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1: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by … ►

Inverse Hyperbolic Sine

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Inverse Hyperbolic Cosine

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Inverse Hyperbolic Tangent

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2: 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) …

3: 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 … ►Hyperbolic umbilic bifurcation set (codimension three): … ►Hyperbolic umbilic cusp line (rib): … ►

§36.4(ii) Visualizations

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4: 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. …

5: 36.3 Visualizations of Canonical Integrals

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Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,0\right)|$.

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Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,1\right)|$.

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Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,2\right)|$.

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Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,3\right)|$.

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Figure 36.3.18: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,0\right)$.

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6: 4.35 Identities

§4.35 Identities

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§4.35(i) Addition Formulas

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§4.35(ii) Squares and Products

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§4.35(iii) Multiples of the Argument

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§4.35(iv) Real and Imaginary Parts; Moduli

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7: 36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;𝐱)$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$; canonical integrals ${\mathrm{\Psi }}_K\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$ generated by the catastrophes. …

8: 20 Theta Functions

Chapter 20 Theta Functions

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9: 36.6 Scaling Relations

§36.6 Scaling Relations

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${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)=k^{{\beta }^{(\mathrm{U})}}{\mathrm{\Psi }}^{(\mathrm{U})}\left({𝐲}^{(\mathrm{U})}(k)\right),$

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$\text{<span class="ltx\_text hilite">umbilics</span>: }{𝐲}^{(\mathrm{U})}(k)=(x{k}^{2/3},y{k}^{2/3},z{k}^{1/3}).$

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Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

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$\text{<span class="ltx\_text hilite">umbilics</span>: }{\beta }^{(\mathrm{U})}={\displaystyle \frac{1}{3}}.$

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10: 4.34 Derivatives and Differential Equations

§4.34 Derivatives and Differential Equations

… ►With $a\ne 0$, the general solutions of the differential equations … ►

4.34.12 $$w=(1/a)\sinh \left(az+c\right),$$

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Symbols:

$\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E12

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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4.34.13 $$w=(1/a)\cosh \left(az+c\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.14 $$w=(1/a)\coth \left(az+c\right),$$

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Symbols:

$\coth z$: hyperbolic cotangent function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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