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16. Jacobian Elliptic Functions and Theta Functions
Jacobian Elliptic Functions
A doubly periodic meromorphic function is
Let m, ml be numbers such that
called an elliptic function.
We call m the parameter, ml the complemkntary
In what follows we shall assume that the param-
eter m is a real number. Without loss of gen-
erality we can then suppose that O<mll (see
We define quarter-periods K and iK’ by
so that K and K’ are real numbers. K is called
the real, iK‘ the imaginary quarter-period.
We note that
16.1.2 K(m) = K’ (m,) = K’ (1 - m) .
We also note that if any one of the numbers m,
ml, K(m), K’(m), K’(m)/K(m) is given, all the
rest are determined. Thus K and K’ can not
both be chosen arbitrarily.
In the Argand diagram denote the points 0, K,
K+iK’, iK‘ by s, c, d, n respectively. These
points are at the vertices of a rectangle. The
translations of this rectangle by XK, piK’, where
A, p are givh all integral values positive or nega-
tive, will lead to the lattice
.S .C .S .C
.n .d .n .d
.n .d .n .d
.S .C .S .C
the pattern being repeated indefinitely on all
Let p, q be any two of the letters s, c, d, n.
Then p, q determine in the lattice a minimum
rectangle whose sides are of length K and K‘ and.
whose vertices s, c, d, n are in counterclockwise
The Jacobian elliptic function pq u is defined by
the following three properties.
(i) pq u has a simple zero at p and a simple
pole at g .
(ii) The step from p to q is a half-period of pq u.
Those of the numbers K, iK‘, K+iK‘ which differ
from this step are only quarter-periods.
(iii) The coefficient of the leading term in the
expansion of pq u in ascending powers of u about
u=O is unity. With regard to (iii) the leading
term is u, llu, 1 according as u=O is a zero, a
pole, or an ordinary point.
Thus the functions with a pole or zero at the
origin (i.e., the functions in which one letter is s)
are odd, and the others are even.
Should we wish to call explicit attention to the
value of the parameter, we write pq (ulm) instead
The Jacobian elliptic functions can also be
of pq u.
defined with respect to certain integrals.
(1 - m sin2 8) 1/2’
the angle p is called the amplitude
16.1.4 p=am u
and we define
sn u=sin p, cn u=cos p,
dn u=(l-m sin2 p)1/2=A(p).
Similarly all the functions pq u can be expressed
in terms of p. This second set of definitions,
although seemingly different, is mathematically
equivalent to the definition previously given in
terms of a lattice. For further explanation of
notations, iwluding the interpretation, of such
expressions as sn (p\a), cn (ulm), dn (u, k), see 17.2.
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