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21: 23.5 Special Lattices

… ►For the case ${\omega }_3={\mathrm{e}}^{\pi \mathrm{i}/3}{\omega }_1$ see §23.5(v). … ►This occurs when ${\omega }_1$ is real and positive and ${\omega }_3={\mathrm{e}}^{\pi \mathrm{i}/3}{\omega }_1$. … ►

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Figure 23.5.2: Equianharmonic lattice. $2{\omega }_3={\mathrm{e}}^{\pi \mathrm{i}/3}2{\omega }_1$, $2{\omega }_1-2{\omega }_3={\mathrm{e}}^{-\pi \mathrm{i}/3}2{\omega }_1$. Magnify

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23.5.6 $${\eta }_1={\mathrm{e}}^{\pi \mathrm{i}/3}{\eta }_3=\frac{\pi }{2\sqrt{3}{\omega }_1},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.5(v), §23.5 and Ch.23

… ►

23.5.7 $$e_1={\mathrm{e}}^{2\pi \mathrm{i}/3}{e}_3={\mathrm{e}}^{-2\pi \mathrm{i}/3}{e}_2=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^6}{2^{14/3}{\pi }^2{\omega }_1^2},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $e_j$: Weierstrass lattice roots, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝕃$: lattice and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.5(v), §23.5 and Ch.23

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22: 4.4 Special Values and Limits

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4.4.6 $${\mathrm{e}}^{\pm \pi \mathrm{i}/2}=\pm \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

A&S Ref:

4.2.27

Permalink:

http://dlmf.nist.gov/4.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

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4.4.8 $${\mathrm{e}}^{\pm \pi \mathrm{i}/3}=\frac{1}{2}\pm \mathrm{i}\frac{\sqrt{3}}{2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

►

4.4.9 $${\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}=-\frac{1}{2}\pm \mathrm{i}\frac{\sqrt{3}}{2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

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4.4.10 $${\mathrm{e}}^{\pm \pi \mathrm{i}/4}=\frac{1}{\sqrt{2}}\pm \mathrm{i}\frac{1}{\sqrt{2}},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

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4.4.11 $${\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}=-\frac{1}{\sqrt{2}}\pm \mathrm{i}\frac{1}{\sqrt{2}},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

…

23: 10.5 Wronskians and Cross-Products

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10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.5 $$𝒲\left\{H_{\nu }^{(1)}\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=H_{\nu +1}^{(1)}\left(z\right)H_{\nu }^{(2)}\left(z\right)-H_{\nu }^{(1)}\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-4\mathrm{i}/(\pi z).$$

ⓘ

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.17

Permalink:

http://dlmf.nist.gov/10.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

24: 9.7 Asymptotic Expansions

… ►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. ►

Table 9.7.1: $\chi (n)$.

► ►►►

$n$	$\chi (n)$	$n$	$\chi (n)$	$n$	$\chi (n)$	$n$	$\chi (n)$
…
5	2.95	10	4.06	15	4.94	20	5.68

►

… ►

9.7.13 $$\mathrm{Bi}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/3}\right)\sim \sqrt{\frac{2}{\pi }}\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}/6}}{z^{1/4}}\left(\cos \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Proof sketch:

See Olver (1997b, pp. 392–393 and 413–414).

Permalink:

http://dlmf.nist.gov/9.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

►

9.7.14 $${\mathrm{Bi}}^{\prime }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/3}\right)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\mp \pi \mathrm{i}/6}{z}^{1/4}\left(-\sin \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}+\cos \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Proof sketch:

See Olver (1997b, pp. 392–393 and 413–414).

Referenced by:

§9.7(iv), §9.7(v)

Permalink:

http://dlmf.nist.gov/9.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

…

25: 7.10 Derivatives

… ►

7.10.2 $$w^{\prime }\left(z\right)=-2zw\left(z\right)+(2\mathrm{i}/\sqrt{\pi }),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.20

Permalink:

http://dlmf.nist.gov/7.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

…

26: 10.63 Recurrence Relations and Derivatives

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$f_{\nu -1}(x)+f_{\nu +1}(x)=-(\nu \sqrt{2}/x)\left(f_{\nu }(x)-g_{\nu }(x)\right),$

… ►

$f_{\nu }^{\prime }(x)=-(1/\sqrt{2})\left(f_{\nu -1}(x)+g_{\nu -1}(x)\right)-(\nu /x)f_{\nu }(x),$

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$f_{\nu }^{\prime }(x)=(1/\sqrt{2})\left(f_{\nu +1}(x)+g_{\nu +1}(x)\right)+(\nu /x)f_{\nu }(x).$

… ►

$p_{\nu +1}=p_{\nu -1}-(4\nu /x)r_{\nu },$

… ►

$r_{\nu +1}=-((\nu +1)/x)p_{\nu +1}+q_{\nu },$

…

27: 10.20 Uniform Asymptotic Expansions for Large Order

… ►

10.20.6 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu z\right)\\ H_{\nu }^{(2)}\left(\nu z\right)\end{array}\}\sim 2{\mathrm{e}}^{\mp \pi \mathrm{i}/3}{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.37

Referenced by:

§10.20(iii), §10.41(v)

Permalink:

http://dlmf.nist.gov/10.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

… ►

10.20.9 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu z\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu z\right)\end{array}\}\sim \frac{4{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.45

Referenced by:

§10.20(ii)

Permalink:

http://dlmf.nist.gov/10.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

… ►

►►

Figure 10.20.2: $\zeta$-plane. $E_1$ and $E_2$ are the points ${\mathrm{e}}^{\mp \pi \mathrm{i}/3}{(3\pi /2)}^{2/3}.$ Magnify

…

28: 10.19 Asymptotic Expansions for Large Order

… ►

10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

… ►

10.19.13 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim -\frac{2^{\frac{5}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{4}{3}}}{{\nu }^{\frac{4}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $R_k(a)$: polynomial coefficient and $S_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

…

29: 9.13 Generalized Airy Functions

… ►

9.13.18 $$w=U_m(t{\mathrm{e}}^{-2j\pi \mathrm{i}/m}),$$ $j=0,\pm 1,\pm 2,\mathrm{\dots }.$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $w$: function, $m$: index and $j$: index

Notes:

See Olver (1978).

Permalink:

http://dlmf.nist.gov/9.13.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►

$A_2(z,0)={\mathrm{e}}^{2\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)$ ,

►

$A_3(z,0)={\mathrm{e}}^{-2\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)$ ,

… ►

$A_2(z,p)={\mathrm{e}}^{-2(p-1)\pi \mathrm{i}/3}{A}_1(z{\mathrm{e}}^{2\pi \mathrm{i}/3},p)$ ,

►

$A_3(z,p)={\mathrm{e}}^{2(p-1)\pi \mathrm{i}/3}{A}_1(z{\mathrm{e}}^{-2\pi \mathrm{i}/3},p)$ .

…

30: 23.22 Methods of Computation

… ►

If $c=0$, then

23.22.3 $$2{\omega }_1=2{\mathrm{e}}^{-\pi \mathrm{i}/3}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^3}{2\pi d^{1/6}}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 6 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when $d>0$ and ${\omega }_1>0$.

… ►Then $\alpha =-1-2\mathrm{i}$, $\beta =1$, $\gamma =2\mathrm{i}$; $k^2=(7+6\mathrm{i})/17$, and $k_{}^{\prime }{}_{}{}^2=(10-6\mathrm{i})/17$. …