naginterfaces.library.specfun.ellipint_​general_​2

naginterfaces.library.specfun.ellipint_general_2(z, akp, a, b)

ellipint_general_2 returns the value of the general elliptic integral of the second kind $F(z,k^{\prime },a,b)$ for a complex argument $z$.

For full information please refer to the NAG Library document for s21da

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s21daf.html

Parameters

zcomplex

The argument $z$ of the function.

akpfloat

The argument $k^{\prime }$ of the function.

afloat

The argument $a$ of the function.

bfloat

The argument $b$ of the function.

Returns

fcomplex

The value of the general elliptic integral of the second kind $F(z,k^{\prime },a,b)$ for a complex argument $z$.

Raises

NagValueError

(errno $1$)

On entry, $\left|Im\left(z\right)\right|$ is too large: $\left|Im\left(z\right)\right|=⟨value⟩$. It must not exceed $⟨value⟩$.

(errno $1$)

On entry, $Re\left(z\right)$ is too large: $Re\left(z\right)=⟨value⟩$. It must not exceed $⟨value⟩$.

(errno $1$)

On entry, $Re\left(z\right)<0.0$: $Re\left(z\right)=⟨value⟩$.

(errno $1$)

On entry, $\left|akp\right|$ is too large: $\left|akp\right|=⟨value⟩$. It must not exceed $⟨value⟩$.

(errno $2$)

The iterative procedure used to evaluate the integral has failed to converge.

Notes

ellipint_general_2 evaluates an approximation to the general elliptic integral of the second kind $F(z,k^{\prime },a,b)$ given by

$$F(z,k^{\prime },a,b)={\int }_0^z\frac{a+b{\zeta }^2}{(1+{\zeta }^2)\sqrt{(1+{\zeta }^2)(1+{k^{\prime }}^2{\zeta }^2)}}d\zeta \text{,}$$

where $a$ and $b$ are real arguments, $z$ is a complex argument whose real part is non-negative and $k^{\prime }$ is a real argument (the complementary modulus). The evaluation of $F$ is based on the Gauss transformation. Further details, in particular for the conformal mapping provided by $F$, can be found in Bulirsch (1960).

Special values include

$$F(z,k^{\prime },1,1)={\int }_0^z\frac{d\zeta }{\sqrt{(1+{\zeta }^2)(1+k^{\prime }2{\zeta }^2)}}\text{,}$$

or $F_1(z,k^{\prime })$ (the elliptic integral of the first kind) and

$$F(z,k^{\prime },1,{k^{\prime }}^2)={\int }_0^z\frac{\sqrt{1+{k^{\prime }}^2{\zeta }^2}}{(1+{\zeta }^2)\sqrt{1+{\zeta }^2}}d\zeta \text{,}$$

or $F_2(z,k^{\prime })$ (the elliptic integral of the second kind). Note that the values of $F_1(z,k^{\prime })$ and $F_2(z,k^{\prime })$ are equal to ${tan}^{-1}\left(z\right)$ in the trivial case $k^{\prime }=1$.

ellipint_general_2 is derived from an Algol 60 procedure given by Bulirsch (1960). Constraints are placed on the values of $z$ and $k^{\prime }$ in order to avoid the possibility of machine overflow.

References

Bulirsch, R, 1960, Numerical calculation of elliptic integrals and elliptic functions, Numer. Math. (7), 76–90