Jacobian elliptic function

In mathematics , a Jacobian elliptic function is one of twelve special elliptic functions . The Jacobian elliptical functions have some analogies to the trigonometric functions and find numerous applications in mathematical physics , with elliptical filters and in geometry , in particular for the pendulum equation and the arc length of an ellipse . Carl Gustav Jakob Jacobi introduced it around 1830. However, Carl Friedrich Gauß had already investigated two special Jacobian functions with the lemniscate sine and cosine in 1796 , but did not publish his notes on them. For the general theory of elliptic functions today, however, the Jacobian and more the Weierstrasse elliptic functions play a role.

The three basic Jacobian functions

There are twelve Jacobian elliptic functions, nine of which can be formed from three basic functions. Given a parameter , the elliptical module, which satisfies the inequality . It is also often specified as, where , or as a modular angle , where . In addition, the so-called complementary parameters are often used as well . The three basic Jacobian elliptic functions are then ${\displaystyle k}$ ${\displaystyle 0<k<1}$ ${\displaystyle m}$ ${\displaystyle m=k^{2}}$  ${\displaystyle \alpha }$ ${\displaystyle \sin ^{2}\alpha =k^{2}}$ ${\displaystyle k'={\sqrt {1-k^{2}}}}$ ${\displaystyle m_{1}={k'}^{2}}$

• the sinus amplitudinis , ${\displaystyle \operatorname {sn} \,(z;k)}$
• the cosine amplitudinis , ${\displaystyle \operatorname {cn} \,(z;k)}$
• the delta amplitudinis . ${\displaystyle \operatorname {dn} \,(z;k)}$

They are elliptical functions and accordingly have two periods. Overall, the following properties apply to them:

function	Periods	Zero	Pole position
${\displaystyle \operatorname {sn} \,(z;k)}$	${\displaystyle 4\,K,\ 2\,\mathrm {i} K'}$	${\displaystyle 2mK+2\,n\,\mathrm {i} \,K'}$	${\displaystyle 2\,mK+(2n+1)\,\mathrm {i} \,K'}$
${\displaystyle \operatorname {cn} \,(z;k)}$	${\displaystyle 4\,K,\ 2\,(K+\mathrm {i} K')}$	${\displaystyle (2m+1)\,K+2\,n\,\mathrm {i} \,K'}$	${\displaystyle 2\,mK+(2n+1)\,\mathrm {i} \,K'}$
${\displaystyle \operatorname {dn} \,(z;k)}$	${\displaystyle 2\,K,\ 4\,\mathrm {i} K'}$	${\displaystyle (2\,m+1)\,K+(2n+1)\,\mathrm {i} \,K'}$	${\displaystyle 2\,mK+(2n+1)\,\mathrm {i} \,K'}$
n and m are whole numbers

Here the real numbers and with the parameter depend on the elliptic integrals ${\displaystyle K}$ ${\displaystyle K'}$ ${\displaystyle k}$

${\displaystyle K=K(k)=\int _{0}^{\pi /2}{\frac {\mathrm {d} \,\varphi }{\sqrt {1-k^{2}\sin ^{2}\varphi }}},{\mbox{ }}K'(k)=\int _{0}^{\pi /2}{\frac {\mathrm {d} \,\varphi }{\sqrt {1+k^{2}\cos ^{2}\varphi }}}}$

together. For example, has zeros at and and poles at and . ${\displaystyle sn}$ ${\displaystyle z=0}$ ${\displaystyle z=2K}$ ${\displaystyle z=\mathrm {i} \,K'}$ ${\displaystyle z=2K+\mathrm {i} \,K'}$

Especially for the three basic Jacobian functions, the lemniscate sine and cosine functions introduced by Gauss result as follows: ${\displaystyle m=k^{2}=1/2}$

${\displaystyle \textstyle \mathrm {sl} \,(z)={\frac {1}{\sqrt {2}}}\mathrm {sn} \,({\sqrt {2}}z;{\frac {1}{2}})/\mathrm {dn} \,({\sqrt {2}}z;{\frac {1}{2}}),\qquad \textstyle \mathrm {cl} \,(z)=\,\mathrm {cn} \,({\sqrt {2}}z;{\frac {1}{2}}).}$

For the borderline cases and the Jacobi functions result in the (non-elliptical) trigonometric functions or hyperbolic functions : ${\displaystyle k=0}$ ${\displaystyle k=1}$

function	k = 0	k = 1
${\displaystyle \operatorname {sn} \,(z;k)}$	${\displaystyle \sin z}$	${\displaystyle \tanh z}$
${\displaystyle \operatorname {cn} \,(z;k)}$	${\displaystyle \cos z}$	${\displaystyle \mathrm {sech} \,z={\tfrac {1}{\cosh z}}}$
${\displaystyle \operatorname {dn} \,(z;k)}$	${\displaystyle 1}$	${\displaystyle \operatorname {sech} z={\tfrac {1}{\cosh z}}}$

Definitions

There are several equivalent definitions of the Jacobian functions.

Abstract definition as special meromorphic functions

Construction Aid

The free parameters given are the elliptical module with and the real numbers and with, which depend on them as above ${\displaystyle k}$ ${\displaystyle 0<k<1}$ ${\displaystyle K}$ ${\displaystyle K'}$

${\displaystyle {\begin{aligned}K(k)&=\int _{0}^{\pi /2}{\frac {\mathrm {d} \,\varphi }{\sqrt {1-k^{2}\sin ^{2}\varphi }}},\\K'(k)&=\int _{0}^{\pi /2}{\frac {\mathrm {d} \,\varphi }{\sqrt {1+k^{2}\cos ^{2}\varphi }}}.\end{aligned}}}$

Furthermore, a rectangle with the side lengths and in the complex plane with the corners is given, the corner of which lies at the origin. The sides of the length are parallel to the real axis, those of the length parallel to the imaginary axis. Let the corner be the point of the point and the point on the imaginary axis. The twelve Jacobian elliptical functions are then formed from a combination of letters , where and are each one of the letters . ${\displaystyle K}$ ${\displaystyle K'}$ ${\displaystyle s,c,d,n}$ ${\displaystyle s}$ ${\displaystyle K}$ ${\displaystyle K'}$ ${\displaystyle c}$ ${\displaystyle K,d}$ ${\displaystyle K+iK'}$ ${\displaystyle n}$ ${\displaystyle iK'}$ ${\displaystyle pq}$ ${\displaystyle p}$ ${\displaystyle q\neq p}$ ${\displaystyle s,c,d,n}$

A Jacobian elliptic function is then the unique double-periodic meromorphic function that fulfills the following three properties: ${\displaystyle \operatorname {pq} \,(z;k)}$

• The function has at a simple zero and at a simple pole. ${\displaystyle \operatorname {pq} \,(z;k)}$ ${\displaystyle p}$ ${\displaystyle q}$
• The function is periodic in direction , where the period is twice the distance from to . Similarly is periodic in the other two directions, but with a period four times the distance from to the other point. ${\displaystyle \operatorname {pq} \,(z;k)}$ ${\displaystyle p-q}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle \operatorname {pq} \,(z;k)}$ ${\displaystyle p}$
• If the function is expanded around the corner point , the leading term is simple (with the coefficient 1), the leading term of the development around the point is , and the leading term of the development around the other two corner points is 1 each. ${\displaystyle \operatorname {pq} \,(z;k)}$ ${\displaystyle p}$ ${\displaystyle z}$ ${\displaystyle q}$ ${\displaystyle 1/z}$

Definition as inverse functions of elliptic integrals

The above definition as a unique meromorphic function is very abstract. Equivalently, a Jacobian elliptic function can be defined as a unique inverse function of the incomplete elliptic integral of the first kind. This is the most common and perhaps most understandable definition. Let be a given parameter with , and be ${\displaystyle k}$ ${\displaystyle 0<k<1}$

${\displaystyle z=\int _{0}^{\phi }{\frac {\mathrm {d} \,\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}},}$

so . Then the Jacobian elliptic functions and are given by ${\displaystyle z=z(\phi )}$ ${\displaystyle \operatorname {sn} ,\operatorname {cn} }$ ${\displaystyle \operatorname {dn} }$

${\displaystyle \operatorname {sn} \;(z;k)=\sin \phi ,}$

${\displaystyle \operatorname {cn} \;(z;k)=\cos \phi }$

and

${\displaystyle \operatorname {dn} \;(z;k)={\sqrt {1-k^{2}\sin ^{2}\phi }}.}$

The angle is the amplitude, for it is called delta amplitude. Furthermore, the free parameter satisfies the inequality . For is the quarter period . ${\displaystyle \phi =\phi (z)}$ ${\displaystyle \operatorname {dn} \;z=\Delta (z)}$ ${\displaystyle k}$ ${\displaystyle 0\leq k^{2}\leq 1}$ ${\displaystyle \phi =\pi /2}$ ${\displaystyle z}$ ${\displaystyle K}$

The other nine Jacobian elliptic functions are built from these three basic functions, see next section.

Definition using the theta functions

Another definition of Jacobian functions uses the theta functions . Let and be two real constants with and . Then the three basic Jacobian functions are: ${\displaystyle k}$ ${\displaystyle k'}$ ${\displaystyle 0<k,k'<1}$ ${\displaystyle k^{2}+k'^{2}=1}$

${\displaystyle \operatorname {sn} (z;k)={\frac {1}{\sqrt {k}}}\ {\vartheta _{1}\left({\frac {z}{2K}};\tau \right) \over \vartheta _{0}\left({\frac {z}{2K}};\tau \right)}}$

${\displaystyle \operatorname {cn} (z;k)={\sqrt {\frac {k'}{k}}}\ {\vartheta _{2}\left({\frac {z}{2K}};\tau \right) \over \vartheta _{0}\left({\frac {z}{2K}};\tau \right)}}$

${\displaystyle \operatorname {dn} (z;k)={\sqrt {k'}}\ {\vartheta _{3}\left({\frac {z}{2K}};\tau \right) \over \vartheta _{0}\left({\frac {z}{2K}};\tau \right)}}$

The following applies:

${\displaystyle K(k)=\int _{0}^{\pi /2}{\frac {\mathrm {d} \,\varphi }{\sqrt {1-k^{2}\sin ^{2}\varphi }}}}$

${\displaystyle K'(k)=K(k')}$

${\displaystyle \tau =\mathrm {i} K'(k)/K(k)}$

The derived Jacobi functions

Usually the reciprocal values ​​of the three basic Jacobi functions are denoted by reversing the order of the letters, i.e.:

${\displaystyle \operatorname {ns} (z;k)=1/\operatorname {sn} (z;k)}$

${\displaystyle \operatorname {nc} (z;k)=1/\operatorname {cn} (z;k)}$

${\displaystyle \operatorname {nd} (z;k)=1/\operatorname {dn} (z;k)}$

The relationships between the three basic Jacobi functions are indicated by the first letter of the numerator and the denominator, i.e.:

${\displaystyle \operatorname {sc} (z;k)=\operatorname {sn} (z;k)/\operatorname {cn} (z;k)}$

${\displaystyle \operatorname {sd} (z;k)=\operatorname {sn} (z;k)/\operatorname {dn} (z;k)}$

${\displaystyle \operatorname {dc} (z;k)=\operatorname {dn} (z;k)/\operatorname {cn} (z;k)}$

${\displaystyle \operatorname {ds} (z;k)=\operatorname {dn} (z;k)/\operatorname {sn} (z;k)}$

${\displaystyle \operatorname {cs} (z;k)=\operatorname {cn} (z;k)/\operatorname {sn} (z;k)}$

${\displaystyle \operatorname {cd} (z;k)=\operatorname {cn} (z;k)/\operatorname {dn} (z;k)}$

So we can write shortened

${\displaystyle \operatorname {pq} (z;k)={\frac {\operatorname {pr} (z;k)}{\operatorname {qr} (z;k)}},}$

where and are each one of the letters and are set. ${\displaystyle p,q}$ ${\displaystyle r}$ ${\displaystyle s,c,d,n}$ ${\displaystyle ss=cc=dd=nn=1}$

Addition theorems

The Jacobi functions satisfy the two algebraic relationships

${\displaystyle \operatorname {cn} ^{2}+\operatorname {sn} ^{2}=1}$

${\displaystyle \operatorname {dn} ^{2}+k^{2}\operatorname {sn} ^{2}=1}$

Thus, parameterize an elliptic curve that represents the intersection of the two quadrics defined by the equations above . Furthermore, with the addition theorems, we can define a group law for points on this curve: ${\displaystyle (\operatorname {cn,sn,dn} )}$

${\displaystyle \operatorname {cn} (x+y)={\operatorname {cn} (x)\;\operatorname {cn} (y)-\operatorname {sn} (x)\;\operatorname {sn} (y)\;\operatorname {dn} (x)\;\operatorname {dn} (y) \over {1-k^{2}\;\operatorname {sn} ^{2}(x)\;\operatorname {sn} ^{2}(y)}}}$

${\displaystyle \operatorname {sn} (x+y)={\operatorname {sn} (x)\;\operatorname {cn} (y)\;\operatorname {dn} (y)+\operatorname {sn} (y)\;\operatorname {cn} (x)\;\operatorname {dn} (x) \over {1-k^{2}\;\operatorname {sn} ^{2}(x)\;\operatorname {sn} ^{2}(y)}}}$

${\displaystyle \operatorname {dn} (x+y)={\operatorname {dn} (x)\;\operatorname {dn} (y)-k^{2}\;\operatorname {sn} (x)\;\operatorname {sn} (y)\;\operatorname {cn} (x)\;\operatorname {cn} (y) \over {1-k^{2}\;\operatorname {sn} ^{2}(x)\;\operatorname {sn} ^{2}(y)}}}$

Quadratic relationships

${\displaystyle {\begin{aligned}-\operatorname {dn} ^{2}(z;k)+{k'}^{2}&\quad =\qquad -k^{2}\;\operatorname {cn} ^{2}(z;k)&&=k^{2}\;\operatorname {sn} ^{2}(z;k)-k^{2}\\-{k'}^{2}\;\operatorname {nd} ^{2}(z;k)+k'^{2}&\quad =\qquad -k^{2}{k'}^{2}\;\operatorname {sd} ^{2}(z;k)&&=k^{2}\;\operatorname {cd} ^{2}(z;k)-k^{2}\\{k'}^{2}\;\operatorname {sc} ^{2}(z;k)+{k'}^{2}&\quad =\qquad {k'}^{2}\;\operatorname {nc} ^{2}(z;k)&&=\operatorname {dc} ^{2}(z;k)-k^{2}\\\operatorname {cs} ^{2}(z;k)+{k'}^{2}&\quad =\qquad \operatorname {ds} ^{2}(z;k)&&=\operatorname {ns} ^{2}(z;k)-k^{2}\\\end{aligned}}}$

with . Further quadratic relationships can be formed with and , where and are each one of the letters and are set. ${\displaystyle k^{2}+k'^{2}=1}$ ${\displaystyle \operatorname {pq} ^{2}\cdot \operatorname {qp} ^{2}=1}$ ${\displaystyle \operatorname {pq} =\operatorname {pr} /\operatorname {qr} }$ ${\displaystyle p,q}$ ${\displaystyle r}$ ${\displaystyle s,c,d,n}$ ${\displaystyle ss=cc=dd=nn=1}$

Development as a Lambert series

With (in English nome ) and the argument , the functions can be developed into a Lambert series : ${\displaystyle q=\exp(-\pi K'/K)}$ ${\displaystyle v=\pi u/(2K(k))}$

${\displaystyle \operatorname {sn} (u;k)={\frac {2\pi }{K(k){\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1-q^{2n+1}}}\sin(2n+1)v}$

${\displaystyle \operatorname {cn} (u;k)={\frac {2\pi }{K{\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1+q^{2n+1}}}\cos(2n+1)v}$

${\displaystyle \operatorname {dn} (u;k)={\frac {\pi }{2K}}+{\frac {2\pi }{K}}\sum _{n=0}^{\infty }{\frac {q^{n}}{1+q^{2n}}}\cos 2nv}$

The elliptic Jacobi functions as solutions to nonlinear differential equations

The derivatives of the three basic Jacobian elliptic functions are:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {sn} \,(z;k)=\mathrm {cn} \,(z;k)\,\mathrm {dn} \,(z;k)}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {cn} \,(z;k)=-\mathrm {sn} \,(z;k)\,\mathrm {dn} \,(z;k)}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {dn} \,(z;k)=-k^{2}\mathrm {sn} \,(z;k)\,\mathrm {cn} \,(z;k)}$

With the above addition theorems they are therefore for a given with solutions of the following nonlinear differential equations : ${\displaystyle k}$ ${\displaystyle 0<k<1}$

• ${\displaystyle \operatorname {sn} \,(x;k)}$solves and ${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}+(1+k^{2})y-2k^{2}y^{3}=0}$ ${\displaystyle \left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}=(1-y^{2})(1-k^{2}y^{2})}$

• ${\displaystyle \operatorname {cn} \,(x;k)}$solves and ${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}+(1-2k^{2})y+2k^{2}y^{3}=0}$ ${\displaystyle \left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}=(1-y^{2})(1-k^{2}+k^{2}y^{2})}$

• ${\displaystyle \operatorname {dn} \,(x;k)}$solves and ${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}-(2-k^{2})y+2y^{3}=0}$ ${\displaystyle \left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}=(y^{2}-1)(1-k^{2}-y^{2})}$

Web links

• Jacobian Elliptic Functions in NIST Digital Library of Mathematical Functions
• Definition in Abramowitz & Stegun (English)
• Eric W. Weisstein : Jacobi Elliptic Functions . In: MathWorld (English).

literature

• Heinrich Durège: Theory of the elliptical functions. B. G. Teubner, Leipzig 1861.
• Charles Hermite : Overview of the theory of elliptic functions. Wiegandt & Hempel, Berlin 1863.
• Carl Gustav Jakob Jacobi : C. G. J. Jacobi's collected works. G. Reimer, Berlin 1881-1891.
• Leo Koenigsberger : Lectures on the theory of elliptical functions, together with an introduction to the general theory of functions. B. G. Teubner, Leipzig 1874.
• Karl Weierstrass : Formulas and theorems for the use of the elliptical functions. W. Fr. Kaestner, Göttingen 1883–1885.
• Robert Fricke : The elliptical functions and their applications. Part 2. B. G. Teubner, Leipzig 1922.
• Milton Abramowitz , Irene Stegun : Handbook of Mathematical Functions . Dover Publications, New York 1964, ISBN 0-486-61272-4 , Chapter 16.
• Naum Iljitsch Achijeser : Elements of the Theory of Elliptic Functions. Moscow 1970, translated into English as AMS Translations of Mathematical Monographs, Volume 79, AMS, Rhode Island 1990. ISBN 0-8218-4532-2 .
• ET Whittaker, GN Watson: A Course of Modern Analysis. Cambridge University Press, 1940/1996, ISBN 0-521-58807-3 .
• E. Zeidler (Ed.): Teubner-Taschenbuch der Mathematik. Teubner, Stuttgart / Leipzig 1996, ISBN 3-8154-2001-6 .