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1: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►A fourth integral that is symmetric in only two variables is defined by … ►which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\mathrm{\dots },n$. … … ►

§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

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2: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta }\right)$ as $t\to \mathrm{\infty }$, where $\delta >0$. … ►If the integral converges, then it converges uniformly in any compact domain in the complex $s$-plane not containing any point of the interval $(-\mathrm{\infty },0]$. … ►If $f(t)$ is absolutely integrable on $[0,R]$ for every finite $R$, and the integral (1.14.47) converges, then … ►If $f(t)$ is piecewise continuous on $[0,\mathrm{\infty })$ and the integral (1.14.47) converges, then …

3: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

… ►Unless $p$ is a nonpositive integer, $E_p\left(z\right)$ has a branch point at $z=0$. For $z\ne 0$ each branch of $E_p\left(z\right)$ is an entire function of $p$. … ►For $n=1,2,3,\mathrm{\dots }$ and $x>0$, … ►

§8.19(x) Integrals

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4: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ►As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\mathrm{\infty },0]$ and the principal value is two-valued on $(-\mathrm{\infty },0)$. … ►In the next three equations $x>0$. …($\mathrm{Ei}\left(x\right)$ is undefined when $x=0$, or when $x$ is not real.) … ►

§6.2(ii) Sine and Cosine Integrals

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5: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ is a multivalued function of $z$ with branch point at $z=0$. Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. … ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►

6: 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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7: 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

… ►and for $n=0,1,2,\mathrm{\dots }$, … ► … ►

Hermite Polynomials

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8: 36.2 Catastrophes and Canonical Integrals

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Canonical Integrals

… ►with the contour passing to the lower right of $u=0$. …with the contour passing to the upper right of $u=0$. … ► … ►

§36.2(iii) Symmetries

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9: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

… ►If , then the integral in (19.2.11) is a Cauchy principal value. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

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10: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ► ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …