Euler–Tricomi equation

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1: 9.16 Physical Applications

… ►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)). …

2: 8.15 Sums

… ►

8.15.1 $$\gamma (a,\lambda x)={\lambda }^a\sum _{k=0}^{\mathrm{\infty }}\gamma (a+k,x)\frac{{(1-\lambda )}^k}{k!}.$$

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Symbols:

$!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.15 and Ch.8

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8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

…

3: Bibliography C

… ►

Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.

Encodings:

BibTeX

… ►

A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.

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External Links:

MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(iv).

Encodings:

BibTeX

… ►

Elliptic Integrals: Symmetry and Symbolic Integration. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti dei Convegni Lincei, Vol. 147, pp. 161–181.

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External Links:

MathReview (S. L. Kalla), ZentralBlatt

Referenced by:

§19.26(iii), §19.29(i).

Encodings:

BibTeX

… ►

Elementary Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

… ►

On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.

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Notes:

Includes Fortran code

External Links:

Document, ZentralBlatt

Referenced by:

§8.28(vi).

Encodings:

BibTeX

…

4: 8.11 Asymptotic Approximations and Expansions

… ►This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$. … ►

8.11.6 $$\gamma (a,z)\sim -z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ , $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Notes:

See Paris (2002b)

Referenced by:

§8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, . It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where and $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

►

8.11.7 $$\mathrm{\Gamma }(a,z)\sim z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Referenced by:

§8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, $\lambda >1$. It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where $\lambda >1$ and $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

… ►See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. … ►

8.11.12 $$\mathrm{\Gamma }(z,z)\sim z^{z-1}{\mathrm{e}}^{-z}\left(\sqrt{\frac{\pi }{2}}{z}^{\frac{1}{2}}-\frac{1}{3}+\frac{\sqrt{2\pi }}{24z^{\frac{1}{2}}}-\frac{4}{135z}+\frac{\sqrt{2\pi }}{576z^{\frac{3}{2}}}+\frac{8}{2835z^2}+\mathrm{\dots }\right),$$ $|\mathrm{ph}z|\le 2\pi -\delta$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

6.5.35 (Modified form)

Referenced by:

§8.11(v), Firefox 3 users should upgrade, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

The sector of validity for the above equation was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$.

See also:

Annotations for §8.11(v), §8.11 and Ch.8

…

5: 13.11 Series

… ►

13.11.1 $$M(a,b,z)=\mathrm{\Gamma }\left(a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{\frac{1}{2}-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(2a-1\right)}_s{\left(2a-b\right)}_s}{{\left(b\right)}_{s}s!}\left(a-\frac{1}{2}+s\right)I_{a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(iii), §13.6(iii), §8.7

Permalink:

http://dlmf.nist.gov/13.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.11 and Ch.13

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13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

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Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

… ►

13.11.3 $$𝐌(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}\sum _{s=0}^{\mathrm{\infty }}{A}_s{\left(b-2a\right)}^{\frac{1}{2}(1-b-s)}{\left(\frac{1}{2}z\right)}^{\frac{1}{2}(1-b+s)}{J}_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer, $z$: complex variable and $A_n$: coefficients

Source:

Slater (1960, §3.8)

A&S Ref:

13.3.7

Referenced by:

§13.11, Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

… ►For other series expansions see Tricomi (1954, §1.8), Hansen (1975, §§66 and 87), Prudnikov et al. (1990, §6.6), López and Temme (2010a) and López and Pérez Sinusía (2014). …

6: Bibliography E

… ►

Tables of Integral Transforms. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Table errata: Math. Comp. v. 66 (1997), no. 220, p. 1766–1767, v. 65 (1996), no. 215, p. 1384, v. 50 (1988), no. 182, p. 653, v. 41 (1983), no. 164, p. 778–779, v. 27 (1973), no. 122, p. 451, v. 26 (1972), no. 118, p. 599, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239-240.

External Links:

MathReview (H. Kober), ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.32(iv), §10.43(vi), §10.9(iv), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.10(iii), §13.10(iv), §13.23(i), §13.23(ii), §14.17(v), §15.14, §16.20, §16.5, (18.17.21_1), (18.17.34_5), §18.17(ix), §20.10(ii), §20.10(iii), §25.5(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.14.

Encodings:

BibTeX

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Tables of Integral Transforms. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1385, v. 41 (1983), no. 164, pp. 779–780, v. 31 (1977), no. 138, p. 614, v. 31 (1977), no. 137, pp. 328–329, v. 26 (1972), no. 118, p. 599, v. 25 (1971), no. 113, p. 199, v. 23 (1969), no. 106, p. 468.

External Links:

MathReview (H. Kober), ZentralBlatt

Referenced by:

§1.14(viii), §10.22(v), §10.22(vi), §10.43(v), §10.43(vi), §12.12, §13.10(v), §13.10(v), §13.10(vi), §13.23(iii), §13.23(v), §15.14, §15.14, §18.17(ix), §5.13, §8.14.

Encodings:

BibTeX

… ►

Higher Transcendental Functions. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1385, v. 41 (1983), no. 164, p. 778, v. 30 (1976), no. 135, p. 675, v. 25 (1971), no. 115, p. 635, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 112, p. 999, v. 24 (1970), no. 110, p. 504, v. 17 (1963), no. 84, p. 485.

External Links:

MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§13.1, §13.10(ii), §13.10(iv), §13.12, §13.13(i), §13.16(ii), §13.16(iii), §13.23(i), §13.23(ii), §13.25, §13.4(i), §13.4(ii), §13.9(i), §13.9(ii), Ch.13, §14.1, §14.10, §14.12(ii), §14.12(i), §14.12(ii), §14.13, §14.13, §14.17(ii), §14.17(iv), §14.18(ii), §14.18(iii), §14.18(iv), §14.19(i), §14.19(ii), §14.19(iii), §14.19(iv), §14.20(v), §14.21(iii), §14.25, §14.28(i), §14.28(ii), §14.3(iii), §14.3(iii), §14.3(iv), §14.5(iii), §14.5(iv), §14.6(i), §14.6(ii), §14.7(iv), §14.8(i), §14.8(ii), Ch.14, §15.16, §15.4(iii), §15.5(i), §15.6, §15.8(v), §15.8(ii), §15.8(v), §15.9(iii), Ch.15, §16.12, §16.12, §16.13, §16.14(i), §16.14(ii), (16.15.3), §16.15, §16.16(i), §16.16(ii), §16.18, §16.20, §16.20, §16.21, §16.4(ii), §16.5, §16.6, Ch.16, §19.21(i), §19.28, §19.5, §24.16(iii), §24.5(i), §24.7(i), §24.7(ii), §24.8(i), Ch.24, (25.11.12), (25.11.18), (25.11.29), (25.11.30), (25.12.12), (25.12.13), (25.14.1), (25.14.4), (25.14.5), (25.14.6), §25.14(i), §25.14(i), §25.14(ii), §25.14(ii), §25.14, (25.5.1), (25.5.10), (25.5.11), (25.5.20), (25.5.21), (25.5.3), (25.5.8), (25.5.9), §25.5(i), §25.5(iii), §25.5, (25.8.5), (25.8.6), §25.8, Ch.25, §5.7(i), Ch.5, Notations P, Notations P, Notations Q.

Encodings:

BibTeX

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Higher Transcendental Functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 41 (1983), no. 164, p. 778, v. 36 (1981), no. 153, p. 315, v. 30 (1976), no. 135, p. 675, v. 29 (1975), no. 130, p. 670, v. 26 (1972), no. 118, p. 598, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239, v. 17 (1963), no. 84, p. 485.

External Links:

MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§10.22(iv), §10.22(vi), §10.23(iii), §10.23(iii), §10.23(iv), §10.32(i), §10.32(iii), §10.32(iv), §10.43(iv), §10.43(vi), §10.44(iv), §10.60(iv), §10.9(i), §10.9(iii), §10.9(iv), §11.10(vi), §11.4(i), Ch.11, §12.12, §12.12, §12.13(i), §12.5(iv), Ch.12, §18.16(vi), (18.17.21_1), §18.17(iv), §18.17(i), §18.17(viii), (18.35.1), (18.35.2), (18.35.4_5), §18.35(i), (18.5.11_1), (18.5.11_3), §18.5(ii), §18.9(iii), §19.1, §19.10(i), §19.14(ii), (19.25.40), §19.25(vi), §19.5, §19.7(ii), §19.9(i), Ch.19, §20.9(i), §22.15(ii), §23.6(iv), Ch.23, §8.13(i), §8.14, §8.3(i), §8.4, §8.6(iii), §8.7, Ch.8, Notations P, Notations P.

Encodings:

BibTeX

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Higher Transcendental Functions. Vol. III. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 41 (1983), no. 164, p. 778.

External Links:

MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§10.46, §27.2(i), Ch.27, Table 28.1.1, §28.10(iii), §28.2(iii), §28.2(v), §29.1, §29.10, §29.11, §29.14, §29.17(i), §29.17(iii), §29.18(i), §29.18(ii), §29.18(iii), §29.18(iii), §29.2(i), §29.2(ii), §29.3(i), §29.3(ii), §29.3(ii), §29.3(iv), §29.5, §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), §29.8, §29.8, Ch.29, §30.1, §30.10, §30.11(v), §30.11(vi), §30.13(i), §30.14(i), §30.9(iii), Ch.30, §31.2(i), §31.4, §31.5, Ch.31.

Encodings:

BibTeX

…

7: 13.9 Zeros

… ►

13.9.9 $$z=\pm (2n+a)\pi \mathrm{i}+\ln \left(-\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(b-a\right)}{\left(\pm 2n\pi \mathrm{i}\right)}^{b-2a}\right)+O\left(n^{-1}\ln n\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(i)

Permalink:

http://dlmf.nist.gov/13.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(i), §13.9 and Ch.13

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13.9.11 $$T(a,b)=\lfloor -a\rfloor +1,$$ , $\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\lfloor x\rfloor$: floor of $x$ and $T(a,b)$: number of zeros

Permalink:

http://dlmf.nist.gov/13.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

►

13.9.12 $$T(a,b)=\lfloor -a\rfloor ,$$ , ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\lfloor x\rfloor$: floor of $x$ and $T(a,b)$: number of zeros

Permalink:

http://dlmf.nist.gov/13.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

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13.9.16 $$a=-n-\frac{2}{\pi }\sqrt{zn}-\frac{2z}{{\pi }^2}+\frac{1}{2}b+\frac{1}{4}+\frac{z^2\left(\frac{1}{3}-4{\pi }^{-2}\right)+z-{(b-1)}^2+\frac{1}{4}}{4\pi \sqrt{zn}}+O\left(\frac{1}{n}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii), DLMF Update; Version 1.0.13, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16)

Permalink:

http://dlmf.nist.gov/13.9.E16

Encodings:

pMML, png, TeX

Errata (effective with 1.0.13):

In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.

Reported 2016-09-12 by Adri Olde Daalhuis

Errata (effective with 1.0.12):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right hand side, $\sim$ was replaced by $=$.

Reported 2016-07-11 by Rudi Weikard

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

…

8: Bibliography T

… ►

Über die Abzählung der Nullstellen der konfluenten hypergeometrischen Funktionen. Math. Z. 52, pp. 669–675 (German).

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External Links:

ISSN 0025-5874, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.9(i), §13.9(ii).

Encodings:

BibTeX

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Asymptotische Eigenschaften der unvollständigen Gammafunktion. Math. Z. 53, pp. 136–148 (German).

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External Links:

ISSN 0025-5874, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§8.11(iii), §8.11(iv), §8.13(i), §8.13(i), §8.13(iii), §8.15.

Encodings:

BibTeX

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Sugli zeri delle funzioni di cui si conosce una rappresentazione asintotica. Ann. Mat. Pura Appl. (4) 26, pp. 283–300 (Italian).

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External Links:

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

Referenced by:

§13.9(i).

Encodings:

BibTeX

… ►

Funzioni Ellittiche. 2nd edition, Nicola Zanichelli Editore, Bologna (Italian).

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Notes:

German edition (M. Krafft, Ed.) Akad. Verlag., Leipzig, 1948

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

Ch.19.

Encodings:

BibTeX

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Funzioni ipergeometriche confluenti. Edizioni Cremonese, Roma (Italian).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.11.

Encodings:

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9: Bibliography

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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:

English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47

External Links:

ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

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Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.

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External Links:

ISSN 0167-2789, Document, MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.2(v).

Encodings:

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Numerical computation of Tricomi’s psi function by the trapezoidal rule. Computing 39 (3), pp. 271–279.

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External Links:

ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§13.29(iii).

Encodings:

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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

Encodings:

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt

Referenced by:

§3.7(i).

Encodings:

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