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1: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►The Fourier transform of a real- or complex-valued function $f(t)$ is defined by … ►The Laplace transform of $f$ is defined by … ►The Mellin transform of a real- or complex-valued function $f(x)$ is defined by … ►The Stieltjes transform of a real-valued function $f(t)$ is defined by …

2: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

… ►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_p\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed. ►

Other Integral Representations

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§8.19(x) Integrals

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3: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

►The principal value of the exponential integral $E_1\left(z\right)$ is defined by … ►This is also true of the functions $\mathrm{Ci}\left(z\right)$ and $\mathrm{Chi}\left(z\right)$ defined in §6.2(ii). … ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

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4: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►With $\gamma$ and $\mathrm{\Gamma }$ denoting here the general values of the incomplete gamma functions (§8.2(i)), we define … ►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)). Elsewhere in the sector $|\mathrm{ph}z|\le \pi$ the principal values are defined by analytic continuation from $\mathrm{ph}z=0$; compare §4.2(i). … ►

5: 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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6: 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 … ► … ►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: …

7: 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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8: 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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9: 20 Theta Functions

Chapter 20 Theta Functions

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10: 3.4 Differentiation

… ►If $f^{(n+2)}(x)$ is continuous on the interval $I$ defined in §3.3(i), then the remainder in (3.4.1) is given by … ►where $c_n$ is defined by (3.3.12), with numerical values as in §3.3(ii). … ►where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and $x_0$ is interior to $C$. …The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►The integral (3.4.18) becomes …