de Branges–Wilson beta integral

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1: 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 $a,b\in ℂ$ …

2: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►Note: If $f(x)$ is continuous and $\alpha$ and $\beta$ are real numbers such that $f(x)=O\left(x^{\alpha }\right)$ as $x\to 0+$ and $f(x)=O\left(x^{\beta }\right)$ as $x\to \mathrm{\infty }$, then $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma \in (-\alpha ,-\beta )$. … ►

§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).

3: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(iii) Integral Representation

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§8.17(iv) Recurrence Relations

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§8.17(v) Continued Fraction

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§8.17(vi) Sums

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4: 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(ii) Graphics

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§8.19(x) Integrals

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5: 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

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6: 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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§8.21(iv) Interrelations

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§8.21(v) Special Values

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7: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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Values at Infinity

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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8: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ► …where $\mathrm{B}(x,y)$ is the beta function (§5.12) and … ►All other elliptic cases are integrals of the second kind. … ►Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …

9: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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§7.18(i) Definition

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§7.18(iii) Properties

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Hermite Polynomials

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10: 5.13 Integrals

§5.13 Integrals

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Barnes’ Beta Integral

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Ramanujan’s Beta Integral

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de Branges–Wilson Beta Integral

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