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CHAPTER 2. THETA FUNCTIONS
§1. The Theta Functions of Jacobi
To introduce the theta function of Jacobi, we start with two motivation.
The first motivation is to have a function to play the same role in the case of
elliptic functions as the coordinate function in the case of rational functions.
The inhomogeneous coordinate function w of the Riemann sphere P1 = C ∪
{∞} serves as the building block for rational functions on P1 in the sense
that every rational function can be written, up to a nonzero constant factor,
as the quotient
∏
(w − aj )
∏j
k (w − bk )
∏
of
∏ a product j (w − aj ) of translates w − aj of w by another product
k (w − bk ) of translates w − bk of w. The theta function of Jacobi ϑ(w)
is the analogue of the inhomogeneous coordinate function w on P1 when P1
is replaced by C /(Zω1 + Zω2 ) and rational functions are replaced by elliptic
functions which are meromorphic functions on C with periods ω1 and ω2 .
The second motivation is for the integration of elliptic integrals. For
p+1
the integration of xp to get xp+1 , the result is the product of the original
function xp times the constant multiple p1 x of the coordinate function x.
At the beginning of our discussion of elliptic functions we mentioned two
motivations, one from the problem of the arc-length of an ellipse and the other
from the motion of a simple pendulum. The introduction of the Jacobi elliptic
sine function sn w provides a solution of the motion of a simple pendulum,
but is not sufficient to give a solution of the problem of the arc-length of an
ellipse, which will require the use of the Jacobi theta functions, just as the
constant multiple p1 x of the coordinate function x is needed for the integration
of xp .
Let us go into the details of the first motivation for the introduction of
the Jacobi theta function. The theta function of Jacobi ϑ(w) is an entire
function on C with the property that every elliptic function with periods ω1
and ω2 can be written, up to a nonzero constant factor, as the quotient
∏
ϑ(w − aj )
∏j
k ϑ(w − bk )
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∏
of a product j ϑ(w − aj ) of translates ϑ(w − aj ) of ϑ(w) by another product
∏
k ϑ(w − bk ) of translates ϑ(w − bk ) of ϑ(w). Since ϑ(w) is entire, it cannot
be doubly periodic, otherwise it would be constant. However, since we need
the quotient
∏
ϑ(w − aj )
∏j
k ϑ(w − bk )
of the products of its translates to doubly periodicity, it must have some
modified form of periodicity for the two periods ω1 , ω2 .
We consider the case of two normalized periods ω1 , ω2 which are set to be
equal to π, πτ with Im τ > 0. We would like to make ϑ(w) as close to doubly
periodic for the two periods π, πτ as possible. First let us look for an entire
function f (w) with period π so that we have the Fourier series expansion
f (w) =
∞
∑
cn e2niw
n=−∞
with cn ∈ C. One way to make certain quotients of products of translates of
f (w) periodic also for the period πτ is to require that f (w) is periodic for
πτ with some periodicity factor in the sense that
f (w + πτ ) = eaw+b f (w)
for some a, b ∈ C. The easiest way to write down an example of such an
entire function f (w) is to use a = 2i or a = −2i, because then the condition
f (w + πτ ) = eaw+b f (w)
becomes
f (w + πτ ) = e2iw+b f (w)
or
f (w + πτ ) = e−2iw+b f (w)
so that we have
∞
∑
2ni(w+πτ )
cn e
=e
n=−∞
or
∞
∑
n=−∞
cn e
∞
∑
2iw+b
cn e2niw
n=−∞
2ni(w+πτ )
−2iw+b
=e
∞
∑
n=−∞
cn e2niw
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and the equating of Fourier coefficients on both sides would give us a relation
between cn and cn+1 , which is a recurrent relation for the Fourier coefficients
and determines all the Fourier coefficients after one is given from normalization. The complex number b can be chosen to ensure the convergence of the
Fourier series ans to make the final result more elegant. For the precise definition of the Jacobi theta function we choose a = −2i and b = −πτ i + πi so
that the periodicity factor eawb is e−2iw−πτ i+πi = −e−2iw−πτ i . The condition
∞
∑
= −e
2ni(w+πτ )
cn e
−2iw−πτ i
n=−∞
can be rewritten as
∞
∑
∞
∑
cn e2niw
n=−∞
2ni(w+πτ )
cn e
= −e
−πτ i
n=−∞
∞
∑
cn+1 e2niw
n=−∞
and gives the recurrent relation cn+1 = −e(2n+1)πτ i cn . We choose the simplest
initial condition c0 = 1. Let q = eπτ i so that cn+1 = −q 2n+1 cn . Since the sum
2
of 1, 3, · · · , 2n − 1 is n2 , it follows that cn = (−1)n q n . This way we arrive
this way at the function
∞
∑
2
(−1)n q n e2niw
n=−∞
and we denote it by ϑ(w) (also by ϑ(w|τ ) to emphasize its dependence on τ ).
It is the simplest Jacobi theta function. The equation with the periodicity
factor is
ϑ(w + πτ ) = −e−2iw−iπτ ϑ(w).
2
Note that the absolute value of the coefficient (−1)n q n of the Fourier series
ϑ(w) =
∞
∑
2
(−1)n q n e2niw
n=−∞
is
2
2
2
(−1)n q n = en πRe(τ i) = e−n πIm τ
which decays exponentially as a function of n2 and, with Im τ > 0, guarantees
the convergence of the Fourier series
ϑ(w) =
∞
∑
2
(−1)n q n e2niw .
n=−∞
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We can use the argument principle to determine the number of zeroes of
ϑ(w) in the fundamental parallelogram spanned by the two complex numbers
π and πτ . Because of the periodicity of ϑ(w) with period π and also because
of its periodicity with the periodicity factor e−2iw−πτ i+πi for the period πτ ,
the change of the argument of ϑ (which is the change of the imaginary part
of its logarithm) is equal to the change of the imaginary part of the exponent
−2iw − πτ i + πi of the periodicity factor e−2iw−πτ i+πi along the line-segment
from w = π + πτ to w = πτ and is therefore equal to
w=πτ
Im (−2iw − πτ i + πi) w=π+πτ
= Im ((−2iπτ − πτ i + πi) − (−2i(π + πτ ) − πτ i + πi))
= Im (2iπ) = 2π.
This means that the number of zeroes of ϑ(w) on a fundamental parallelogram
by the two complex numbers π and πτ is precisely 1.
We need to locate the single zero w = c of ϑ in the fundamental parallelgram spanned by π and πτ in order explicitly express an elliptic function
with zeroes at aj and poles at bk in the form
∏
ϑ(w − c − aj )
∏j
k ϑ(w − c − bk )
up to a nonzero constant factor. One way to locate the single zero is to
use symmetry properties. For example, if a function f (w) is odd, then we
know that w = 0 is a zero of f (w). Let us explore the question of oddness
and evenness by grouping together two exponential terms e2niw and e−2niw
(related by the index change n 7→ −n) to form a sine or cosine function. In
our case we have
∞
∑
2
ϑ(w) =
(−1)n q n e2niw
n=−∞
so that such groupings would yield
ϑ(w) = 1 + 2
∞
∑
2
(−1)n q n cos(2nw),
n=1
because both (−1)n and q n are unchanged under the index change n 7→ −n.
The evenness of ϑ(w) does not give us any help in locating its zero. If there is
2
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some way to use index change to change the sign of (−1)n , we would be able
to get a sine function instead of a cosine function to conclude the oddness of
the infinite series. One obvious index change to achieve this is n 7→ −n − 1
so that (−1)n becomes (−1)−n−1 = −(−1)n . Of course, we have to worry
2
about what happens to the factor q n in the index change n 7→ −n − 1 and
2
the ideal situation is to modify q n so that its modified form is unchanged in
2
n2
(n+ 12 )
the index( change
n
→
7
−n
−
1.
One
easier
way
is
to
modify
q
to
q
(
(
)2
)2
)2
because n + 21 becomes −n − 1 + 12 = n + 12 in the index change
1 2
2
n 7→ −n − 1. With such a modification of q n to q (n+ 2 ) , in order to get a
1
sine function we should also modify e2niw to e2(n+ 2 )iw . In other words, we
should consider the new infinite series
∞
∑
∞
∑
2
1
1
(−1) q (n+ 2 ) e2(n+ 2 )iw =
n
n=−∞
=
1 2
(−1)n e(n+ 2 )
n=−∞
∞
∑
πτ i 2(n+ 12 )iw
e
2 πτ i+nπτ i+ 1 πτ i+2niw+2iw
4
(−1)n en
.
n=−∞
After we remove to the front the factor inside the infinite sum which is
independent of n, we get
∞
∑
2
1
1
1
(−1) q (n+ 2 ) e2(n+ 2 )iw = e 4 πτ i+2iw
n
n=−∞
∞
∑
(−1)n en
2 πτ i+nπτ i+2niw
,
n=−∞
because such a factor does not affect the determination of the zero of the
series. We then compare the resulting series
∞
∑
(−1)n en
2 πτ i+nπτ i+2niw
n=−∞
with the infinite series of our theta function
ϑ(w) =
∞
∑
n n2 2niw
(−1) q e
n=−∞
=
∞
∑
n=−∞
2 πτ i+2niw
(−1)n en
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to conclude that
∞
∑
n n2 πτ i+nπτ i+2niw
(−1) e
∞
∑
=
n=−∞
=
2
(−1)n q n enπτ i+2niw
n=−∞
∞
∑
1
(−1)n q n e2ni(w+ 2 πτ )
2
n=−∞
(
)
1
= ϑ w + πτ
2
so that our new infinite series can expressed in terms of the Jacobi theta
function ϑ as
)
(
∞
∑
2
1
1
n (n+ 12 ) 2(n+ 21 )iw
πτ
i+2iw
(−1) q
e
= e4
ϑ w + πτ .
2
n=−∞
Now we group together terms in the infinite series on the left-hand side in
the index change n 7→ −n − 1 to get
∞
∑
n
(−1) q
n=−∞
2
(n+ 12 )
2(n+ 12 )iw
e
= 2i
∞
∑
n=0
n
(−1) q
2
(n+ 12 )
( (
) )
1
sin 2 n +
w
2
so that
(
)
∞
∑
1 2
1
− 41 πτ i−2iw
ϑ w + πτ = 2i e
(−1)n q (n+ 2 ) sin ((2n + 1)w) .
2
n=0
Our final conclusion is that the single zero of ϑ(w) in the fundamental parallelogram spanned by π and πτ is at the second half-period 21 πτ . This
conclusion is obtained by the use of the translate of θ(w) by a half-period
1
πτ . This prompts the definition of two other functions defined by translat2
ing θ(w) by half-periods. We label them as ϑ1 , ϑ2 , ϑ3 , ϑ4 so that their zeroes
are at the origin and three half-periods, arranged in the order of the location of their zeroes in the counterclockwise sense. To make the expressions
in terms of infinite series simpler, we remove a nonzero factor if necessary
and come up with the following precise definition of the four Jacobi theta
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functions.
ϑ1 (w) = −i e
=2
iw+ 14 πiτ
∞
∑
7
(
)
1
ϑ w + πτ
2
1 2
(−1)n q (n+ 2 ) sin(2n + 1)w,
n=0
(
∞
∑
1 2
π)
ϑ2 (w) = ϑ1 w +
=2
q (n+ 2 ) cos(2n + 1)w,
2
n=0
∞
(
)
∑
π
2
ϑ3 (w) = ϑ w +
=1+2
q n cos 2nw,
2
n=1
ϑ4 (w) = ϑ(w) = 1 + 2
∞
∑
2
(−1)n q n cos 2nw.
n=1
§2. Periodicity Factors of Four Jacobi Theta Functions. The periodicity factors of the four Jacobian theta functions are given in the following
table
π
πτ
ϑ1 (w) ϑ2 (w)
−1
−1
−N
N
Here
N=
ϑ3 (w)
1
N
ϑ4 (w)
1
−N
1 −2iw
e
= e−2iw−πiτ .
q
Note that the periodicity factors of π for ϑ1 (w) and ϑ2 (w) are not 1. They
are −1. We could easily have made them 1, but we want to stick with the
historical notations. The two periodicity factors for ϑ4 (w) = ϑ(w) are from
the construction of ϑ(w). The verification of the other three Jacobi theta
functions is simply from their definitions of being the translates of ϑ(w) by
half-periods (with normalization factors if necessary). For ϑ1 (w), from its
definition
) (
)
(
1
1
ϑ1 (w) = −i exp iw + πiτ ϑ w + πτ
4
2
Math 213a (2014-2015) Yum-Tong Siu
we have the relations
(
)
1
ϑ1 (w + π) = −i e
ϑ (w + π) + πτ
2
(
)
1
1
= i eiw+ 4 πiτ ϑ w + πτ
2
= −ϑ1 (w),
(
)
1
i(w+πτ )+ 41 πiτ
ϑ1 (w + πτ ) = −i e
ϑ (w + πτ ) + πτ
2
)
(
)
(
) (
1
−2i(w+ 12 πτ )−iπτ
iπτ iw+ 14 πiτ
= −i e e
−e
ϑ w + πτ
2
(
)
1
1
= e−2iw−πτ i eiw+ 4 πiτ ϑ w + πτ
2
= −N ϑ1 (w).
(
)
For ϑ2 (w), from its definition ϑ2 (w) = ϑ1 w + π2 we get the relations
(
π)
ϑ2 (w + π) = ϑ1 (w + π) +
2
(
π)
= −ϑ1 w +
= −ϑ2 (w),
2
(
π)
ϑ2 (w + πτ ) = ϑ1 (w + πτ ) +
2(
π)
−2i(w+ π2 )−iπτ
= −e
ϑ1 w +
2
= N ϑ2 (w).
(
)
For ϑ3 (w), from its definition ϑ3 (w) = ϑ w + π2 we get the relations
(
π)
ϑ3 (w + π) = ϑ (w + π) +
2
(
π)
=ϑ w+
= ϑ3 (w),
2
(
π)
ϑ3 (w + πτ ) = ϑ (w + πτ ) +
2(
π
π)
= −e−2i(w+ 2 )−iπτ ϑ w +
2
= N ϑ3 (w).
i(w+π)+ 14 πiτ
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§3. Quadratic Relations Among Four Jacobi Theta Functions.
There are two ways to construct Jacobi elliptic functions from Jacobi theta
functions. The first one is just from the first motivation for the introduction
of Jacobi theta functions, namely we use quotients of products of translates
of Jacobi theta functions. Let us consider the case of the Jacobi elliptic sine
function sn w for the parallelogram spanned by π and πτ . Since there are
two simple zeroes of sn w in the parallelogram spanned by π and πτ , one at
0 and one at the half-period π2 and since there are two simple poles of sn w
in the parallelogram spanned by π and πτ , one at the half-period πτ
and one
2
at the half-period π+πτ
,
we
should
use
the
following
quotient
of
products
of
2
translates of ϑ1 (w).
(
)
ϑ1 (w)ϑ1 w − π2
(
) (
) .
ϑ1 w − πτ
ϑ1 w − π+πτ
2
2
This kind of construction would give us π and πτ as the two periods of sn w.
We know that the usual trigonometric sine function sin w should correspond
to the limit of sn w as the imaginary part Im πτ of the other period πτ
becomes +∞ so that sin w is left only with the other remaining period,
which from this kind of construction would be π and is different from the
period 2π of sin w.
Because of this consideration, perhaps we should try the second kind of
construction to get the Jacobi elliptic sine function sn w from the Jacobi
theta functions. We should use the quotient
ϑ1 (w)
ϑ4 (w)
on the horizontally-doubled-up parallelogram spanned by 2π and πτ instead
of just the parallelogram spanned by π and πτ . In the horizontally-doubledup parallelogram spanned by 2π and πτ , the Jacobi theta function ϑ1 (w) has
two simple zeroes at 0 and at the half-period π = 2π
and the Jacobi theta
2
function ϑ1 (w) = ϑ(w) has two simple zeroes at the two half-periods πτ
and
2
2π+πτ
πτ
+π = 2 .
2
For the second kind of construction of using
ϑ1 (w)
ϑ4 (w)
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to obtain sn w, we would have 4K corresponding to the period√2π and 2iK ′
corresponding to the period πτ . From the definition cn w = 1 − sn2 w of
cn w and the special value sn K = 1 of sn w and the oddness of sn w it follows
that the zeroes of cn w are at ±K and the poles of cn w are the same as the
poles of sn w, namely at iK ′ and 2K + iK ′ . When we apply the second kind
of construction to get the Jacobi elliptic cosine function cn w from the Jacobi
theta functions, we should use
ϑ2 (w)
ϑ4 (w)
to obtain cn w,√up to some normalizing constant. Similarly, from the definition dn w = 1 − k 2 sn2 w of dn w and the special value sn (K + iK ′ ) = k1
of sn w and the oddness of sn w it follows that the zeroes of dn w are at
± (K + iK ′ ) and the poles of dn w are the same as the poles of sn w, namely
at iK ′ and 2K + iK ′ . When we apply the second kind of construction to get
the Jacobi delta amplitude function dn w from the Jacobi theta functions,
we should use
ϑ3 (w)
ϑ4 (w)
to obtain dn w, up to some normalizing constant.
Because of this second kind of construction to get the Jacobi elliptic functions sn w, cn w, and dn w from the quotients of the Jacobi theta functions
ϑj (w) for 1 ≤ j ≤ 4, the two quadratic relations
cn2 w + sn2 w = 1 and dn2 w + k 2 sn2 w = 1
correspond to some quadratic relations of the form
A ϑ1 (w)2 + B ϑ2 (w)2 + C ϑ4 (w)2 = 0,
A′ ϑ1 (w)2 + B ′ ϑ3 (w)2 + C ′ ϑ4 (w)2 = 0
for some complex constants A, B, C, A′ , B ′ , C ′ . We now derive these two
quadratic equations and determine the precise values for the complex constants A, B, C, A′ , B ′ , C ′ by using the table of periodicity factors and the
three fundamental properties of elliptic functions.
First we conclude from the table of periodicity factors that
ϑ2 (w)2
ϑ1 (w)2
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and
ϑ4 (w)2
ϑ1 (w)2
are both elliptic functions for the periods π and πτ . They have at most
double poles at the zeroes of ϑ1 (w). We can make the double poles at most
simple poles by considering
a ϑ2 (w)2 + b ϑ4 (w)2
ϑ1 (w)2
so that a ϑ2 (w)2 + b ϑ4 (w)2 vanishes at the zeroes of ϑ1 (w). The zeroes of
ϑ4 (w) are at points congruent to πτ
modulo Zπ + Zπτ . The simplest choices
2
2
for a and b are respectively ϑ4 (0) and −ϑ2 (0)2 . Hence we know that
ϑ4 (0)2 ϑ2 (w)2 − ϑ2 (0)2 ϑ4 (w)2
ϑ1 (w)2
is an elliptic function having at most a simple pole in a fundamental parallelogram with sides π and πτ and so must be a constant function. We still
have to determine this constant. We do this by evaluating the function at
the point πτ
. At that point ϑ4 (w) is zero. Moreover,
2
)
(
(π
)
( πτ )
π+πτ
1
π + πτ
= ϑ1
= −iei( 2 )+ 4 πiτ ϑ4
+ πτ
ϑ2
2
2
2
π+πτ
π+πτ
1
1
1
1
= −iei( 2 )+ 4 πiτ ϑ3 (πτ ) = −iei( 2 )+ 4 πiτ e−2iπτ ϑ3 (0) = q − 4 ϑ3 (0)
q
Likewise ϑ1 ( πτ
) = iq − 4 ϑ4 (0). Hence the constant is −ϑ3 (0)2 and we get the
2
quadratic relation
1
ϑ4 (0)2 ϑ2 (w)2 − ϑ2 (0)2 ϑ4 (w)2
= −ϑ3 (0)2 ,
2
ϑ1 (w)
or
ϑ1 (w)2 ϑ3 (0)2 + ϑ2 (w)2 ϑ4 (0)2 = ϑ4 (w)2 ϑ2 (0)2 .
Similarly, we obtain the quadratic relation
ϑ1 (w)2 ϑ2 (0)2 + ϑ3 (w)2 ϑ4 (0)2 = ϑ4 (w)2 ϑ3 (0)2
by choosing complex constants a′ , b′ so that
a′ ϑ23 (w) + b′ ϑ24 (w)
ϑ21 (w)
so that a′ ϑ23 (w) + bϑ24 (w) vanishes at the zeroes of ϑ1 (w).
Math 213a (2014-2015) Yum-Tong Siu
By replacing w by w +
π
2
12
in the above two algebraic relations
ϑ1 (w)2 ϑ3 (0)2 + ϑ2 (w)2 ϑ4 (0)2 = ϑ24 (w)ϑ2 (0)2 ,
(∗)
ϑ1 (w)2 ϑ2 (0)2 + ϑ3 (w)2 ϑ4 (0)2 = ϑ4 (w)2 ϑ3 (0)2 ,
we get the following two algebraic relations
ϑ2 (w)2 ϑ3 (0)2 + ϑ1 (w)2 ϑ4 (0)2 = ϑ3 (w)2 ϑ2 (0)2 ,
ϑ2 (w)2 ϑ2 (0)2 + ϑ4 (w)2 ϑ4 (0)2 = ϑ3 (w)2 ϑ3 (0)2 .
§4. Differential Equations for Quotients of Jacobi Theta Functions.
In the above discussion of making quotients of Jacobi theta functions correspond to Jacobi elliptic functions, some normalizing constants need to be
used. We are now going to determine such normalizing constants by comparing the differential equations for Jacobi elliptic functions and the differential
equations for quotients of Jacobi theta functions.
There are two key ideas for getting the differential equations for quotients
of Jacobi theta functions. The first key idea is that we can use the table of
periodicity factors to construct a quotient of two chosen Jacobi theta functions which has constant periodicity factors with respect to the two periods
π and πτ so that its derivative also has the same constant periodicity factors. The second key idea is that from the table of periodicity factors we
can construct a quotient of appropriate products of Jacobi theta functions
which, with respect to one period and half of the other period, has the same
periodicity factors as the derivative of the quotient of the two chosen Jacobi
theta functions. Then the use of the same periodicity factors with respect to
one period and half of the other implies from the three fundamental properties of elliptic functions that the derivative of the quotient of the two chosen
Jacobi theta functions differs from the quotient of appropriate products of
Jacobi theta functions only by a constant factor (as a result of considering the
quotient of the two) and as a consequence yields a differential equation for
quotients of Jacobi theta functions. We now give the details of the derivation
of such differential equations.
From the table of periodicity factors we conclude that the two quotients
ϑ1 (w)
ϑ4 (w)
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and
13
ϑ2 (w)ϑ3 (w)
ϑ24 (w)
both have periodicity factors −1 and 1 for the periods π and πτ . The first
key idea is that the derivative of
ϑ1 (w)
ϑ4 (w)
which is
ϑ′1 (w)ϑ4 (w) − ϑ′4 (w)ϑ1 (w)
ϑ24 (w)
also have the same periodicity factors −1 and 1 for the periods π and πτ .
To get rid of all the periodicity factors completely we take the quotient of
ϑ′1 (w)ϑ4 (w) − ϑ′4 (w)ϑ1 (w)
ϑ24 (w)
by
ϑ2 (w)ϑ3 (w)
ϑ24 (w)
and get the elliptic function
ϑ′1 (w)ϑ4 (w) − ϑ′4 (w)ϑ1 (w)
ϑ2 (w)ϑ3 (w)
for the periods π and πτ . Let us call this elliptic function φ.
The only possible poles of φ are the zeroes of ϑ2 (w)ϑ3 (w) and so they
are at most simple poles at points congruent to π2 and π+πτ
. We claim that
2
φ is constant from the use of the second key idea. The second key idea is
to look at the function obtained by translating φ along an appropriate halfperiod. The reason for doing it is that the four Jacobian theta functions are
related by such translation. Since φ is obtained from these four functions,
it is reasonable to expect that the function obtained from φ by translating
along an appropriate half-period is algebraically linked to φ. It turns out that
translation along the half-period 12 πτ leaves φ invariant. The verification is
simple and straightforward and is done as follows. We first express φ(w) in
terms of ϑ(w) and ϑ′ (w) to get
Math 213a (2014-2015) Yum-Tong Siu
14
ϑ′ (w)ϑ4 (w) − ϑ′4 (w)ϑ1 (w)
φ(w) = 1
ϑ2 (w)ϑ3 (w)
(
(
(
(
)
))
(
))
1
1
1
eiw+ 4 πiτ ϑ w + 12 πτ − i eiw+ 4 πiτ ϑ′ w + 12 πτ ϑ(w) − ϑ′ (w) −i eiw+ 4 πiτ ϑ w + 12 πτ
(
=
(
)) (
)
πi
1
−i eiw+ 2 + 4 πiτ ϑ w + π+πτ
ϑ w + π2
2
(
(
(
(
)
)
)
)
1
1
eiw+ 4 πiτ ϑ w + 12 πτ ϑ(w) + i eiw+ 4 πiτ ϑ′ (w)ϑ w + 21 πτ − ϑ′ w + 12 πτ ϑ(w)
=
(
)
(
)
1
eiw+ 4 πiτ ϑ w + π+πτ
ϑ w + π2
2
(
(
)
(
(
)
)
)
ϑ w + 21 πτ ϑ(w) + i ϑ′ (w)ϑ w + 12 πτ − ϑ′ w + 21 πτ ϑ(w)
(
) (
)
=
ϑ w + π+πτ
ϑ w + π2
2
From
ϑ(w + πτ ) = −N ϑ(w)
and
d
d −2iw−πτ
N=
e
= −2i e−2iw−πτ = −2iN
dw
dw
it follows that
and
ϑ′ (w + πτ ) = −N ϑ′ (w) + 2iN ϑ(w)
(
(
(
)
π
π
π)
π)
ϑ w + + πτ = −e−2i(w+ 2 )−πτ i ϑ w +
= Nϑ w +
.
2
2
2
Hence
(
φ
=
=
=
)
( (
)
(
))
(
+ i ϑ′ w + πτ
ϑ(w + πτ ) − ϑ′ (w + πτ )ϑ w + πτ
πτ ) ϑ(w + πτ )ϑ w + πτ
2
2
2
) (
)
(
w+
=
2
ϑ w + π2 + πτ ϑ w + π+πτ
2
(
)
( (
)
(
))
−N ϑ(w)ϑ w + πτ
+ i ϑ′ w + πτ
(−N ϑ(w)) − (−N ϑ′ (w) + 2iN ϑ(w)) ϑ w + πτ
2
2
2
(
) (
)
N ϑ w + π2 ϑ w + π+πτ
2
(
)
(
(
)
(
)
(
))
−ϑ(w)ϑ w + πτ
+ i −ϑ′ w + πτ
ϑ(w) + ϑ′ (w)ϑ w + πτ
− 2iϑ(w)ϑ w + πτ
2
2
2
( 2 π) (
)
ϑ w + 2 ϑ w + π+πτ
2
(
)
(
(
)
(
)
)
πτ
πτ
′
′
+
i
ϑ
(w)ϑ
w
+
−
ϑ
w
+
ϑ(w)
ϑ(w)ϑ w + πτ
2
2
2
(
) (
)
= φ(w).
ϑ w + π2 ϑ w + π+πτ
2
The invariance of φ under translation along the half-period 12 πτ means
that φ is periodic for the two periods π and 12 πτ . In the fundamental parallelogram defined by these two periods φ has at most a simple pole at points
Math 213a (2014-2015) Yum-Tong Siu
15
congruent to π2 . Since the sum of the residues of an elliptic function in a fundamental parallelogram must be zero, we know that φ has no pole and must
be a constant. Evaluating φ at the origin, we conclude that the constant is
ϑ′1 (0)ϑ4 (0)
.
ϑ2 (0)ϑ3 (0)
We have therefore the following differential equation for the Jacobian theta
functions
d ϑ1 (w)
ϑ′ (0)ϑ4 (0) ϑ2 (w)ϑ3 (w)
= 1
.
dw ϑ4 (w)
ϑ2 (0)ϑ3 (0) ϑ4 (w)2
(†)
Let
ξ(w) =
ϑ1 (w)
.
ϑ4 (w)
We are going to use the two algebraic relations (∗) to express
in terms of
ϑ1 (w)
.
ϑ4 (w)
By dividing both sides of the first relation
ϑ1 (w)2 ϑ3 (0)2 + ϑ2 (w)2 ϑ4 (0)2 = ϑ24 (w)ϑ2 (0)2
of (∗) by ϑ4 (w)2 we get
ϑ1 (w)2
ϑ2 (w)2
2
ϑ3 (0) +
ϑ4 (0)2 = ϑ2 (0)2
2
2
ϑ4 (w)
ϑ4 (w)
or
(
)
ϑ2 (w)2
1
ϑ1 (w)2
2
2
=
ϑ2 (0) −
ϑ3 (0)
ϑ4 (w)2
ϑ4 (0)2
ϑ4 (w)2
(
)
ϑ2 (0)2
ϑ3 (0)2 2
=
1−
ξ .
ϑ4 (0)2
ϑ2 (0)2
Similarly, by dividing both sides of the second relation
ϑ1 (w)2 ϑ2 (0)2 + ϑ3 (w)2 ϑ4 (0)2 = ϑ4 (w)2 ϑ3 (0)2
of (∗) by ϑ4 (w)2 we get
ϑ1 (w)2
ϑ3 (w)2
2
ϑ
(0)
+
ϑ4 (0)2 = ϑ3 (0)2
2
ϑ4 (w)2
ϑ4 (w)2
ϑ2 (w)
ϑ4 (w)
and
ϑ3 (w)
ϑ4 (w)
Math 213a (2014-2015) Yum-Tong Siu
16
)
(
ϑ3 (w)2
1
ϑ1 (w)2
2
2
=
ϑ3 (0) −
ϑ2 (0)
ϑ4 (w)2
ϑ4 (0)2
ϑ4 (w)2
(
)
ϑ3 (0)2
ϑ2 (0)2 2
=
1−
ξ .
ϑ4 (0)2
ϑ3 (0)2
or
By taking the square of both sides of the differential equation (†) we get
(
dξ
dw
)2
(
)2 (
)2
ϑ2 (w)
ϑ3 (w)
=
ϑ4 (w)
ϑ4 (w)
( ′
)2
(
)
(
)
2
ϑ1 (0)ϑ4 (0) ϑ2 (0)
ϑ3 (0)2 2 ϑ3 (0)2
ϑ2 (0)2 2
=
1−
ξ
1−
ξ
ϑ2 (0)ϑ3 (0) ϑ4 (0)2
ϑ2 (0)2
ϑ4 (0)2
ϑ3 (0)2
(
)(
)
ϑ′1 (0)2
ϑ3 (0)2 2
ϑ2 (0)2 2
=
1−
ξ
1−
ξ .
ϑ4 (0)2
ϑ2 (0)2
ϑ3 (0)2
ϑ′1 (0)ϑ4 (0)
ϑ2 (0)ϑ3 (0)
)2 (
We have to determine the constant ϑ′1 (0). It turns out that
(&)
ϑ′1 (0) = ϑ2 (0)ϑ3 (0)ϑ4 (0).
It is a very important identity which corresponds to
(
)
d
= 1.
sin x
dx
x=0
We now assume (&) first and continue with our derivation of the differential
equation. Then we will prove (&) later. With (&) we get the differential
equation
( )2
(
)(
)
dξ
ϑ3 (0)2 2
ϑ2 (0)2 2
2
2
= ϑ2 (0) ϑ3 (0) 1 −
ξ
1−
ξ .
dw
ϑ2 (0)2
ϑ3 (0)2
We are going to absorb the factor ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 by rescaling of the
variable w and by adding a factor for the function ξ(w) as follows. We define
(
)
ϑ3 (0)
w
x(w) =
ξ
ϑ2 (0)
ϑ3 (0)2
so that
d
1
x(w) =
ξ′
dw
ϑ2 (0)ϑ3 (0)
(
w
ϑ3 (0)2
)
Math 213a (2014-2015) Yum-Tong Siu
and
17
(
)2
(
)
(
)
dx
ϑ2 (0)4 2
2
= 1−x
1−
x .
dw
ϑ3 (0)4
Since ξ(0) = 0 from ϑ1 (0) = 0, it follows from this differential equation that
)
(
w
ϑ3 (0) ϑ1 ϑ3 (0)2
(
)
sn(w, k) =
w
ϑ2 (0) ϑ
4
with
(
k=
From
ϑ2 (0)
ϑ3 (0)
ϑ2 (0)2
ϑ2 (w)2
=
ϑ4 (w)2
ϑ4 (0)2
ϑ3 (0)2
)2
.
(
)
ϑ3 (0)2 2
1−
ξ
ϑ2 (0)2
(
we conclude that
cn(w, k) =
ϑ4 (0) ϑ2
(
ϑ2 (0) ϑ
4
Similarly from
ϑ3 (w)2
ϑ3 (0)2
=
ϑ4 (w)2
ϑ4 (0)2
w
ϑ3 (0)2
w
ϑ3 (0)2
)
).
(
)
(
)
2
ϑ2 (0)2 2
ϑ3 (0)2
2 ϑ3 (0)
2
1−
ξ =
1−k
ξ
ϑ3 (0)2
ϑ4 (0)2
ϑ2 (0)2
(
we conclude that
dn(w, k) =
ϑ4 (0) ϑ3
(
ϑ3 (0) ϑ
4
w
ϑ3 (0)2
w
ϑ3 (0)2
)
).
§5. Comparison of Fundamental Parallelograms for Jacobi Elliptic
Functions and for Jacobi Theta Functions. The modulus
(
)2
ϑ2 (0)
k=
ϑ3 (0)
is not the modulus of the lattice Zπ + Zπτ . Let us determine the values K
and K ′ for k in terms of τ and ϑ3 (0). We have
(
)
w
ϑ
2
2
ϑ3 (0)
ϑ4 (0)
(
).
cn(w, k) =
w
ϑ2 (0) ϑ
4
ϑ3 (0)2
Math 213a (2014-2015) Yum-Tong Siu
The periodicity factors of
(
ϑ2
ϑ4
(
w
ϑ3 (0)2
w
ϑ3 (0)2
18
)
)
are −1 for both πϑ3 (0)2 and πτ ϑ3 (0)2 . We also know that cn w has periodicity factors −1 for both 2K and 2iK ′ . In a fundamental parallelogram of
2KZ + 2iK ′ Z the function cn w has only one zero. Also in the fundamental
parallelogram of ϑ23 (0)(πZ + πτ Z) the function
(
)
ϑ2 ϑ3w(0)2
(
)
ϑ4 ϑ3w(0)2
has only one zero. Thus we know that πϑ3 (0)2 and πτ ϑ3 (0)2 generate the
same lattice as 2K and 2iK ′ . Consider the case of a purely imaginary τ
with Im τ sufficiently large. Then both ϑ2 (0) and ϑ3 (0) are positive and
ϑ2 (0) < ϑ3 (0). Thus
ϑ2 (0)2
k=
ϑ3 (0)2
is positive and less than 1. Since both 2K and πϑ3 (0)2 are positive and both
2iK ′ and πτ ϑ3 (0)2 are on the positive y-axis, from the fact that
2KZ + 2iK ′ Z = ϑ3 (0)2 (πZ + πτ Z)
we conclude that 2K = πϑ3 (0)2 and 2iK ′ = πτ ϑ3 (0)2 for this special case of a
purely imaginary τ with Im τ sufficiently large. By analytic continuation we
know that they must be the same for all values of τ . The primitive periods
of
(
)
w
ϑ
ϑ3 (0) 1 ϑ3 (0)2
(
)
sn(w, k) =
w
ϑ2 (0) ϑ
4
ϑ3 (0)2
are 4K = 2π ϑ3 (0)2 and 2iK ′ = πτ ϑ3 (0)2 . When the period lattice Zπ +Zπτ
degenerates to the lattice Zπ, sn(w, k) degenerates to the function sin w with
period 2π.
Math 213a (2014-2015) Yum-Tong Siu
19
§6. Derivation of Identity ϑ′1 (0) = ϑ2 (0)ϑ3 (0)ϑ4 (0). This identity is
motivated by the corresponding identity
(
)
d
sin x
=1
dx
x=0
in trigonometry for the limiting case of Im τ → +∞. The four Jacobi theta
functions are actually functions of the two complex variables w and τ . To
highlight this point, we write
ϑ1 (w|τ ) =
ϑ2 (w|τ ) =
ϑ3 (w|τ ) =
ϑ4 (w|τ ) =
∞
∑
n=−∞
∞
∑
n=−∞
∞
∑
n=−∞
∞
∑
1
1
2
e(n− 2 )πi+(n +n+ 4 )πτ i+(2n+1)iw ,
1
2
e(n +n+ 4 )πτ i+(2n+1)iw ,
en
2 πτ i+2niw
(−1)n en
,
2 πiτ +2niw
.
n=−∞
We can use the fact
(
)
d
=1
sin x
dx
x=0
in trigonometry which is the limiting case of our identity
(♮)
ϑ′1 (0|τ ) = ϑ2 (0|τ )ϑ3 (0|τ )ϑ4 (0|τ )
as Im τ → +∞ if the equation obtained by differentiating (♮) once with
respect to the variable τ holds. Instead of applying dτd to (♮), we can get the
same effect by applying dτd log to (♮), which has the advantage of converting
the product on the right-hand side of (♮) to a sum before differentiation. So
we consider the equation
(♭)
1
dϑ′1 (0|τ )
1
dϑ2 (0|τ )
1
dϑ3 (0|τ )
1
dϑ4 (0|τ )
=
+
+
.
′
ϑ1 (0|τ )
dτ
ϑ2 (0|τ )
dτ
ϑ3 (0|τ )
dτ
ϑ4 (0|τ )
dτ
For each term of the infinite sum expressions for the four Jacobi theta functions ϑj (w|τ ) for 1 ≤ j ≤ 4 in the exponent of e the coefficient of τ is
Math 213a (2014-2015) Yum-Tong Siu
20
− πi
times the square of the coefficient of w. This implies that each infinite
4
sum satisfies the following “heat equation” when w is regarded as the “space
variable” and τ is regarded as the “time variable”.
πi ∂ 2
∂
ϑj (w|τ ) +
ϑj (w|τ ) = 0 for 1 ≤ j ≤ 4.
2
4 ∂w
∂τ
This heat equation at w = 0 for the four Jacobi theta functions transforms
the differential equation (♭) to the equation
(♯)
ϑ′′′
ϑ′′2 (0) ϑ′′3 (0) ϑ′′4 (0)
1 (0)
=
+
+
.
ϑ′1 (0)
ϑ2 (0) ϑ3 (0) ϑ4 (0)
Here we have suppressed the variable τ in ϑj (w|τ ) and we simply use ϑj (w)
d2 ϑ (w)
with ϑ′′j (w) denoting dwj 2 . Now that we have used the heat equation to
change time-differentiation to space-differentiation, we want to reverse direction to invert the process of logarithmic differentiation. We consider the
function
ϑ1 (w)ϑ2 (w)ϑ3 (w)ϑ4 (w)
f (w) =
ϑ1 (2w)
so that the vanishing of its logarithmic second derivative
would give us (♯).
d2 log f
dw2
at w = 0
The function f has no zero and no pole in the fundamental parallelogram
spanned by π and πτ , because the four zeroes of ϑ1 (2w) in the parallelogram
spanned by π and πτ are precisely the four zeroes of ϑ1 (w)ϑ2 (w)ϑ3 (w)ϑ4 (w).
Now f (w) is an entire function with periodicity factors 1 for π and q 2 N 2
for πτ . It is actually irrelevant to know the precise expression q 2 N 2 for
the periodicity factor of f for the period πτ . What is relevant is that the
periodicity factor for πτ is equal to the exponential function of a polynomial
in w of degree ≤ 1. It means that f (w) is a theta function in the sense that
f (w) is an entire function on C whose periodicity factors are each equal to
the exponential function of some polynomial in w of degree ≤ 1.
A nowhere zero theta function is called a trivial theta function. The
reason for the terminology is that theta functions are introduced as building
blocks for elliptic functions so that elliptic functions can be expressed as the
quotient of products of their translates when the zero-set and the pole-set of
an elliptic function is created from the zero-set of the theta function. So a
nowhere zero theta function is useless for this purpose and so is called trivial.
Math 213a (2014-2015) Yum-Tong Siu
21
Any trivial theta function is equal to the exponential function of some
polynomial of degree ≤ 2, because the second derivative of its logarithm
is holomorphic and elliptic and so must be constant. In particular, in our
2
case f (w) = ea w +b w+c for some complex numbers a, b, c, because we can
choose a branch log f (w) on C and from f (w + π) = f (w) and f (w + πτ ) =
q 2 N 2 f (w) with N being a linear function of w it follows that the entire
d2
function dw
2 log f (w) has periods π and πτ and thus must be constant, and
2
then integrating and exponentiating yields that f (w) = eaw +bw+c for some
complex numbers a, b, c.
Besides being a trivial theta function, our function f (w) has π as period. This gives us more information concerning the constants a, b, c. More
precisely, from f (w + π) = f (w) it follows that
2 +b(w+π)+c
ea(w+π)
= eaw
2 +bw+c
2
and e2awπ+π +bπ = 1 for all w and the constant a must be (0. Thus f)(w) =
d2
d2
exp(bw + c) and dw
log f (0) is
2 log f is identically zero. In particular,
dw2
zero.
2
d
We now derive (♯) from the vanishing of ( dw
2 log f )(0). From the evenness
′
of ϑν (w) we have the vanishing of ϑν (0) for 2 ≤ ν ≤ 4. Hence for 2 ≤ ν ≤ 4,
( 2
)
d
ϑ′′ν (0)
log
ϑ
.
(0)
=
ν
dw2
ϑν (0)
Let ϑ1 (w) = α(w + β w3 + · · · ) be the power series expansion of the odd
(w)
entire function ϑ1 (w). Let g(w) = ϑϑ11(2w)
. Then
g(w) =
)
α(w + β w3 + · · · )
1(
2
=
1
−
3βw
+
·
·
·
.
α((2w) + β (2w)3 + · · · )
2
(w)
is even and its derivative
Since ϑ1 (w) is odd, it follows that g(w) = ϑϑ11(2w)
′
′
g (w) is odd, implying that g (0) = 0. Hence
( 2
)
d
g ′′ (0) g ′ (0)2
g ′′ (0)
log
g
(0)
=
−
=
dw2
g(0)
g(0)2
g(0)
′′′
ϑ (0)
.
= −6β = − 1′
ϑ1 (0)
Math 213a (2014-2015) Yum-Tong Siu
22
Note that the above argument of computing
( 2
)
d
ϑ′′′
1 (0)
log
g
(0)
=
−
2
dw
ϑ′1 (0)
is simply a modified form of L’Hôpital’s rule in which the limit of the quotient is computed by using power series expansions. That is the reason why
the third-order derivative ϑ′′′
1 (0) appears in the expression for the second
derivative for log g(w). The identity (♯) now follows from the vanishing of
( 2
)
d
g ′′ (0) ϑ′′2 (0) ϑ′′3 (0) ϑ′′4 (0)
log
f
(0)
=
+
+
+
.
dw2
g(0)
ϑ2 (0) ϑ3 (0) ϑ4 (0)
Thus, the differential equation (♭) holds for all τ .
Finally from
ϑ1 (w, q) = 2
ϑ2 (w, q) = 2
∞
∑
1 2
(−1)n q (n+ 2 ) sin(2n + 1)w,
n=0
∞
∑
1 2
q (n+ 2 ) cos(2n + 1)w,
n=0
ϑ3 (w, q) = 1 + 2
ϑ4 (w, q) = 1 + 2
∞
∑
n=1
∞
∑
2
q n cos 2nw,
2
(−1)n q n cos 2nw
n=1
we have
ϑ′1 (w, q) = 2
∞
∑
1 2
(−1)n q (n+ 2 ) (2n + 1) cos(2n + 1)w
n=0
and
lim q − 4 ϑ′1 (0, q) = 2,
1
q→0
lim ϑ3 (0, q) = 1,
q→0
so that
lim q − 4 ϑ2 (0, q) = 2,
1
q→0
lim ϑ4 (0, q) = 1
q→0
ϑ′1 (0, q)
= 1.
q→0 ϑ2 (0, q)ϑ3 (0, q)ϑ4 (0, q)
This limiting value as q → 0 (or equivalently as Im τ → +∞), together with
the differential equation (♭) would imply that the identity (♮) holds for all τ
with Im τ > 0.
lim
Math 213a (2014-2015) Yum-Tong Siu
23
We would like to remark that the identity (♮) which states
ϑ′1 (0) = ϑ2 (0)ϑ3 (0)ϑ4 (0)
is equivalent to the derivative of
(
)
w
ϑ
ϑ3 (0) 1 ϑ3 (0)2
)
(
sn(w, k) =
w
ϑ2 (0) ϑ
4
ϑ3 (0)2
at w = 0 being 1, because the vanishing of ϑ1 (w) at w = 0 means that
d
ϑ′1 (0)
sn(w, k) =
.
dw
ϑ2 (0)ϑ3 (0)ϑ4 (0)
Thus the identity
corresponds to
ϑ′1 (0) = ϑ2 (0)ϑ3 (0)ϑ4 (0)
(
d
sin x
dx
)
= 1.
x=0
To be continued ...