| An elliptic function is a doubly periodic complex function
(much as a trig function is singly periodic) like this:
f(z+m*w1+n*w2) = f(z) for integers m and n, and complex numbers
w1 and w2.
Viewed as vectors, w1 and w2 must be linearly independant;
they define a periodic lattice of paralellograms in the plane.
Elliptic integrals are integrals of algebraic functions of genus one
(functions of a torus); one of their forms gives the arc length of an
ellipse. An elliptic function arises as the inverse function of the
integral.
Here's a simple example of an elliptic function as a continued product:
Sn(z;u) = tanh(z) * PROD(j>1) tanh(j*u+z)*tanh(j*u-z)
It has periods of 2*u and i*pi (u real) and maps the interior of a
rectangle to the upper half plane. The sides of the rectangles are
identified giving the torus mentioned above. The real axis wraps around
the rectangle.
The algebraic function that goes with this is (k is a parameter
related to u and pi above in terms of their arithmetic geometric
mean):
F(z,w) = w^2 - (1-z^2)*(1-k*z^2) = 0
This defines an algebraic curve, just like x^2+y^2-r^2 = 0 defines
a circle. But since z and w are complex, it can be visualized not as
a curve but as a surface - a torus. Because the complex number field
is algebraically closed, such surfaces are always compact. In the same
way sin/cos give a single valued parameterization of a circle,
elliptic functions parameterize a torus.
Even more wondrous are the abelian integrals and functions. They
give tesselations of the non euclidean plane (like an Escher circle-limit
woodcut), analogous to the rectangle case above. The genus of their
Riemann surfaces is higher - there is more than one handle.
For a really beautiful introduction see:
C. L. Siegel - "Topics in complex function theory", Vol 1 and 2.
See Note 667 for a Vaxstation program that draws a minimal surface that is
an embedding of a thrice punctured torus into euclidean space, which is
calculated using elliptic functions.
- Jim
|
| Newsgroups: sci.math
Path: decwrl!labrea!rutgers!att!petsd!cjh
Subject: Re: Elliptic Functions and Solutions of Equations
Posted: 26 Oct 88 00:46:16 GMT
Organization: Perkin-Elmer DSG, Tinton Falls, N.J.
In article <5213@saturn.ucsc.edu> variety@ucscc.UCSC.EDU (Gregory Woodhouse) writes:
>
> I recall reading that it was possible to give explicit expressions
>for the roots of polynomials in terms of elliptic functions, but I
>haven't seen this done in any of the texts on elliptic/abelian
>functions that I know of (e.g. Lang's). Can anyone outline the
>method or post references? Thanks.
The most accessible reference is Felix Klein, "Vorlesungen ueber das
Ikosaeder", translated into English as "Lectures on the
Icosahedron." This was in print in an inexpensive, well-made Dover
paperback for many years; now (sob) out of print.
The book by Fricke & Klein on "Elliptische Modulfunktionen" probably
has even more material, but this is a lot harder to get, and readable
only by someone who climbs up a mountain standing on his head because
it's not enough effort the other way. Also I recently saw the
statement that Fricke wrote a 3-volume Lehrbuch der Algebra, which
was probably the best source for the use of modular functions to
solve polynomial equations. The writer, howeever, regarded all this
as thoroughly obsolete mathematics.
I guess it is. It's utterly impractical for numerical purposes; it
is very far from the usual concerns of "modern" mathematics. But
I DON'T CARE! I LOVE IT ANYWAY!!
I thought I could summarize the method in a few paragraphs of modern
jargon, but my memory of Klein's book is lamentably rusty.
Regards,
Chris
UUCP: ...!rutgers!petsd!cjh
(201)758-7288 106 Apple Street, Tinton Falls,N.J. 07724
|
| Newsgroups: sci.math
Path: decwrl!labrea!agate!gsmith@garnet.berkeley.edu
Subject: Re: Elliptic Functions and Solutions of Equations
Posted: 27 Oct 88 21:44:15 GMT
Organization: Garnet Gang Gems of Wisdom, Inc.
In-reply-to: cjh@petsd.UUCP (Chris Henrich)
In article <1368@petsd.UUCP>, cjh@petsd (Chris Henrich) writes:
>In article <5213@saturn.ucsc.edu> variety@ucscc.UCSC.EDU (Gregory Woodhouse) writes:
>> I recall reading that it was possible to give explicit expressions
>>for the roots of polynomials in terms of elliptic functions, but I
>>haven't seen this done in any of the texts on elliptic/abelian
>>functions that I know of (e.g. Lang's). Can anyone outline the
>>method or post references? Thanks.
>The most accessible reference is Felix Klein, "Vorlesungen ueber das
>Ikosaeder", translated into English as "Lectures on the
>Icosahedron." This was in print in an inexpensive, well-made Dover
>paperback for many years; now (sob) out of print.
>Also I recently saw the statement that Fricke wrote a 3-volume
>Lehrbuch der Algebra, which was probably the best source for the
>use of modular functions to solve polynomial equations. The
>writer, howeever, regarded all this as thoroughly obsolete
>mathematics.
This sounds like Weber's book, actually, one volume of which
does have a lot of good stuff. A lot of what is called 'obsolete'
mathematics is just not popular at the moment: Klein himself
remarked on how abelian functions had gone out of style in his
lifetime; but things like this come back, often with a different
emphasis.
>I thought I could summarize the method in a few paragraphs of modern
>jargon, but my memory of Klein's book is lamentably rusty.
>Regards,
Last time this came up I remarked that I remembered it as Klein
using the rationality of the algebra of invariants of two
dimensional complex pseudo-reflection groups, and then going down
from this to the complex projective line = the Riemann sphere,
and looking at coverings which arise from regular partitions of
the sphere (isocehedrons, etc.) and looking at the function
fields analytically. I think he could have also used the
corresponding real three-dimensional reflection groups but didn't
for some reason. It's been a few years since I have been able to
see a copy (someone swiped it from the UC Berkeley library) but
no one last time told me I was nuts.
Anyway, as before I recommend the appendix to Mumford's 'Theta
II' and its bibliography especially, to see how hyperelliptic
curves and their jacobians can be use to (somewhat artificially)
fudge up an answer to this. Also, it might be noted that Klein's
program could be pushed to six or seven degree equations if
anyone wanted to (i.e., ending with a one-parameter solution, or
in other words one new transcendental function only.)
--
ucbvax!garnet!gsmith Gene Ward Smith/Brahms Gang/Berkeley CA 94720
"A good punch in the nose IS often effective communication"-- Ken Arndt
|
| I have Kleins book on the icosahedron (actually a photocopy);
the title is "Lectures on the Icosahedron and Equations of the Fifth
Degree". The description above is basically how he solves the
problem. I may post a simple description if I remember to bring some
notes in.
Mumfords book (Vol II) is pretty deep-end stuff, but his Vol I is
understandable, mostly. You'd have to be comfortable with modern
algebraic geometry to really handle it. (Not me!)
Some other classical references on elliptic functions that include
algebraic relations would be Cayley's book and Greenhill's book,
both also out of print Dover reprints.
Fricke and Klein is available from Johnson Reprint in New York, but
I never tried to look at a copy.
In his book on elliptic functions, Lang states about this turn of the
century mathematics "...but these things always have a way of returning
to the center of the stage at some point..."
My curiosity about elliptic functions was really for electric filter
synthesis rather than mathematics. For example, to synthesize a
low pass filter with parallel resonant traps (notches) with equal
ripple (Chebyshev) response in the passband and stopbands, you can do
the approximation on the universal covering surface for a pair of
conformally equivalent elliptic functions, generate the values of the
poles and zeroes there, and then relate these to the algebraic function
connecting the two elliptic functions. Since a lumped circuit lowpass
filter has a transfer function given by a rational function, it's then
a matter of matching up the components of the filter with the algebraic
function.
To see how this works, consider an elliptic function which maps the
interior of a rectangle to the upper half plane. The real axis wraps
around the period rectangle. Let the real axis be the frequency axis.
Now place several narrower period rectangles side by side inside the
first period rectangle, and let the real axis of their image be attenuation.
These functions are connected by a rational function, much the way
sine wave harmonics are connected by a polynomial.
Now scan around the frequency axis of the larger rectangle - you'll
meet attenuations of the narrow rectangles which undulate between zero
(flat response) and the passband ripple. When you reach the corner of the
larger rectangle and travel up the side, the attenuation will monotonically
increase - this is the transition band. Then coming back along the top,
you'll see attenuation rippling between infinity (poles) and the value
at the upper corners - equal ripple in both passband and stopbands in
closed form!
To get equiripple phaseshifts, stack the period rectangles vertically
instead.
- Jim
+--C1--+--C2--+
| | |
o--+--L1--+--L2--+--o
Example | | |
filter: In C3 C4 C5 Out
| | |
o--+------+------+--o
Approximation: imag
|
|
X----s----x----s----x----S
(mirror | | | | | |
image) | | | | | |
------------O----p----0----p----0----P----- real
(mirror images)
O = origin of complex plane
X = pole of squat frequency rectangle
x = pole of narrow attenuation rectangle
P = passband frequency
S = stopband frequency
p = passband attenuation
s = stopband attenuation
There are periodic repitions of these rectangles like a checkerboard;
only a fundamental region is shown above.
If you let the period rectangles degenerate to vertical strips, then
all the poles go to infinity, and you get a Chebyshev (all pole)
lowpass filter, which can be expressed in terms of trig functions.
|
| Here is an example of numerically solving a quintic via elliptic
modular functions. The basic idea is to express the solution
in closed form as a function of the coefficients. For example,
a quadratic can be solved with the quadratic formula.
Since the Galois group of a quintic is not solvable in general, no
closed form solution exists involving only radicals. But this doesn't
rule out the existance of a solution involving some transcendental
functions.
Elliptic functions satisfy algebraic relations much as trig functions
do. The technique here is due to Hermite. He was able to algebraically
transform a 5'th order relationship between a pair of modular functions
into a quintic equation in the canonical form y^5 - y - a = 0. A *mess* of
algebra. And those guys didn't have MAPLE!
The modular functions he uses are related via a set of fractional
linear transformations of their argument, t, by (a*t+b)/(c*t+d) where
the transformations run through the residue class of 2 by 2 matrices
with integer coefficients and determinant of 5 modulo integer matrices
with determinant of one (unimodular matrices.)
There are 6 such matrices: (proof left to reader as they say...)
| 5 0 | | 1 j |
| | and | | j = 0,1,2,3,4
| 0 1 | | 0 5 |
This is analogous to expressing p'th roots of unity by exp(2*pi*i*j/p)
by letting j run through the integers modulo p.
The value of the associated elliptic function can be expressed in terms of
the value of constant term "a" of the equation (by radicals). The ratio of
the sides of its period parallelogram can be solved for, much as we'd
solve for an arctangent. I used an arithmetic geometric mean iteration
for this.
Then we transform this ratio, called tau, by the set of 6 fractional
linear functions and combine these results in a set of 5 equations
involving products of their differences, scale by a factor which is a
function of "a" (again only involving radicals) and that's the set of
roots!
This is hardly recomended as a computationally efficient way to solve
such a polynomial, but is pretty remarkable. You'll want to look at
Klein's book for a whole bunch of neat geometric interpretation of
these things. Another fun book on elliptic functions is Borwein and
Borwein's book on "PI and the AGM" - lots of stuff on fast computation
of functions to zillions of digits using elliptic functions.
Here's the output of the program (which follows the formfeed)
applied to the equation that cropped up earlier: z^5 - 5*z + 12 = 0
Elliptic modular functions:
V[0] = ( 0.9861716805827444, 0.0000000000000000)
V[1] = ( 0.2507408063323664, 1.2161539662806384)
V[2] = ( -0.7651511272858174, 0.6991681021219538)
V[3] = ( -0.7651511272858174, -0.6991681021219538)
V[4] = ( 0.2507408063323665, -1.2161539662806385)
V[5] = ( 0.0028439991217688, 0.0000000000000000)
Roots:
z[0] = ( -1.8420859661902544, -0.0000000000000002), |y(z)| = 0.79916E-14
z[1] = ( 1.2728972239224991, 0.7197986814838614), |y(z)| = 0.15632E-14
z[2] = ( -0.3518542408273719, -1.7095610433703288), |y(z)| = 0.42247E-14
z[3] = ( -0.3518542408273722, 1.7095610433703288), |y(z)| = 0.82963E-14
z[4] = ( 1.2728972239224991, -0.7197986814838614), |y(z)| = 0.30288E-14
- Jim
PROGRAM QUINTIC
C
C Numerically solve a quintic by elliptic functions
C
C y^5 - 5*y + 12 = 0
C
C In canonical form this is x^5 - x - a = 0
C where a = -12/5^(5/4) and y = 5^(1/4)*x
C
C Let A = 0.5*5^(5/4)*a
C Then sin(alpha) = 4/A^2 = 1/9, tan(alpha) = 1/sqrt(80)
C tan(alpha/4) = elliptic modulus k
C tau = ratio of sides of period paralellogram
C
C Let B = 1/(2*5^(3/4)*k^(1/4)*k_prime)
C Then roots are expressed as B*(products of modular functions)
C
IMPLICIT REAL*8 (A-H, O-Z)
COMPLEX*16 PHI, TAU, V(0:5), Z(0:4)
REAL*8 K, K_PRIME
ALPHA = ATAN(1.0/SQRT(80.0D0))
K = TAN(0.25*ALPHA)
K_PRIME = SQRT(1.0-K**2)
TAU = (0.0D0, 1.0D0)*AGM(1.0D0, K_PRIME)/AGM(1.0D0, K)
V(5) = PHI(TAU*5.0)
DO 100 J=0,4
V(J) = PHI((TAU+16.0*J)/5.0)
100 CONTINUE
TYPE *
TYPE *, 'Elliptic modular functions:'
DO 200 J=0,5
TYPE 201, J, V(J)
201 FORMAT (' V['I1'] = ('F20.16','F20.16')')
200 CONTINUE
C = 1.0/(2.0*SQRT(5.0D0*SQRT(K))*K_PRIME)
Z(0) = C*(V(5)+V(0))*(V(1)-V(4))*(V(2)-V(3))
Z(1) = C*(V(5)+V(1))*(V(2)-V(0))*(V(3)-V(4))
Z(2) = C*(V(5)+V(2))*(V(3)-V(1))*(V(4)-V(0))
Z(3) = C*(V(5)+V(3))*(V(4)-V(2))*(V(0)-V(1))
Z(4) = C*(V(5)+V(4))*(V(0)-V(3))*(V(1)-V(2))
TYPE *
TYPE *, 'Roots:'
DO 300 J=0,4
TYPE 301, J, Z(J), CDABS(Z(J)**5-5.0*Z(J)+12.0)
301 FORMAT (' z['I1'] = ('F20.16','F20.16'), |y(z)| = 'E12.5)
300 CONTINUE
END
COMPLEX*16 FUNCTION PHI(TAU)
C
C Elliptic modular lambda function
C
C lambda(tau) = k(tau)^2 = 16*q*PROD(j > 0) (1+q^(2*j))/(1+q^(2*j-1))
C
C Where:
C tau = ratio of periods w2/w1, must be nonreal
C q = exp(pi*i*tau)
C k = modulus of Jacobian elliptic function
C
C Return phi(tau) = lambda(tau)^(1/8)
C
COMPLEX*16 TAU, P, Q, Q2, PTMP, QTMP
REAL*8 PI
PI = 3.14159265358979323846264338
Q = CDEXP(PI*(0.0D0, 1.0D0)*TAU)
Q2 = Q**2
P = 1.0
QTMP = 1.0
100 CONTINUE
PTMP = P
P = P*(1.0+QTMP*Q2)/(1.0+QTMP*Q)
QTMP = QTMP*Q2
IF (PTMP.NE.P) GOTO 100
PHI = SQRT(2.0D0)*CDEXP(PI*(0.0D0, 0.125D0)*TAU)*P
RETURN
END
REAL*8 FUNCTION AGM(A,B)
C
C Arithmetic/Geometric mean function
C
C In terms of complete elliptic integrals, K(k) this satisfies
C
C K(k) = PI/(2.*AGM(1,kprime)) = PI/(2.*AGM(1-k,1+k))
C K(kprime) = PI/(2.*AGM(1,k))
C
C where k = modulus, and kprime = complementary modulus.
C
C Ref: AMS 55, "Handbook of Mathematical Functions", Chapter 17, Sec 6.
C
IMPLICIT REAL*8 (A-H,O-Z)
P = A
Q = B
100 CONTINUE
T = P
P = DSQRT(T*Q)
Q = .5*(T+Q)
IF (P.NE.T) GOTO 100
AGM = Q
RETURN
END
|