reduction to basic elliptic integrals

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1: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\mathrm{\dots },n$. …The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … … ►Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …

2: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►If $f(t)$ is absolutely integrable on $(-\mathrm{\infty },\mathrm{\infty })$, then $F(x)$ is continuous, $F(x)\to 0$ as $x\to \pm \mathrm{\infty }$, and … ►If also ${lim}_{t\to 0+}f(t)/t$ exists, then … ►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 $\sigma \in (-\alpha ,-\beta )$. … ►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 $\delta >0$. …

3: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

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§23.2(iii) Periodicity

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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

… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(x) Integrals

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5: 22.15 Inverse Functions

§22.15 Inverse Functions

… ►The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). … ►Equations (22.15.1) and (22.15.4), for $\mathrm{arcsn}(x,k)$, are equivalent to (22.15.12) and also to … ►

§22.15(ii) Representations as Elliptic 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

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7: 36.2 Catastrophes and Canonical Integrals

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Normal Forms for Umbilic Catastrophes with Codimension $K=3$

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Canonical Integrals

… ► ${\mathrm{\Psi }}_1$ is related to the Airy function (§9.2): … … ►

§36.2(iv) Addendum to 36.2(ii) Special Cases

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8: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

… ►is called an elliptic integral. … ►

§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

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

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9: 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(v) Special Values

… ►When $z\to \mathrm{\infty }$ with $|\mathrm{ph}z|\le \pi -\delta$ (), … ►

10: 22.2 Definitions

§22.2 Definitions

… ►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ► … ► $\mathrm{s}\mathrm{s}(z,k)=1$. … ►and on the left-hand side of (22.2.11) $\mathrm{p}$, $\mathrm{q}$ are any pair of the letters $\mathrm{s}$, $\mathrm{c}$, $\mathrm{d}$, $\mathrm{n}$, and on the right-hand side they correspond to the integers $1,2,3,4$.