solutions as trigonometric and hyperbolic functions

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21: Bibliography F

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Complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$ . Math. Comp. 30 (135), pp. 541–545.

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External Links:

FullText, MathReview (Gerhard Merz), ZentralBlatt

Referenced by:

§4.46.

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

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Error bounds for a uniform asymptotic expansion of the Legendre function $Q_n^{-m}(\cosh z)$ . SIAM J. Math. Anal. 21 (2), pp. 523–535.

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ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

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Solution of the transcendental equation $w{e}^w=x$ . Comm. ACM 16 (2), pp. 123–124.

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Notes:

Includes Fortran code.

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ISSN 0001-0782, Document

Referenced by:

§4.48(iv).

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22: 36.4 Bifurcation Sets

… ►These are real solutions $t_j(𝐱)$, $1\le j\le j_{\max }(𝐱)\le K+1$, of … ►These are real solutions $\{s_j(𝐱),t_j(𝐱)\}$, $1\le j\le j_{\max }(𝐱)\le 4$, of … ►Hyperbolic umbilic bifurcation set (codimension three): … ►Hyperbolic umbilic cusp line (rib): … ►

§36.4(ii) Visualizations

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23: 36.7 Zeros

… ►where ${\xi }_n$ is the real solution of … ►The zeros are approximated by solutions of the equation …There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by … ►

§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

►The zeros of these functions are curves in $𝐱=(x,y,z)$ space; see Nye (2007) for ${\mathrm{\Phi }}_3$ and Nye (2006) for ${\mathrm{\Phi }}^{(\mathrm{H})}$.

24: Bibliography S

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Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt

Referenced by:

§4.47(i).

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Numerical evaluation of integrals of the form $I={\int }_{x_1}^{x_2}f(x)e^{i\varphi (x)}𝑑x$ and the tabulation of the function $\mathrm{Gi}(z)=(1/\pi ){\int }_0^{\mathrm{\infty }}\sin (uz+\frac{1}{3}{u}^3)𝑑u$ . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.

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Document, MathReview (S. C. van Veen), ZentralBlatt

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1st item.

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Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent. Phys. D 3 (1-2), pp. 165–184.

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External Links:

Document, ZentralBlatt

Referenced by:

§32.11(ii).

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Error bounds for a uniform asymptotic expansion of the Legendre function $P_n^{-m}(\cosh z)$ . Quart. Appl. Math. 46 (3), pp. 473–488.

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ISSN 0033-569X, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

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The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§1.13(v).

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25: 2.7 Differential Equations

… ►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►has twice-continuously differentiable solutions …Here $F(x)$ is the error-control function … ►

§2.7(iv) Numerically Satisfactory Solutions

… ►Another is $w_3(z)=\cosh z$, $w_4(z)=\sinh z$. …