definite integral

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31: 21.1 Special Notation

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$g,h$	positive integers.
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$𝛀$	$g\times g$ complex, symmetric matrix with $\mathrm{\Im }𝛀$ strictly positive definite, i.e., a Riemann matrix.
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${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

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32: 3.11 Approximation Techniques

… ►They enjoy an orthogonal property with respect to integrals: ►

3.11.8

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/3.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(ii), §3.11 and Ch.3

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3.11.10 $$c_n=\frac{2}{\pi }{\int }_0^{\pi }f(\cos \theta )\cos \left(n\theta \right)d\theta ,$$ $n=0,1,2,\mathrm{\dots }$,

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Defines:

$c_n$: coefficients (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§3.11(ii)

Permalink:

http://dlmf.nist.gov/3.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(ii), §3.11(ii), §3.11 and Ch.3

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3.11.26 $$F(s)=ℒf\left(s\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}f(t)dt$$

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Defines:

$F(s)$: Laplace transform of $f(t)$ (locally)

Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Keywords:

Laplace transform

A&S Ref:

29.1.1

Permalink:

http://dlmf.nist.gov/3.11.E26

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(z\right)$ from $ℒ(f;z)$.

See also:

Annotations for §3.11(iv), §3.11(iv), §3.11 and Ch.3

… ►The matrix is symmetric and positive definite, but the system is ill-conditioned when $n$ is large because the lower rows of the matrix are approximately proportional to one another. …