|
|
NAG[s21cbc] NAG[nag_jacobian_elliptic] - Jacobian elliptic functions sn, cn and dn of complex argument
|
|
Calling Sequence
s21cbc(z, ak2, sn, cn, dn, 'fail'=fail)
nag_jacobian_elliptic(. . .)
Parameters
|
|
z - complex;
|
|
|
|
On entry: the argument  of the functions.
|
|
|
 .
|
|
|
|
ak2 - float;
|
|
|
|
On entry: the value of  .
|
|
|
Constraint:  . .
|
|
|
|
sn - assignable;
cn - assignable;
dn - assignable;
|
|
|
|
Note: On exit the variable sn will have a value of type complex, cn will have a value of type complex, dn will have a value of type complex.
|
|
|
|
'fail'=fail - table; (optional)
|
|
|
|
The NAG error argument, see the documentation for NagError.
|
|
|
|
|
Description
|
|
|
|
Description
|
|
nag_jacobian_elliptic (s21cbc) evaluates the Jacobian elliptic functions  ,  and  given by
![[[sn(z,|,k),=sin`ϕ`],[cn(z,|,k),=cos`ϕ`],[dn(z,|,k),=sqrt(1-k^2sin^2`ϕ`) ]]](/support/helpjp/helpview.aspx?si=9728/file08692/math101.png)
where  is a complex argument,  is a real argument (the modulus) with  and  (the amplitude of  ) is defined by the integral
![z=(∫)[0]^`ϕ`(dtheta)/(sqrt(1-k^2sin^2theta))](/support/helpjp/helpview.aspx?si=9728/file08692/math115.png)
The above definitions can be extended for values of  (see Salzer (1962)) by means of the formulae
![[[sn(z,|,k),=k[1]sn(kz,|,k[1])],[cn(z,|,k),=dn(kz,|,k[1])],[dn(z,|,k),=cn(kz,|,k[1])]]](/support/helpjp/helpview.aspx?si=9728/file08692/math121.png)
where ![k[1] = 1/k](/support/helpjp/helpview.aspx?si=9728/file08692/math124.png) .
Special values include
![[[sn(z,|,0),=sinz],[cn(z,|,0),=cosz],[dn(z,|,0),=1],[sn(z,|,1),=tanhz],[cn(z,|,1),=sechz],[dn(z,|,1),=sechz ]]](/support/helpjp/helpview.aspx?si=9728/file08692/math129.png)
These functions are often simply written as  ,  and  , thereby avoiding explicit reference to the argument  . They can also be expressed in terms of Jacobian theta functions (see s21ccc (nag_jacobian_theta)).
Another nine elliptic functions may be computed via the formulae
![[[cdz,=cnz/dnz],[sdz,=snz/dnz],[ndz,=1/dnz]]](/support/helpjp/helpview.aspx?si=9728/file08692/math146.png)
![[[dcz,=dnz/cnz],[ncz,=1/cnz],[scz,=snz/cnz],[nsz,=1/snz],[dsz,=dnz/snz],[csz,=cnz/snz]]](/support/helpjp/helpview.aspx?si=9728/file08692/math148.png)
(see Abramowitz and Stegun (1972)).
|
|
|
Error Indicators and Warnings
|
|
"NE_COMPLEX"
On entry,  . Constraint:  and  , where  .
"NE_INTERNAL_ERROR"
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.
"NE_REAL"
On entry,  . Constraint:  .
|
|
|
Accuracy
|
|
In principle nag_jacobian_elliptic (s21cbc) is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the C standard library elementary functions such as sin and cos.
|
|
|
Further Comments
|
|
The values of  ,  and  are computed via the formulae
![[[snz,=(sn(u,,,k)dn(v,,,k'))/(1-dn^2(u,,,k)sn^2(v,,,k'))+i(cn(u,,,k)dn(u,,,k)sn(v,,,k')cn(v,,,k'))/(1-dn^2(u,,,k)sn^2(v,,,k'))],[cnz,=(cn(u,,,k)cn(v,,,k'))/(1-dn^2(u,,,k)sn^2(v,,,k'))-i(sn(u,,,k)dn(u,,,k)sn(v,,,k')dn(v,,,k'))/(1-dn^2(u,,,k)sn^2(v,,,k'))],[dnz,=(dn(u,,,k)cn(v,,,k')dn(v,,,k'))/(1-dn^2(u,,,k)sn^2(v,,,k'))-i(k^2sn(u,,,k)cn(u,,,k)sn(v,,,k'))/(1-dn^2(u,,,k)sn^2(v,,,k')) ]]](/support/helpjp/helpview.aspx?si=9728/file08692/math198.png)
where  and  (the complementary modulus).
|
|
|
|
Examples
|
|
|
>
|
z := -2 +3*I:
ak2 := 0.25:
NAG:-s21cbc(z, ak2, sn, cn, dn):
|
|
|
