integrals of vector-valued functions

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►If $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma$ in , then the integral (1.14.32) converges and $ℳf\left(s\right)$ is an analytic function of $s$ in the vertical strip . … ►

§1.14(viii) Compendia

►For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

2: 7.2 Definitions

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§7.2(i) Error Functions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

… ► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. … ►

§7.2(iv) Auxiliary Functions

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3: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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4: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►Just as the elementary function $R_C(x,y)$ (§19.2(iv)) is the degenerate case … ►

§19.16(ii) $R_{-a}(𝐛;𝐳)$

►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The $R$-function is often used to make a unified statement of a property of several elliptic integrals. …

5: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

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Other Integral Representations

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§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(x) Integrals

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6: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

… ► $\mathrm{Si}\left(z\right)$ is an odd entire function. … ►

§6.2(iii) Auxiliary Functions

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7: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(iii) Integral Representations

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Spherical-Bessel-Function Expansions

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§8.21(vii) Auxiliary Functions

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§8.21(viii) Asymptotic Expansions

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8: 5.12 Beta Function

§5.12 Beta Function

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Euler’s Beta Integral

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Figure 5.12.1: $t$-plane. Contour for first loop integral for the beta function. Magnify

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Pochhammer’s Integral

►When $a,b\in ℂ$ …

9: 9.12 Scorer Functions

§9.12 Scorer Functions

… ►where … ►

§9.12(vii) Integral Representations

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Integrals

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10: 5.2 Definitions

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§5.2(i) Gamma and Psi Functions

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Euler’s Integral

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5.2.1 $$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $\mathrm{\Re }z>0$.

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Defines:

$\mathrm{\Gamma }\left(z\right)$: gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

6.1.1

Referenced by:

§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)

Permalink:

http://dlmf.nist.gov/5.2.E1

Encodings:

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See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

… ►It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^n/n!$ at $z=-n$. … ►

5.2.2 $$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$ $z\ne 0,-1,-2,\mathrm{\dots }$.

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Defines:

$\psi \left(z\right)$: psi (or digamma) function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable

A&S Ref:

6.3.1

Permalink:

http://dlmf.nist.gov/5.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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