EllipticF - Maple Programming Help

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EllipticF

Incomplete elliptic integral of the first kind

EllipticK

Complete elliptic integral of the first kind

EllipticCK

Complementary complete elliptic integral of the first kind

 Calling Sequence EllipticF(z, k) EllipticK(k) EllipticCK(k)

Parameters

 z - algebraic expression (the sine of the amplitude) k - algebraic expression (the parameter)

Description

 EllipticF is the Incomplete Elliptic integral of the first kind and is defined by
 $\left[{\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{{∫}}_{{0}}^{{z}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{}{{k}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (1)
 EllipticK and EllipticCK are respectively the Complete and the Complementary Elliptic integrals of the first kind and are defined by
 $\left[{\mathrm{EllipticK}}{}\left({k}\right){=}{{∫}}_{{0}}^{{1}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{}{{k}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({k}\right)\right]$ (2)
 $\left[{\mathrm{EllipticCK}}{}\left({k}\right){=}{{∫}}_{{0}}^{{1}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{1}{+}\left({{k}}^{{2}}{-}{1}\right){}{{\mathrm{_α1}}}^{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({k}\right)\right]$ (3)
 EllipticK, EllipticCK and EllipticF are related by
 ${\mathrm{EllipticK}}{}\left({k}\right){=}{\mathrm{EllipticF}}{}\left({1}{,}{k}\right)$ (4)
 ${\mathrm{EllipticK}}{}\left({k}\right){=}{\mathrm{EllipticCK}}{}\left(\sqrt{{-}{{k}}^{{2}}{+}{1}}\right)$ (5)
 EllipticF is also identical to the InverseJacobiSN function
 ${\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{\mathrm{InverseJacobiSN}}{}\left({z}{,}{k}\right)$ (6)
 and therefore can be used to represent all the InverseJacobiPQ functions provided some restrictions on the function parameters hold.
 Elliptic integrals and the related functions are well described in the Table of Integrals Series and Products, Gradshteyn and Ryzhik (G&R) and in the popular Handbook of Mathematical Functions edited by Abramowitz and Stegun (A&S). In A&S, these functions are expressed in terms of a parameter m, representing the square of the modulus k entering the definition of the Elliptic, JacobiPQ and InverseJacobiPQ functions in Maple and G&R. For example, the $K\left(m\right)$ function shown in A&S is numerically equal to the Maple $\mathrm{EllipticK}\left(\sqrt{m}\right)$ command.
 It is worth noting the difference between the Legendre normal form of the Incomplete Elliptic integral of the first kind (see A&S 17.2.7), in Maple represented by EllipticF(z,k) but for the splitting of the square root in the denominator of the integrand (see definition lines above), and the normal trigonometric form of this elliptic integral (see A&S 17.2.6), in Maple represented by the InverseJacobiAM function
 > InverseJacobiAM(phi,k);
 ${\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right)$ (7)
 > (7) = convert((7), Int);
 ${\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{{∫}}_{{0}}^{{\mathrm{φ}}}\frac{{1}}{\sqrt{{1}{-}{{k}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{_θ1}}\right)}^{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_θ1}}$ (8)
 For instance, for -Pi/2 <= phi <= Pi/2 these two forms can be related with ease by changing variables:
 > EllipticF(z,k);
 ${\mathrm{EllipticF}}{}\left({z}{,}{k}\right)$ (9)
 > (9) = convert((9), Int);
 ${\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{{∫}}_{{0}}^{{z}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{}{{k}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}$ (10)
 > {z=sin(phi), _alpha1=sin(_theta1)};     #  -1 <= z <= 1
 $\left\{{\mathrm{_α1}}{=}{\mathrm{sin}}{}\left({\mathrm{_θ1}}\right){,}{z}{=}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\right\}$ (11)
 > PDEtools[dchange]((11), (10));
 ${\mathrm{EllipticF}}{}\left({\mathrm{sin}}{}\left({\mathrm{φ}}\right){,}{k}\right){=}{{∫}}_{{0}}^{{\mathrm{arcsin}}{}\left({\mathrm{sin}}{}\left({\mathrm{φ}}\right)\right)}\frac{{\mathrm{cos}}{}\left({\mathrm{_θ1}}\right)}{\sqrt{{-}{{\mathrm{sin}}{}\left({\mathrm{_θ1}}\right)}^{{2}}{+}{1}}{}\sqrt{{1}{-}{{k}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{_θ1}}\right)}^{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_θ1}}$ (12)
 > simplify((12)) assuming phi in RealRange(-Pi/2, Pi/2);
 ${\mathrm{EllipticF}}{}\left({\mathrm{sin}}{}\left({\mathrm{φ}}\right){,}{k}\right){=}{{∫}}_{{0}}^{{\mathrm{φ}}}\frac{{1}}{\sqrt{{{\mathrm{cos}}{}\left({\mathrm{_θ1}}\right)}^{{2}}{}{{k}}^{{2}}{-}{{k}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_θ1}}$ (13)
 where the right-hand side is actually equal to the trigonometric form $\mathrm{InverseJacobiAM}\left(\mathrm{phi},k\right)$. The general relationship between these two forms and the restriction on the values of the parameters such that the relation is valid are given by
 $\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{\mathrm{EllipticF}}{}\left({\mathrm{sin}}{}\left({\mathrm{π}}{}{\mathrm{floor}}{}\left(\frac{{1}}{{2}}{-}\frac{{\mathrm{ℜ}}{}\left({\mathrm{φ}}\right)}{{\mathrm{π}}}\right){+}{\mathrm{φ}}\right){,}{k}\right){-}{2}{}{\mathrm{floor}}{}\left(\frac{{1}}{{2}}{-}\frac{{\mathrm{ℜ}}{}\left({\mathrm{φ}}\right)}{{\mathrm{π}}}\right){}{\mathrm{EllipticK}}{}\left({k}\right){,}{\mathrm{with no restrictions on}}{}\left({\mathrm{φ}}{,}{k}\right)\right]{,}\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{\mathrm{EllipticF}}{}\left({\mathrm{sin}}{}\left({\mathrm{φ}}\right){,}{k}\right){,}{-}\frac{{\mathrm{\pi }}}{{2}}{<}{\mathrm{\Re }}{}\left({\mathrm{\phi }}\right){<}\frac{{\mathrm{\pi }}}{{2}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{-}\frac{{\mathrm{\pi }}}{{2}}{=}{\mathrm{\Re }}{}\left({\mathrm{\phi }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{\le }{\mathrm{\Im }}{}\left({\mathrm{\phi }}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\frac{{\mathrm{\pi }}}{{2}}{=}{\mathrm{\Re }}{}\left({\mathrm{\phi }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\Im }}{}\left({\mathrm{\phi }}\right){\le }{0}\right]$ (14)
 $\left[{\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{\mathrm{InverseJacobiAM}}{}\left({\mathrm{arcsin}}{}\left({z}\right){,}{k}\right){,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (15)

Examples

Reflection symmetry and special values for EllipticK and EllipticF

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{EllipticK}\right)$
 $\left[{\mathrm{EllipticK}}{}\left({-}{k}\right){=}{\mathrm{EllipticK}}{}\left({k}\right){,}{\mathrm{EllipticK}}{}\left({0}\right){=}\frac{{1}}{{2}}{}{\mathrm{π}}{,}{\mathrm{EllipticK}}{}\left({\mathrm{∞}}\right){=}{0}{,}{\mathrm{EllipticK}}{}\left({\mathrm{∞}}{}{I}\right){=}{0}\right]$ (16)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{EllipticF}\right)$
 $\left[{\mathrm{EllipticF}}{}\left({0}{,}{k}\right){=}{0}{,}{\mathrm{EllipticF}}{}\left({1}{,}{k}\right){=}{\mathrm{EllipticK}}{}\left({k}\right){,}{\mathrm{EllipticF}}{}\left({z}{,}{0}\right){=}{\mathrm{arcsin}}{}\left({z}\right){,}{\mathrm{EllipticF}}{}\left({z}{,}{1}\right){=}{\mathrm{arctanh}}{}\left({z}\right){,}{\mathrm{EllipticF}}{}\left({z}{,}{\mathrm{∞}}\right){=}{0}{,}{\mathrm{EllipticF}}{}\left({z}{,}{-}{\mathrm{∞}}\right){=}{0}\right]$ (17)

Branch points for EllipticF

 > $\mathrm{FunctionAdvisor}\left(\mathrm{branch_points},\mathrm{EllipticF}\right)$
 $\left[{\mathrm{EllipticF}}{}\left({z}{,}{k}\right){,}{z}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left[{-}{1}{,}{1}{,}{-}\frac{{1}}{{k}}{,}\frac{{1}}{{k}}{,}{\mathrm{∞}}{+}{\mathrm{∞}}{}{I}\right]\right]$ (18)

Branch points and the branch cut for EllipticK

 > $\mathrm{FunctionAdvisor}\left(\mathrm{branch_points},\mathrm{EllipticK}\right)$
 $\left[{\mathrm{EllipticK}}{}\left({k}\right){,}{k}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left[{-}{\mathrm{∞}}{-}{\mathrm{∞}}{}{I}{,}{-}{1}{,}{1}{,}{\mathrm{∞}}{+}{\mathrm{∞}}{}{I}\right]\right]$ (19)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{branch_cuts},\mathrm{EllipticK}\right)$
 $\left[{\mathrm{EllipticK}}{}\left({k}\right){,}{k}{<}{-}{1}{,}{1}{<}{k}\right]$ (20)

For $\mathrm{Re}\left(k\right)$ in the cut, so for $1<=\mathrm{Re}\left(k\right)<=\mathrm{infinity}$, EllipticK is continuous from below.

 > $\mathrm{EllipticK}\left(2-\frac{1I}{100000}\right)$
 ${\mathrm{EllipticK}}{}\left({2}{-}\frac{{1}}{{100000}}{}{I}\right)$ (21)
 > $=\mathrm{evalf}\left(\right)$
 ${\mathrm{EllipticK}}{}\left({2}{-}\frac{{1}}{{100000}}{}{I}\right){=}{0.8428783289}{-}{1.078252932}{}{I}$ (22)
 > $\mathrm{EllipticK}\left(2\right)$
 ${\mathrm{EllipticK}}{}\left({2}\right)$ (23)
 > $=\mathrm{evalf}\left(\right)$
 ${\mathrm{EllipticK}}{}\left({2}\right){=}{0.8428751774}{-}{1.078257824}{}{I}$ (24)
 > $\mathrm{EllipticK}\left(2+\frac{1I}{100000}\right)$
 ${\mathrm{EllipticK}}{}\left({2}{+}\frac{{1}}{{100000}}{}{I}\right)$ (25)
 > $=\mathrm{evalf}\left(\right)$
 ${\mathrm{EllipticK}}{}\left({2}{+}\frac{{1}}{{100000}}{}{I}\right){=}{0.8428783289}{+}{1.078252932}{}{I}$ (26)
 >