: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

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§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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