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31: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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28.5.1 $${\mathrm{fe}}_n(z,q)=C_n(q)\left(z{\mathrm{ce}}_n(z,q)+f_n(z,q)\right),$$

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Defines:

${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $f_n(z,q)$: functions and $C_n(q)$: factor

A&S Ref:

20.3.6 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.2 $${\mathrm{ge}}_n(z,q)=S_n(q)\left(z{\mathrm{se}}_n(z,q)+g_n(z,q)\right),$$

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Defines:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $g_n(z,q)$: functions and $S_n(q)$: factor

A&S Ref:

20.3.7 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

… ►The factors $C_n(q)$ and $S_n(q)$ in (28.5.1) and (28.5.2) are normalized so that ►

28.5.5 $${(C_n(q))}^2{\int }_0^{2\pi }{(f_n(x,q))}^{2}dx={(S_n(q))}^2{\int }_0^{2\pi }{(g_n(x,q))}^{2}dx=\pi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $x$: real variable, $f_n(z,q)$: functions, $g_n(z,q)$: functions, $C_n(q)$: factor and $S_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

… ►(Other normalizations for $C_n(q)$ and $S_n(q)$ can be found in the literature, but most formulas—including connection formulas—are unaffected since ${\mathrm{fe}}_n(z,q)/C_n(q)$ and ${\mathrm{ge}}_n(z,q)/S_n(q)$ are invariant.) …

32: 3.7 Ordinary Differential Equations

… ►The remaining two equations are supplied by boundary conditions of the form … ►The eigenvalues ${\lambda }_k$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …

33: 20.13 Physical Applications

… ►For $\tau =\mathrm{i}t$, with $\alpha ,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation …These two apparently different solutions differ only in their normalization and boundary conditions. … ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

34: 29.15 Fourier Series and Chebyshev Series

… ►be the eigenvector corresponding to $H_m$ and normalized so that … ►Since (29.2.5) implies that $\cos \varphi =\mathrm{sn}(z,k)$, (29.15.1) can be rewritten in the form …The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an $(n+1)\times (n+1)$ tridiagonal matrix; see Arscott and Khabaza (1962). …

35: 28.15 Expansions for Small $q$

§28.15 Expansions for Small $q$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

… ►

28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(ii), §28.15 and Ch.28

…

36: DLMF Project News

error generating summary

37: 21.7 Riemann Surfaces

… ►The ${\omega }_j$ are normalized so that … ►

21.7.8 $$\zeta ={\sum _{j=1}^g{\omega }_j\frac{\partial }{\partial z_j}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}}.$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector and $\omega$: differential

Permalink:

http://dlmf.nist.gov/21.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

►Then the prime form on the corresponding compact Riemann surface $\mathrm{\Gamma }$ is defined by ►

21.7.9 $$E(P_1,P_2)=\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left({\int }_{P_1}^{P_2}𝝎\right|𝛀)/\left(\sqrt{\zeta (P_1)}\sqrt{\zeta (P_2)}\right),$$

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Defines:

$E(P_1,P_2)$: prime form (locally)

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\int$: integral, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $P_1,P_2$: points and $𝝎$: vector of differentials

Permalink:

http://dlmf.nist.gov/21.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

… ►These are Riemann surfaces that may be obtained from algebraic curves of the form …

38: 30.8 Expansions in Series of Ferrers Functions

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30.8.1 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-R}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: coefficients

A&S Ref:

21.7.1 (in different form)

Referenced by:

§30.16(ii), §30.16(ii)

Permalink:

http://dlmf.nist.gov/30.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

… ►(note that $A_{-R}=0$) that satisfies the normalizing condition … ►

30.8.9 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-\mathrm{\infty }}^{-N-1}{(-1)}^{k}a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right)+\sum _{k=-N}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖰}_{n+2k}^m\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients and $N$: upper limit

A&S Ref:

21.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/30.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(ii), §30.8 and Ch.30

… ►The set of coefficients $a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2)$, $k=-N-1,-N-2,\mathrm{\dots }$, is the recessive solution of (30.8.4) as $k\to -\mathrm{\infty }$ that is normalized by …

39: 18.28 Askey–Wilson Class

… ►The Askey–Wilson polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. The $q$-Racah polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots },N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+c{q}^{y+1}\}$, $y=0,1,\mathrm{\dots },N$, with $c$ a constant. … ►Assume $a,b,c,d$ are all real, or two of them are real and two form a conjugate pair, or none of them are real but they form two conjugate pairs. … ►

18.28.29 $$\underset{q\to 1}{lim}\frac{p_n(1-\frac{1}{2}x{(1-q)}^2;q^a,q^b,q^c,q^d|q)}{{(1-q)}^{3n}}=W_n(x;a,b,c,d).$$

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Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, $W_n(x;a,b,c,d)$: Wilson polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Substitute (18.28.1) in the left-hand side, take the limit, and use (18.26.1).

Notes:

Another form of this limit is given in Koekoek et al. (2010, (14.1.21)).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

…

40: 33.9 Expansions in Series of Bessel Functions

… ►

33.9.3 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\eta )}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{I}_k\left(2\sqrt{t}\right),$$ $\eta >0$,

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

A&S Ref:

14.4.1

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

►

33.9.4 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\left|\eta \right|)}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{J}_k\left(2\sqrt{t}\right),$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

… ►

33.9.6 $$G_{\mathrm{\ell }}(\eta ,\rho )\sim \frac{{\rho }^{-\mathrm{\ell }}}{(\mathrm{\ell }+\frac{1}{2}){\lambda }_{\mathrm{\ell }}(\eta )C_{\mathrm{\ell }}\left(\eta \right)}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{(-1)}^k{b}_k{t}^{k/2}{K}_k\left(2\sqrt{t}\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter, $b_k$: coefficient and ${\lambda }_{\mathrm{\ell }}(\eta )$: coefficient

A&S Ref:

14.4.2

Referenced by:

§33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

…