22 Jacobian Elliptic Functions Properties 22.5 Special Values 22.7 Landen Transformations

§22.6 Elementary Identities

Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.6

§22.6(i) Sums of Squares
§22.6(ii) Double Argument
§22.6(iii) Half Argument
§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
§22.6(v) Change of Modulus

§22.6(i) Sums of Squares

Notes:: See Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 491–494).
Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.6.i

22.6.1 ${\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right)+{\mathop{\mathrm{cn}\/% }\nolimits^{{2}}}\left(z,k\right)=k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}% \left(z,k\right)+{\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z,k\right)=1,$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex and : modulus
A&S Ref:: 16.9.1
Referenced by:: §22.11, §22.11, §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.6.E1
Encodings:: TeX, pMML, png

22.6.2 $1+{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(z,k\right)=k^{2}+{\mathop{% \mathrm{ds}\/}\nolimits^{{2}}}\left(z,k\right)={\mathop{\mathrm{ns}\/}% \nolimits^{{2}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex and : modulus
A&S Ref:: 16.9.4
Permalink:: http://dlmf.nist.gov/22.6.E2
Encodings:: TeX, pMML, png

22.6.3 ${k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(z,k\right)+1={% \mathop{\mathrm{dc}\/}\nolimits^{{2}}}\left(z,k\right)={k^{{\prime}}}^{2}{% \mathop{\mathrm{nc}\/}\nolimits^{{2}}}\left(z,k\right)+k^{2},$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.9.3
Permalink:: http://dlmf.nist.gov/22.6.E3
Encodings:: TeX, pMML, png

22.6.4 $-k^{2}{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}\left(z,k\right% )=k^{2}({\mathop{\mathrm{cd}\/}\nolimits^{{2}}}\left(z,k\right)-1)={k^{{\prime% }}}^{2}(1-{\mathop{\mathrm{nd}\/}\nolimits^{{2}}}\left(z,k\right)).$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.9.2
Permalink:: http://dlmf.nist.gov/22.6.E4
Encodings:: TeX, pMML, png

§22.6(ii) Double Argument

Notes:: See Lawden (1989, pp. 33–36). For (22.6.6) and (22.6.7) set in (22.8.2) and use (22.6.1) repeatedly.
Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.6.ii

22.6.5 $\mathop{\mathrm{sn}\/}\nolimits\left(2z,k\right)=\frac{2\mathop{\mathrm{sn}\/}% \nolimits\left(z,k\right)\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)% \mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)}{1-k^{2}{\mathop{\mathrm{sn}\/% }\nolimits^{{4}}}\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex and : modulus
A&S Ref:: 16.18.1
Permalink:: http://dlmf.nist.gov/22.6.E5
Encodings:: TeX, pMML, png

22.6.6 $\mathop{\mathrm{cn}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{cn}\/}% \nolimits^{{2}}}\left(z,k\right)-{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(% z,k\right){\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z,k\right)}{1-k^{2}{% \mathop{\mathrm{sn}\/}\nolimits^{{4}}}\left(z,k\right)}=\frac{{\mathop{\mathrm% {cn}\/}\nolimits^{{4}}}\left(z,k\right)-{k^{{\prime}}}^{2}{\mathop{\mathrm{sn}% \/}\nolimits^{{4}}}\left(z,k\right)}{1-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{% {4}}}\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.18.2
Referenced by:: §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.6.E6
Encodings:: TeX, pMML, png

22.6.7 $\mathop{\mathrm{dn}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{dn}\/}% \nolimits^{{2}}}\left(z,k\right)-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}% \left(z,k\right){\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z,k\right)}{1-k^{% 2}{\mathop{\mathrm{sn}\/}\nolimits^{{4}}}\left(z,k\right)}=\frac{{\mathop{% \mathrm{dn}\/}\nolimits^{{4}}}\left(z,k\right)+k^{2}{k^{{\prime}}}^{2}{\mathop% {\mathrm{sn}\/}\nolimits^{{4}}}\left(z,k\right)}{1-k^{2}{\mathop{\mathrm{sn}\/% }\nolimits^{{4}}}\left(z,k\right)}.$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.18.3
Referenced by:: §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.6.E7
Encodings:: TeX, pMML, png

22.6.8 $\mathop{\mathrm{cd}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{cd}\/}% \nolimits^{{2}}}\left(z,k\right)-{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}% \nolimits^{{2}}}\left(z,k\right){\mathop{\mathrm{nd}\/}\nolimits^{{2}}}\left(z% ,k\right)}{1+k^{2}{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{4}}}% \left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E8
Encodings:: TeX, pMML, png

22.6.9 $\mathop{\mathrm{sd}\/}\nolimits\left(2z,k\right)=\frac{2\mathop{\mathrm{sd}\/}% \nolimits\left(z,k\right)\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)% \mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)}{1+k^{2}{k^{{\prime}}}^{2}{% \mathop{\mathrm{sd}\/}\nolimits^{{4}}}\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E9
Encodings:: TeX, pMML, png

22.6.10 $\mathop{\mathrm{nd}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{nd}\/}% \nolimits^{{2}}}\left(z,k\right)+k^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}% \left(z,k\right){\mathop{\mathrm{cd}\/}\nolimits^{{2}}}\left(z,k\right)}{1+k^{% 2}{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{4}}}\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E10
Encodings:: TeX, pMML, png

22.6.11 $\mathop{\mathrm{dc}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{dc}\/}% \nolimits^{{2}}}\left(z,k\right)+{k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}% \nolimits^{{2}}}\left(z,k\right){\mathop{\mathrm{nc}\/}\nolimits^{{2}}}\left(z% ,k\right)}{1-{k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{4}}}\left(z,% k\right)},$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E11
Encodings:: TeX, pMML, png

22.6.12 $\mathop{\mathrm{nc}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{nc}\/}% \nolimits^{{2}}}\left(z,k\right)+{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(% z,k\right){\mathop{\mathrm{dc}\/}\nolimits^{{2}}}\left(z,k\right)}{1-{k^{{% \prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{4}}}\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E12
Encodings:: TeX, pMML, png

22.6.13 $\mathop{\mathrm{sc}\/}\nolimits\left(2z,k\right)=\frac{2\mathop{\mathrm{sc}\/}% \nolimits\left(z,k\right)\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)% \mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)}{1-{k^{{\prime}}}^{2}{\mathop{% \mathrm{sc}\/}\nolimits^{{4}}}\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E13
Encodings:: TeX, pMML, png

22.6.14 $\mathop{\mathrm{ns}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{ns}\/}% \nolimits^{{4}}}\left(z,k\right)-k^{2}}{2\mathop{\mathrm{cs}\/}\nolimits\left(% z,k\right)\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)\mathop{\mathrm{ns}\/% }\nolimits\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.6.E14
Encodings:: TeX, pMML, png

22.6.15 $\mathop{\mathrm{ds}\/}\nolimits\left(2z,k\right)=\frac{k^{2}{k^{{\prime}}}^{2}% +{\mathop{\mathrm{ds}\/}\nolimits^{{4}}}\left(z,k\right)}{2\mathop{\mathrm{cs}% \/}\nolimits\left(z,k\right)\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)% \mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E15
Encodings:: TeX, pMML, png

22.6.16 $\mathop{\mathrm{cs}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{cs}\/}% \nolimits^{{4}}}\left(z,k\right)-{k^{{\prime}}}^{2}}{2\mathop{\mathrm{cs}\/}% \nolimits\left(z,k\right)\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)% \mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)}.$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E16
Encodings:: TeX, pMML, png

§22.6(iii) Half Argument

Notes:: See Lawden (1989, pp. 33–36).
Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.6.iii

22.6.19 ${\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(\tfrac{1}{2}z,k\right)=\frac{1-% \mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)}{1+\mathop{\mathrm{dn}\/}% \nolimits\left(z,k\right)}=\frac{1-\mathop{\mathrm{dn}\/}\nolimits\left(z,k% \right)}{k^{2}(1+\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right))}=\frac{% \mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)-k^{2}\mathop{\mathrm{cn}\/}% \nolimits\left(z,k\right)-{k^{{\prime}}}^{2}}{k^{2}(\mathop{\mathrm{dn}\/}% \nolimits\left(z,k\right)-\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right))},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.19.1
Permalink:: http://dlmf.nist.gov/22.6.E19
Encodings:: TeX, pMML, png

22.6.20 ${\mathop{\mathrm{cn}\/}\nolimits^{{2}}}\left(\tfrac{1}{2}z,k\right)=\frac{-{k^% {{\prime}}}^{2}+\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)+k^{2}\mathop{% \mathrm{cn}\/}\nolimits\left(z,k\right)}{k^{2}(1+\mathop{\mathrm{cn}\/}% \nolimits\left(z,k\right))}=\frac{{k^{{\prime}}}^{2}(1-\mathop{\mathrm{dn}\/}% \nolimits\left(z,k\right))}{k^{2}(\mathop{\mathrm{dn}\/}\nolimits\left(z,k% \right)-\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right))}=\frac{{k^{{\prime}}}% ^{2}(1+\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right))}{{k^{{\prime}}}^{2}+% \mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)-k^{2}\mathop{\mathrm{cn}\/}% \nolimits\left(z,k\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.19.2
Permalink:: http://dlmf.nist.gov/22.6.E20
Encodings:: TeX, pMML, png

22.6.21 ${\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(\tfrac{1}{2}z,k\right)=\frac{k^{2% }\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)+\mathop{\mathrm{dn}\/}% \nolimits\left(z,k\right)+{k^{{\prime}}}^{2}}{1+\mathop{\mathrm{dn}\/}% \nolimits\left(z,k\right)}=\frac{{k^{{\prime}}}^{2}(1-\mathop{\mathrm{cn}\/}% \nolimits\left(z,k\right))}{\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)-% \mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)}=\frac{{k^{{\prime}}}^{2}(1+% \mathop{\mathrm{dn}\/}\nolimits\left(z,k\right))}{{k^{{\prime}}}^{2}+\mathop{% \mathrm{dn}\/}\nolimits\left(z,k\right)-k^{2}\mathop{\mathrm{cn}\/}\nolimits% \left(z,k\right)}.$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.19.3
Permalink:: http://dlmf.nist.gov/22.6.E21
Encodings:: TeX, pMML, png

If $\{\mbox{p,q,r}\}$ is any permutation of $\{\mbox{c,d,n}\}$ , then

22.6.22 ${\mathop{\mathrm{pq}\/}\nolimits^{{2}}}\left(\tfrac{1}{2}z,k\right)=\frac{% \mathop{\mathrm{ps}\/}\nolimits\left(z,k\right)+\mathop{\mathrm{rs}\/}% \nolimits\left(z,k\right)}{\mathop{\mathrm{qs}\/}\nolimits\left(z,k\right)+% \mathop{\mathrm{rs}\/}\nolimits\left(z,k\right)}=\frac{\mathop{\mathrm{pq}\/}% \nolimits\left(z,k\right)+\mathop{\mathrm{rq}\/}\nolimits\left(z,k\right)}{1+% \mathop{\mathrm{rq}\/}\nolimits\left(z,k\right)}=\frac{\mathop{\mathrm{pr}\/}% \nolimits\left(z,k\right)+1}{\mathop{\mathrm{qr}\/}\nolimits\left(z,k\right)+1}.$

Symbols:: $\mathop{\mathrm{pq}\/}\nolimits\left(z,k\right)$ : generic Jacobian elliptic function, : complex and : modulus
Referenced by:: §22.6(iii)
Permalink:: http://dlmf.nist.gov/22.6.E22
Encodings:: TeX, pMML, png

For (22.6.22) and similar results, see Carlson (2004).

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

Notes:: See Lawden (1989, pp. 38–40), Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 505–508).
Keywords:: Jacobian elliptic functions, Jacobi’s imaginary transformation
Permalink:: http://dlmf.nist.gov/22.6.iv

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.

$\mathop{\mathrm{sn}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$i\mathop{\mathrm{sc}\/}\nolimits\left(z,k^{{\prime}}\right)$	$\mathop{\mathrm{dc}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$\mathop{\mathrm{dn}\/}\nolimits\left(z,k^{{\prime}}\right)$
$\mathop{\mathrm{cn}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$\mathop{\mathrm{nc}\/}\nolimits\left(z,k^{{\prime}}\right)$	$\mathop{\mathrm{nc}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$\mathop{\mathrm{cn}\/}\nolimits\left(z,k^{{\prime}}\right)$
$\mathop{\mathrm{dn}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$\mathop{\mathrm{dc}\/}\nolimits\left(z,k^{{\prime}}\right)$	$\mathop{\mathrm{sc}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$i\mathop{\mathrm{sn}\/}\nolimits\left(z,k^{{\prime}}\right)$
$\mathop{\mathrm{cd}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$\mathop{\mathrm{nd}\/}\nolimits\left(z,k^{{\prime}}\right)$	$\mathop{\mathrm{ns}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$-i\mathop{\mathrm{cs}\/}\nolimits\left(z,k^{{\prime}}\right)$
$\mathop{\mathrm{sd}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$i\mathop{\mathrm{sd}\/}\nolimits\left(z,k^{{\prime}}\right)$	$\mathop{\mathrm{ds}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$-i\mathop{\mathrm{ds}\/}\nolimits\left(z,k^{{\prime}}\right)$
$\mathop{\mathrm{nd}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$\mathop{\mathrm{cd}\/}\nolimits\left(z,k^{{\prime}}\right)$	$\mathop{\mathrm{cs}\/}\nolimits\left(iz,k\right)$ $\;=\;$	$-i\mathop{\mathrm{ns}\/}\nolimits\left(z,k^{{\prime}}\right)$