Jacobi elliptic functions: Difference between revisions

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In the most general setting, <math>\operatorname{am}(u,m)</math> is a [[multivalued function]] (in <math>u</math>) with infinitely many [[Branch point|logarithmic branch points]] (the branches differ by integer multiples of <math>2\pi</math>), namely the points <math>2sK(m)+(4t+1)K(1-m)i</math> and <math>2sK(m)+(4t+3)K(1-m)i</math> where <math>s,t\in\mathbb{Z}</math>.<ref name="sala">{{cite journal |last=Sala |first=Kenneth L. |date=November 1989 |title=Transformations of the Jacobian Amplitude Function and Its Calculation via the Arithmetic-Geometric Mean|url=https://epubs.siam.org/doi/abs/10.1137/0520100 |journal=SIAM Journal on Mathematical Analysis|volume=20|issue=6|pages=1514–1528|doi=10.1137/0520100 }}</ref> This multivalued function can be made single-valued by cutting the complex plane along the line segments joining these branch points (the cutting can be done in non-equivalent ways, giving non-equivalent single-valued functions), thus making <math>\operatorname{am}(u,m)</math> [[Analytic function|analytic]] everywhere except on the [[Branch point#Branch cuts|branch cuts]]. In contrast, <math>\sin\operatorname{am}(u,m)</math> and other elliptic functions have no branch points, give consistent values for every branch of <math>\operatorname{am}</math>, and are [[meromorphic function|meromorphic]] in the whole complex plane. Since every elliptic function is meromorphic in the whole complex plane (by definition), <math>\operatorname{am}(u,m)</math> (when considered as a single-valued function) is not an elliptic function.

However, a particular cutting for <math>\operatorname{am}(u,m)</math> can be made in the <math>u</math>-plane by line segments from <math>2sK(m)+(4t+1)K (1-m)i</math> to <math>2sK(m)+(4t+3)K(1-m)i</math> with <math>s,t\in\mathbb{Z}</math>; then it only remains to define <math>\operatorname{am}(u,m)</math> at the branch cuts by continuity from some direction. Then <math>\operatorname{am}(u,m)</math> becomes single-valued and singly-periodic in <math>u</math> with the minimal period <math>4iK(1-m)</math> and it has singularities at the logarithmic branch points mentioned above. If <math>m\in\mathbb{R}</math> and <math>m\le 1</math>, <math>\operatorname{am}(u,m)</math> is continuous in <math>u</math> on the real line. When <math>m>1</math>, the branch cuts of <math>\operatorname{am}(u,m)</math> in the <math>u</math>-plane cross the real line at <math>2(2s+1)K(1/m)/\sqrt{m}</math> for <math>s\in\mathbb{Z}</math>; therefore for <math>m>1</math>, <math>\operatorname{am}(u,m)</math> is not continuous in <math>u</math> on the real line and jumps by <math>2\pi</math> on the discontinuities.

Let

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* <math>\operatorname{dn}(x)</math> solves the differential equations <math>\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - (2 - m) y + 2 y^3 = 0</math> and <math> \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (y^2 - 1) (1 - m - y^2)</math>

The Jacobifunction amplitudewhich providesexactly a non-trivial solution ofsolves the differential equation describing the exact motion of a [[Pendulum (mechanics)#Simple gravity pendulum|simplependulum pendulumdifferential equation]]. In particular,

:<math>\frac{\mathrm d^2 \theta}{\mathrm dt^2}+c\sin \theta=0,</math>

with initial angle <math>\theta_0</math> and zero initial angular velocity is

:<math>\begin{align}\theta&=2\arcsin (\sqrt{m}\operatorname{cd}(\sqrt{c}t,m))\\

:<math>\theta =2\operatorname{am}\left(\frac{t\sqrt{2c}}{2},2\right)\rightarrow \frac{\mathrm d^2 \theta}{\mathrm dt^2}+c\sin \theta=0,\quad -2\varpi < t\sqrt{2c}<2\varpi</math>

&=2\operatorname{am}\left(\frac{1+\sqrt{m}}{2}(\sqrt{c}t+K),\frac{4\sqrt{m}}{(1+\sqrt{m})^2}\right)-2\operatorname{am}\left(\frac{1+\sqrt{m}}{2}(\sqrt{c}t-K),\frac{4\sqrt{m}}{(1+\sqrt{m})^2}\right)-\pi\end{align}</math>

where <math>\varpi</math> is the [[lemniscate constant]].

where <math>m=\sin (\theta_0/2)^2</math>, <math>c>0</math> and <math>t\in\mathbb{R}</math>.

A function which solves the above pendulum differential equation with initial angle <math>\theta_0</math> is

:<math>\theta=2\arcsin (k\operatorname{cd}(\sqrt{c}t,k^2)),\quad k=\sin\frac{\theta_0}{2},\quad c>0,\,t\in\mathbb{R}.</math>

===Derivatives with respect to the second variable===