Jacobi imaginary transformation

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21: 31.2 Differential Equations

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Jacobi’s Elliptic Form

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$F$-Homotopic Transformations

… ►By composing these three steps, there result $2^3=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). ►

Homographic Transformations

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Composite Transformations

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22: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Assume that $w({\mathrm{e}}^{\mathrm{i}\varphi })=w({\mathrm{e}}^{-\mathrm{i}\varphi })$. … ►

$w_1(x)={(1-x^2)}^{-\frac{1}{2}}w\left(x+\mathrm{i}{(1-x^2)}^{\frac{1}{2}}\right),$

… ►After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi }_n$. … ►Askey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle. … ►

18.33.19 $$d\mu (z)=\frac{1}{2\pi \mathrm{i}}w(z)\frac{dz}{z}$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $z$: complex variable

Referenced by:

§18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33 and Ch.18

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23: Errata

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23.6.11 $$\sigma \left({\omega }_2\right)=-2{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1({\omega }_1{\tau }^2+{\omega }_3-{\omega }_2)\right){\theta }_3(0,q)}{\pi q^{1/4}{\theta }_1^{\prime }(0,q)}$$

The factor $2{\omega }_1\mathrm{i}\exp \left(\frac{1}{2}{\eta }_1{\omega }_1{\tau }^2\right)$ has been corrected to be $-2{\omega }_1\exp \left(\frac{1}{2}{\eta }_1({\omega }_1{\tau }^2+{\omega }_3-{\omega }_2)\right)$.

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23.6.12 $$\sigma \left({\omega }_3\right)=2\mathrm{i}{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1{\omega }_1{\tau }^2\right){\theta }_4(0,q)}{\pi q^{1/4}{\theta }_1^{\prime }(0,q)}$$

The factor $-2{\omega }_1\exp \left(\frac{1}{2}{\eta }_1{\omega }_1\right)$ has been corrected to be $2\mathrm{i}{\omega }_1\exp \left(\frac{1}{2}{\eta }_1{\omega }_1{\tau }^2\right)$.

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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$ℱ\left(f\right)\left(x\right)$	$ℱf\left(x\right)$
Fourier Cosine	${ℱ}_c\left(f\right)\left(x\right)$	${ℱ}_{c}f\left(x\right)$
Fourier Sine	${ℱ}_s\left(f\right)\left(x\right)$	${ℱ}_{s}f\left(x\right)$
Laplace	$ℒ\left(f\right)\left(s\right)$	$ℒf\left(s\right)$	$ℒ(f(t);s)$
Mellin	$ℳ\left(f\right)\left(s\right)$	$ℳf\left(s\right)$	$ℳ(f;s)$
Hilbert	$\mathscr{H}\left(f\right)\left(s\right)$	$\mathscr{H}f\left(s\right)$	$\mathscr{H}(f;s)$
Stieltjes	$𝒮\left(f\right)\left(s\right)$	$𝒮f\left(s\right)$	$𝒮(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime }$). The correct entries are $-k^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}{K}^{\prime }$	$z+\mathrm{i}{K}^{\prime }$	$z+2K$	$z+2K+2\mathrm{i}{K}^{\prime }$	$z+2\mathrm{i}{K}^{\prime }$
$\mathrm{⋮}$
$\mathrm{cd}u$	$-\mathrm{sn}z$	$-k^{-1}\mathrm{ns}z$	$k^{-1}\mathrm{dc}z$	$-\mathrm{cd}z$	$-\mathrm{cd}z$	$\mathrm{cd}z$
$\mathrm{⋮}$
$\mathrm{dc}u$	$-\mathrm{ns}z$	$-k\mathrm{sn}z$	$k\mathrm{cd}z$	$-\mathrm{dc}z$	$-\mathrm{dc}z$	$\mathrm{dc}z$
$\mathrm{⋮}$

Reported 2014-02-28 by Svante Janson.

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The entry for $\mathrm{sn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$ has been corrected. The correct entry is $(1+\mathrm{i})({(1+k^{\prime })}^{1/2}-\mathrm{i}{(1-k^{\prime })}^{1/2})/(2k^{1/2})$. Originally the terms ${(1+k^{\prime })}^{1/2}$ and ${(1-k^{\prime })}^{1/2}$ were given incorrectly as ${(1+k)}^{1/2}$ and ${(1-k)}^{1/2}$.

Similarly, the entry for $\mathrm{dn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$ has been corrected. The correct entry is $(-1+\mathrm{i})k_{}^{\prime }{}_{}{}^{1/2}({(1+k)}^{1/2}+i{(1-k)}^{1/2})/2$. Originally the terms ${(1+k)}^{1/2}$ and ${(1-k)}^{1/2}$ were given incorrectly as ${(1+k^{\prime })}^{1/2}$ and ${(1-k^{\prime })}^{1/2}$

Reported 2014-02-28 by Svante Janson.

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24: 15.12 Asymptotic Approximations

… ►See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1). … ►

15.12.9 $${(z+1)}^{3\lambda /2}{(2\lambda )}^{c-1}𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b+2\lambda }{c};-z)={\lambda }^{-1/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_0(\zeta )+O({\lambda }^{-1})\right)+{\lambda }^{-2/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_1(\zeta )+O({\lambda }^{-1})\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $a_j(\zeta )$: function

Referenced by:

§15.12(iii)

Permalink:

http://dlmf.nist.gov/15.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F(a+e_1\lambda ,b+e_2\lambda ;c+e_3\lambda ;z)$ can be obtained with $e_j=\pm 1$ or $0$, $j=1,2,3$. …