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1: 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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2: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

… ►The principal values satisfy … ►

§22.15(ii) Representations as Elliptic Integrals

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3: 22.2 Definitions

§22.2 Definitions

… ►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ► $\mathrm{s}\mathrm{s}(z,k)=1$. …

4: 36.2 Catastrophes and Canonical Integrals

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

… ►(elliptic umbilic). …(hyperbolic umbilic). ►

Canonical Integrals

… ►Addendum: For further special cases see §36.2(iv) …

5: 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. … … ►All other elliptic cases are integrals of the second kind. …

6: 22.16 Related Functions

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§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

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Definition

… ►With $q$ as in (22.2.1) and $\zeta =\pi x/(2K)$, … ►

Relation to Elliptic Integrals

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Relation to the Elliptic Integral $E(\varphi ,k)$

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7: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

… ►is called an elliptic integral. …Thus the elliptic part of (19.2.1) is … ►

§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

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8: 36.4 Bifurcation Sets

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Bifurcation (Catastrophe) Set for Umbilics

… ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►Elliptic umbilic cusp lines (ribs): … ►Hyperbolic umbilic bifurcation set (codimension three): … ►

§36.4(ii) Visualizations

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9: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . … ►

§36.5(iii) Umbilics

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Elliptic Umbilic Stokes Set (Codimension three)

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Hyperbolic Umbilic Stokes Set (Codimension three)

►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. …

10: 36.3 Visualizations of Canonical Integrals

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Figure 36.3.6: Modulus of elliptic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,0\right)|$.

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Figure 36.3.7: Modulus of elliptic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,2\right)|$.

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Figure 36.3.8: Modulus of elliptic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,4\right)|$.

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Figure 36.3.15: Phase of elliptic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,0\right)$.

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Figure 36.3.16: Phase of elliptic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,2\right)$.

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