fundamental parallelogram

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21: 28.10 Integral Equations

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28.10.9 $${\int }_0^{\pi /2}{J}_0\left(2\sqrt{q({\cos }^2\tau -{\sin }^2\zeta )}\right){\mathrm{ce}}_{2n}(\tau ,q)d\tau =w_{\text{II}}(\frac{1}{2}\pi ;a_{2n}\left(q\right),q){\mathrm{ce}}_{2n}(\zeta ,q),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $q=h^2$: parameter, $n$: integer and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.10.E9

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See also:

Annotations for §28.10(ii), §28.10 and Ch.28

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28.10.10 $${\int }_0^{\pi }{J}_0\left(2\sqrt{q}(\cos \tau +\cos \zeta )\right){\mathrm{ce}}_n(\tau ,q)d\tau =w_{\text{II}}(\pi ;a_n\left(q\right),q){\mathrm{ce}}_n(\zeta ,q).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.10.E10

Encodings:

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See also:

Annotations for §28.10(ii), §28.10 and Ch.28

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22: Bibliography V

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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23: 1.4 Calculus of One Variable

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Fundamental Theorem of Calculus

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24: 4.23 Inverse Trigonometric Functions

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§4.23(v) Fundamental Property

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25: 10.73 Physical Applications

… ►It is fundamental in the study of electromagnetic wave transmission. …

26: 13.14 Definitions and Basic Properties

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§13.14(v) Numerically Satisfactory Solutions

►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … ►A fundamental pair of solutions that is numerically satisfactory in the sector $|\mathrm{ph}z|\le \pi$ near the origin is …

27: 28.4 Fourier Series

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28.4.24 $$\frac{A_{2m}^{2n}(q)}{A_0^{2n}(q)}=\frac{{(-1)}^m}{{(m!)}^2}{\left(\frac{q}{4}\right)}^m\frac{\pi \left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}(\frac{1}{2}\pi ;a_{2n}\left(q\right),q)},$$

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Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $n$: integer, $w_{\text{II}}(z;a,q)$: fundamental solution and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E24

Encodings:

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See also:

Annotations for §28.4(vii), §28.4 and Ch.28

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28.4.26 $$\frac{B_{2m+1}^{2n+1}(q)}{B_1^{2n+1}(q)}=\frac{{(-1)}^m}{{\left({\left(\frac{1}{2}\right)}_{m+1}\right)}^2}{\left(\frac{q}{4}\right)}^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}{w_{\text{I}}(\frac{1}{2}\pi ;b_{2n+1}\left(q\right),q)},$$

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Symbols:

$O\left(x\right)$: order not exceeding, $b_n\left(q\right)$: eigenvalues of Mathieu equation, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer, $w_{\text{I}}(z;a,q)$: fundamental solution and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E26

Encodings:

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See also:

Annotations for §28.4(vii), §28.4 and Ch.28

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28: Bibliography G

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Quantum mechanics: fundamentals. Second edition, Springer-Verlag, New York.

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External Links:

ISBN 0-387-22023-2, MathReview (Howard E. Brandt), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

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29: Bibliography K

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The Art of Computer Programming. Vol. 1: Fundamental Algorithms. 1st edition, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont.

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Notes:

Second printing of second edition by Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam, 1975.

External Links:

MathReview (M. Muller), ZentralBlatt

Referenced by:

§1.4(iii), (25.11.33).

Encodings:

BibTeX

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30: 18.38 Mathematical Applications

… ►Classical OP’s play a fundamental role in Gaussian quadrature. …