Euler homogeneity relation

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1: 19.18 Derivatives and Differential Equations

… ►and two similar equations obtained by permuting $x,y,z$ in (19.18.10). ►More concisely, if $v=R_{-a}(𝐛;𝐳)$, then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation: …

2: 31.14 General Fuchsian Equation

… ►The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …The three sets of parameters comprise the singularity parameters $a_j$, the exponent parameters $\alpha ,\beta ,{\gamma }_j$, and the $N-2$ free accessory parameters $q_j$. … ►

Normal Form

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${\tilde{\gamma }}_j={\displaystyle \frac{{\gamma }_j}{2}}\left({\displaystyle \frac{{\gamma }_j}{2}}-1\right).$

… ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …

3: 9.16 Physical Applications

… ►The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)). … ►The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)). … ►An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of $\mathrm{Ai}\left(x\right)$. …

4: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

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§13.12.

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Recursion relations for the $3$-$j$ symbols. Nuclear Physics A 113 (1), pp. 215–220.

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External Links:

Document

Referenced by:

§34.3(iii).

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

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On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.

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External Links:

Document, MathReview (W. R. Wasow), ZentralBlatt

Referenced by:

§12.2(vi), §2.7(iv), §2.7(iv).

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