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1: 31.2 Differential Equations

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Jacobi’s Elliptic Form

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2: 32.2 Differential Equations

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§32.2(iv) Elliptic Form

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3: 22.18 Mathematical Applications

… ►Algebraic curves of the form $y^2=P(x)$, where $P$ is a nonsingular polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are elliptic curves, which are also considered in §23.20(ii). …For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …

4: 22.15 Inverse Functions

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§22.15(ii) Representations as Elliptic Integrals

… ►The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. … ►

5: 29.2 Differential Equations

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§29.2(ii) Other Forms

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6: 29.12 Definitions

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§29.12(i) Elliptic-Function Form

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Table 29.12.1: Lamé polynomials.

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$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
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… ►With the substitution $\xi ={\mathrm{sn}}^2(z,k)$ every Lamé polynomial in Table 29.12.1 can be written in the form …

7: 22.5 Special Values

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§22.5(ii) Limiting Values of $k$

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Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
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Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$.

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$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
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8: 36.2 Catastrophes and Canonical Integrals

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

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9: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form … ► $K$ always has the form $T\times {\mathbb{Z}}^r$ (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the rank of $K$, raises questions of great difficulty, many of which are still open. …

10: 22.4 Periods, Poles, and Zeros

… ►Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form $2mK+2n\mathrm{i}{K}^{\prime }$, where $m,n\in \mathbb{Z}$. …