§23.6 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/23.6

§23.6(i) Theta Functions
§23.6(ii) Jacobian Elliptic Functions
§23.6(iii) General Elliptic Functions
§23.6(iv) Elliptic Integrals

§23.6(i) Theta Functions

Notes:: For (23.6.2)–(23.6.7) see Walker (1996, pp. 94 and 103). For (23.6.8) and (23.6.9) see Whittaker and Watson (1927, §21.43). For (23.6.10)–(23.6.12) use (23.6.9). For (23.6.13) and (23.6.14) see Lawden (1989, §6.6). For (23.6.15) combine (20.2.6) and (23.6.9).
Keywords:: Weierstrass elliptic functions, theta functions
Referenced by:: §20.9(ii)
Permalink:: http://dlmf.nist.gov/23.6.i

In this subsection $2\omega _{1}$ , $2\omega _{3}$ are any pair of generators of the lattice $\mathbb{L}$ , and the lattice roots $e_{1}$ , $e_{2}$ , $e_{3}$ are given by (23.3.9).

23.6.1

$q=e^{{i\pi\tau}},$

$\tau=\omega _{3}/\omega _{1}.$

Symbols:: $e$ : base of exponential function, $q$ : nome, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\tau$ : complex variable
A&S Ref:: 18.10.1, 18.10.2, and 18.9.7
Permalink:: http://dlmf.nist.gov/23.6.E1
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23.6.2 $e_{1}=\frac{\pi^{2}}{12\omega _{1}^{2}}\left({\mathop{\theta _{{2}}\/}\nolimits^{{4}}}\!\left(0,q\right)+2\!{\mathop{\theta _{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right)\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $q$ : nome, $e_{j}$ : zeros and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
A&S Ref:: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11
Referenced by:: ¶ ‣ §23.22(ii), §23.22(i), §23.22(ii), §23.6(i), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E2
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23.6.3 $e_{2}=\frac{\pi^{2}}{12\omega _{1}^{2}}\left({\mathop{\theta _{{2}}\/}\nolimits^{{4}}}\!\left(0,q\right)-{\mathop{\theta _{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right)\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $q$ : nome, $e_{j}$ : zeros and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
A&S Ref:: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11
Permalink:: http://dlmf.nist.gov/23.6.E3
Encodings:: TeX, pMML, png

23.6.4 $e_{3}=-\frac{\pi^{2}}{12\omega _{1}^{2}}\left(2\!{\mathop{\theta _{{2}}\/}\nolimits^{{4}}}\!\left(0,q\right)+{\mathop{\theta _{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right)\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $q$ : nome, $e_{j}$ : zeros and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
A&S Ref:: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11
Referenced by:: ¶ ‣ §23.22(ii), §23.22(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E4
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23.6.5 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{1}=\left(\frac{\pi\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(\pi z/(2\omega _{1}),q\right)}{2\omega _{1}\mathop{\theta _{{1}}\/}\nolimits\!\left(\pi z/(2\omega _{1}),q\right)}\right)^{2},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathbb{L}$ : lattice, $q$ : nome, $e_{j}$ : zeros, $z$ : complex and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
A&S Ref:: 18.10.5
Referenced by:: §20.9(i), §23.22(i), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E5
Encodings:: TeX, pMML, png

23.6.6 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{2}=\left(\frac{\pi\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(\pi z/(2\omega _{1}),q\right)}{2\omega _{1}\mathop{\theta _{{1}}\/}\nolimits\!\left(\pi z/(2\omega _{1}),q\right)}\right)^{2},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathbb{L}$ : lattice, $q$ : nome, $e_{j}$ : zeros, $z$ : complex and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
A&S Ref:: 18.10.5
Permalink:: http://dlmf.nist.gov/23.6.E6
Encodings:: TeX, pMML, png

23.6.7 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{3}=\left(\frac{\pi\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(\pi z/(2\omega _{1}),q\right)}{2\omega _{1}\mathop{\theta _{{1}}\/}\nolimits\!\left(\pi z/(2\omega _{1}),q\right)}\right)^{2}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathbb{L}$ : lattice, $q$ : nome, $e_{j}$ : zeros, $z$ : complex and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
A&S Ref:: 18.10.5
Referenced by:: §20.9(i), §23.6(i), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E7
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23.6.8 $\eta _{1}=-\frac{\pi^{2}}{12\omega _{1}}\frac{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(0,q\right)}{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $q$ : nome, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Referenced by:: §23.22(i), §23.6(i)
Permalink:: http://dlmf.nist.gov/23.6.E8
Encodings:: TeX, pMML, png

23.6.9 $\mathop{\sigma\/}\nolimits\!\left(z\right)=2\omega _{1}\mathop{\exp\/}\nolimits\!\left(\frac{\eta _{1}z^{2}}{2\omega _{1}}\right)\frac{\mathop{\theta _{{1}}\/}\nolimits\!\left(\pi z/(2\omega _{1}),q\right)}{\pi{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
A&S Ref:: 18.10.8
Referenced by:: §23.22(i), §23.6(i)
Permalink:: http://dlmf.nist.gov/23.6.E9
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23.6.10 $\mathop{\sigma\/}\nolimits\!\left(\omega _{1}\right)=2\omega _{1}\frac{\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{2}\eta _{1}\omega _{1}\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)}{\pi{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Referenced by:: §23.6(i)
Permalink:: http://dlmf.nist.gov/23.6.E10
Encodings:: TeX, pMML, png

23.6.11 $\mathop{\sigma\/}\nolimits\!\left(\omega _{2}\right)=2\omega _{1}i\frac{\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{2}\eta _{1}\omega _{1}\tau^{2}\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)}{\pi q^{{1/4}}{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $\tau$ : complex variable and $\eta _{j}$
Permalink:: http://dlmf.nist.gov/23.6.E11
Encodings:: TeX, pMML, png

23.6.12 $\mathop{\sigma\/}\nolimits\!\left(\omega _{3}\right)=-2\omega _{1}\frac{\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{2}\eta _{1}\omega _{1}\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right)}{\pi q^{{1/4}}{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Referenced by:: §23.6(i)
Permalink:: http://dlmf.nist.gov/23.6.E12
Encodings:: TeX, pMML, png

With $z=\ifrac{\pi u}{(2\omega _{1})}$ ,

23.6.13 $\mathop{\zeta\/}\nolimits\!\left(u\right)=\frac{\eta _{1}}{\omega _{1}}u+\frac{\pi}{2\omega _{1}}\frac{d}{dz}\mathop{\ln\/}\nolimits\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $\eta _{j}$ and $u$ : change of variable
Referenced by:: §23.22(i), §23.6(i)
Permalink:: http://dlmf.nist.gov/23.6.E13
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23.6.14 $\mathop{\wp\/}\nolimits\!\left(u\right)=\left(\frac{\pi}{2\omega _{1}}\right)^{2}\left(\frac{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(0,q\right)}{3\!{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}-\frac{{d}^{2}}{{dz}^{2}}\mathop{\ln\/}\nolimits\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q\right)\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $u$ : change of variable
Referenced by:: §23.6(i)
Permalink:: http://dlmf.nist.gov/23.6.E14
Encodings:: TeX, pMML, png

23.6.15 $\frac{\mathop{\sigma\/}\nolimits\!\left(u+\omega _{j}\right)}{\mathop{\sigma\/}\nolimits\!\left(\omega _{j}\right)}=\mathop{\exp\/}\nolimits\!\left(\eta _{j}u+\frac{\eta _{j}u^{2}}{2\omega _{1}}\right)\frac{\mathop{\theta _{{j+1}}\/}\nolimits\!\left(z,q\right)}{\mathop{\theta _{{j+1}}\/}\nolimits\!\left(0,q\right)},$ $j=1,2,3$ .

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $\eta _{j}$ and $u$ : change of variable
Referenced by:: §23.6(i)
Permalink:: http://dlmf.nist.gov/23.6.E15
Encodings:: TeX, pMML, png

For further results for the $\mathop{\sigma\/}\nolimits$ -function see Lawden (1989, §6.2).

§23.6(ii) Jacobian Elliptic Functions

Notes:: For (23.6.16) and (23.6.17) combine (23.6.2)–(23.6.4) with (22.2.2) and (20.7.5). For (23.6.18)–(23.6.20) combine (20.9.1) and (20.9.2) with (23.6.2)–(23.6.4). For (23.6.21)–(23.6.23) combine (23.6.5)–(23.6.7) with (22.2.4)–(22.2.9). For (23.6.24)–(23.6.26) combine (23.6.21)–(23.6.23) with §22.4(iii). (23.6.27)–(23.6.29) can be verified by matching periods, poles, and residues as in Lawden (1989, §8.11).
Keywords:: Jacobian elliptic functions, Weierstrass elliptic functions
Referenced by:: §20.11(iii)
Permalink:: http://dlmf.nist.gov/23.6.ii

Again, in Equations (23.6.16)–(23.6.26), $2\omega _{1},2\omega _{3}$ are any pair of generators of the lattice $\mathbb{L}$ and $e_{1},e_{2},e_{3}$ are given by (23.3.9).

23.6.16

$k^{2}=\frac{e_{2}-e_{3}}{e_{1}-e_{3}},$

${k^{{\prime}}}^{2}=\frac{e_{1}-e_{2}}{e_{1}-e_{3}},$

Symbols:: $e_{j}$ : zeros, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 18.9.9
Referenced by:: ¶ ‣ §23.22(ii), §23.6(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E16
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23.6.17

${\mathop{K\/}\nolimits^{{2}}}=(\mathop{K\/}\nolimits\!\left(k\right))^{2}=\omega _{1}^{2}(e_{1}-e_{3}),$

${\mathop{{K^{{\prime}}}\/}\nolimits^{{2}}}=(\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right))^{2}=\omega _{3}^{2}(e_{3}-e_{1}).$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 18.9.8
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E17
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23.6.18 $e_{1}=\frac{{\mathop{K\/}\nolimits^{{2}}}}{3\omega _{1}^{2}}(1+{k^{{\prime}}}^{2}),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 18.9.1
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E18
Encodings:: TeX, pMML, png

23.6.19 $e_{2}=\frac{{\mathop{K\/}\nolimits^{{2}}}}{3\omega _{1}^{2}}(k^{2}-{k^{{\prime}}}^{2}),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 18.9.2
Permalink:: http://dlmf.nist.gov/23.6.E19
Encodings:: TeX, pMML, png

23.6.20 $e_{3}=-\frac{{\mathop{K\/}\nolimits^{{2}}}}{3\omega _{1}^{2}}(1+k^{2}).$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.9.3
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E20
Encodings:: TeX, pMML, png

23.6.21 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{1}=\frac{{\mathop{K\/}\nolimits^{{2}}}}{\omega _{1}^{2}}{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(\frac{\mathop{K\/}\nolimits\! z}{\omega _{1}},k\right),$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E21
Encodings:: TeX, pMML, png

23.6.22 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{2}=\frac{{\mathop{K\/}\nolimits^{{2}}}}{\omega _{1}^{2}}{\mathop{\mathrm{ds}\/}\nolimits^{{2}}}\left(\frac{\mathop{K\/}\nolimits\! z}{\omega _{1}},k\right),$

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
Permalink:: http://dlmf.nist.gov/23.6.E22
Encodings:: TeX, pMML, png

23.6.23 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{3}=\frac{{\mathop{K\/}\nolimits^{{2}}}}{\omega _{1}^{2}}{\mathop{\mathrm{ns}\/}\nolimits^{{2}}}\left(\frac{\mathop{K\/}\nolimits\! z}{\omega _{1}},k\right).$

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.9.11
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E23
Encodings:: TeX, pMML, png

23.6.24 $\mathop{\wp\/}\nolimits\!\left(z+\omega _{1}\right)-e_{1}=\left(\frac{\mathop{K\/}\nolimits\! k^{{\prime}}}{\omega _{1}}\right)^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(\frac{\mathop{K\/}\nolimits\! z}{\omega _{1}},k\right),$

Symbols:: $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E24
Encodings:: TeX, pMML, png

23.6.25 $\mathop{\wp\/}\nolimits\!\left(z+\omega _{2}\right)-e_{2}=-\left(\frac{\mathop{K\/}\nolimits\! kk^{{\prime}}}{\omega _{1}}\right)^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}\left(\frac{\mathop{K\/}\nolimits\! z}{\omega _{1}},k\right),$

Symbols:: $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/23.6.E25
Encodings:: TeX, pMML, png

23.6.26 $\mathop{\wp\/}\nolimits\!\left(z+\omega _{3}\right)-e_{3}=\left(\frac{\mathop{K\/}\nolimits\! k}{\omega _{1}}\right)^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(\frac{\mathop{K\/}\nolimits\! z}{\omega _{1}},k\right).$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
Referenced by:: §23.6(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E26
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In (23.6.27)–(23.6.29) the modulus $k$ is given and $\mathop{K\/}\nolimits=\mathop{K\/}\nolimits\!\left(k\right)$ , $\mathop{{K^{{\prime}}}\/}\nolimits=\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right)$ are the corresponding complete elliptic integrals (§19.2(ii)). Also, $\mathbb{L}_{{\mspace{1.0mu}1}}$ , $\mathbb{L}_{{\mspace{1.0mu}2}}$ , $\mathbb{L}_{{\mspace{1.0mu}3}}$ are the lattices with generators $(4\!\mathop{K\/}\nolimits,2i\mathop{{K^{{\prime}}}\/}\nolimits)$ , $(2\!\mathop{K\/}\nolimits-2i\mathop{{K^{{\prime}}}\/}\nolimits,2\!\mathop{K\/}\nolimits+2i\mathop{{K^{{\prime}}}\/}\nolimits)$ , $(2\!\mathop{K\/}\nolimits,4i\mathop{{K^{{\prime}}}\/}\nolimits)$ , respectively.

23.6.27 $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}_{{\mspace{1.0mu}1}}\right)-\mathop{\zeta\/}\nolimits\!\left(z+2\!\mathop{K\/}\nolimits|\mathbb{L}_{{\mspace{1.0mu}1}}\right)+\mathop{\zeta\/}\nolimits\!\left(2\!\mathop{K\/}\nolimits|\mathbb{L}_{{\mspace{1.0mu}1}}\right)=\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right),$

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex and $k$ : modulus
Referenced by:: §23.6(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E27
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23.6.28 $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}_{{\mspace{1.0mu}2}}\right)-\mathop{\zeta\/}\nolimits\!\left(z+2\!\mathop{K\/}\nolimits|\mathbb{L}_{{\mspace{1.0mu}2}}\right)+\mathop{\zeta\/}\nolimits\!\left(2\!\mathop{K\/}\nolimits|\mathbb{L}_{{\mspace{1.0mu}2}}\right)=\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right),$

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/23.6.E28
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23.6.29 $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}_{{\mspace{1.0mu}3}}\right)-\mathop{\zeta\/}\nolimits\!\left(z+2i\!\mathop{{K^{{\prime}}}\/}\nolimits|\mathbb{L}_{{\mspace{1.0mu}3}}\right)-\mathop{\zeta\/}\nolimits\!\left(2i\!\mathop{{K^{{\prime}}}\/}\nolimits|\mathbb{L}_{{\mspace{1.0mu}3}}\right)=\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right).$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex and $k$ : modulus
Referenced by:: §23.6(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/23.6.E29
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Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3.

For representations of the Jacobi functions $\mathop{\mathrm{sn}\/}\nolimits$ , $\mathop{\mathrm{cn}\/}\nolimits$ , and $\mathop{\mathrm{dn}\/}\nolimits$ as quotients of $\mathop{\sigma\/}\nolimits$ -functions see Lawden (1989, §§6.2, 6.3).

§23.6(iii) General Elliptic Functions

Keywords:: Weierstrass elliptic functions, elliptic functions
Permalink:: http://dlmf.nist.gov/23.6.iii

For representations of general elliptic functions (§23.2(iii)) in terms of $\mathop{\sigma\/}\nolimits\!\left(z\right)$ and $\mathop{\wp\/}\nolimits\!\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\mathop{\zeta\/}\nolimits\!\left(z\right)$ see Lawden (1989, §8.11).

§23.6(iv) Elliptic Integrals

Keywords:: Weierstrass elliptic functions, elliptic integrals
Referenced by:: §19.10(i)
Permalink:: http://dlmf.nist.gov/23.6.iv

¶ Rectangular Lattice

Let $z$ be on the perimeter of the rectangle with vertices $0,2\omega _{1},2\omega _{1}+2\omega _{3},2\omega _{3}$ . Then $t=\mathop{\wp\/}\nolimits\!\left(z\right)$ is real (§§23.5(i)–23.5(ii)), and

23.6.30 $z=\frac{1}{2}\int _{t}^{\infty}\frac{du}{\sqrt{(u-e_{1})(u-e_{2})(u-e_{3})}},$ $t\geq e_{1}$ , $z\in(0,\omega _{1}]$ ,

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(a,b]$ : half-closed interval, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $t$
Referenced by:: ¶ ‣ §23.6(iv)
Permalink:: http://dlmf.nist.gov/23.6.E30
Encodings:: TeX, pMML, png

23.6.31 $z-\omega _{1}=\frac{i}{2}\int _{t}^{{e_{1}}}\frac{du}{\sqrt{(e_{1}-u)(u-e_{2})(u-e_{3})}},$ $e_{2}\leq t\leq e_{1}$ , $z\in[\omega _{1},\omega _{1}+\omega _{3}]$ ,

Symbols:: $[a,b]$ : closed interval, $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $t$
Permalink:: http://dlmf.nist.gov/23.6.E31
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23.6.32 $z-\omega _{3}=\frac{1}{2}\int _{{e_{3}}}^{t}\frac{du}{\sqrt{(e_{1}-u)(e_{2}-u)(u-e_{3})}},$ $e_{3}\leq t\leq e_{2}$ , $z\in[\omega _{3},\omega _{1}+\omega _{3}]$ ,

Symbols:: $[a,b]$ : closed interval, $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $t$
Permalink:: http://dlmf.nist.gov/23.6.E32
Encodings:: TeX, pMML, png

23.6.33 $z=\frac{i}{2}\int _{{-\infty}}^{t}\frac{du}{\sqrt{(e_{1}-u)(e_{2}-u)(e_{3}-u)}},$ $t\leq e_{3}$ , $z\in(0,\omega _{3}]$ .

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(a,b]$ : half-closed interval, $e_{j}$ : zeros, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $t$
Permalink:: http://dlmf.nist.gov/23.6.E33
Encodings:: TeX, pMML, png

23.6.34 $2\omega _{1}=\int _{{e_{1}}}^{\infty}\frac{du}{\sqrt{(u-e_{1})(u-e_{2})(u-e_{3})}}=\int _{{e_{3}}}^{{e_{2}}}\frac{du}{\sqrt{(e_{1}-u)(e_{2}-u)(u-e_{3})}},$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $e_{j}$ : zeros and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
Permalink:: http://dlmf.nist.gov/23.6.E34
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23.6.35 $2\omega _{3}=i\int _{{e_{2}}}^{{e_{1}}}\frac{du}{\sqrt{(e_{1}-u)(u-e_{2})(u-e_{3})}}=i\int _{{-\infty}}^{{e_{3}}}\frac{du}{\sqrt{(e_{1}-u)(e_{2}-u)(e_{3}-u)}}.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $e_{j}$ : zeros and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
Referenced by:: ¶ ‣ §23.6(iv)
Permalink:: http://dlmf.nist.gov/23.6.E35
Encodings:: TeX, pMML, png

For (23.6.30)–(23.6.35) and further identities see Lawden (1989, §6.12).

See also §§19.2(i), 19.14, and Erdélyi et al. (1953b, §13.14).

For relations to symmetric elliptic integrals see §19.25(vi).

¶ General Lattice

Keywords:: Weierstrass elliptic functions, elliptic integrals

Let $z$ be a point of $\Complex$ different from $e_{1},e_{2},e_{3}$ , and define $w$ by

23.6.36 $w=\int _{z}^{\infty}\frac{du}{\sqrt{4u^{3}-g_{2}u-g_{3}}}=\frac{1}{2}\int _{z}^{\infty}\frac{du}{\sqrt{(u-e_{1})(u-e_{2})(u-e_{3})}},$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $g_{2}$ , $g_{3}$ : lattice invariants, $e_{j}$ : zeros and $z$ : complex
Referenced by:: §19.25(vi)
Permalink:: http://dlmf.nist.gov/23.6.E36
Encodings:: TeX, pMML, png

where the integral is taken along any path from $z$ to $\infty$ that does not pass through any of $e_{1},e_{2},e_{3}$ . Then $z=\mathop{\wp\/}\nolimits\!\left(w\right)$ , where the value of $w$ depends on the choice of path and determination of the square root; see McKean and Moll (1999, pp. 87–88 and §2.5).