# reduction to basic elliptic integrals

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##### 1: 19.16 Definitions
###### §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 $1,\dots,n$. …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 $(-\infty,\infty)$, then $F(x)$ is continuous, $F(x)\to 0$ as $x\to\pm\infty$, 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 $x\to\infty$, then $x^{\sigma-1}f(x)$ is integrable on $(0,\infty)$ 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 $t\to\infty$, where $\delta>0$. …
##### 4: 8.19 Generalized Exponential Integral
###### §8.19(i) Definition and Integral Representations
In Figures 8.19.28.19.5, height corresponds to the absolute value of the function and color to the phase. …
##### 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 $\operatorname{arcsn}\left(x,k\right)$, are equivalent to (22.15.12) and also to
##### 6: 6.2 Definitions and Interrelations
###### §6.2(i) Exponential and Logarithmic Integrals
The logarithmic integral is defined by …
##### 7: 36.2 Catastrophes and Canonical Integrals
###### Canonical Integrals
$\Psi_{1}$ is related to the Airy function (§9.2): … …
##### 8: 19.2 Definitions
###### §19.2(i) General EllipticIntegrals
is called an elliptic integral. …
##### 9: 8.21 Generalized Sine and Cosine Integrals
###### §8.21(v) Special Values
When $z\to\infty$ with $|\operatorname{ph}z|\leq\pi-\delta$ ($<\pi$), …
##### 10: 22.2 Definitions
###### §22.2 Definitions
where $K\left(k\right)$, ${K^{\prime}}\left(k\right)$ are defined in §19.2(ii). … $\operatorname{ss}\left(z,k\right)=1$. … 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 $1,2,3,4$.