A Logarithmic Derivative of Theta Function and Implication
Yaling Men1, Jiaolian Zhao2
1School of Mathematics, Xianyang Vocational and Technical College, Xianyang, P. R. China
2School of Mathematics and Informatics, Weinan Teacher`s University, Weinan, P. R. China
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To cite this article:
Yaling Men, Jiaolian Zhao. A Logarithmic Derivative of Theta Function and Implication. Pure and Applied Mathematics Journal. Special Issue: Mathematical Aspects of Engineering Disciplines. Vol. 4, No. 5-1, 2015, pp. 55-59. doi: 10.11648/j.pamj.s.2015040501.21
Received: June 26, 2015; Accepted: June 28, 2015; Published: June 30, 2016
Abstract: In this paper we establish an identity involving logarithmic derivative of theta function by the theory of elliptic functions. Using these identities we introduce Ramanujan’s modular identities, and also re-derive the product identity, and many other new interesting identities.
Keywords: Theta Function, Elliptic Function, Logarithmic Derivative
1. Introduction and Definitions
Assume throughout this paper that, when
. As usual, the classical Jacobi theta functions are defined as follow[1-3],
(1.1)
(1.2)
(1.3)
(1.4)
The shifed factorial is defined by
and some times write
With above notation, the celebrated Jacobi triple product identity can be expressed as follow
(1.5)
Employing the Jacobi triple product identity, we can derive the infinite product expressions for theta function
Proposition1.1. (Infinite product representations for theta functions)
When there is no confusion, We will usefor
,
for
to denote the partial derivative with respect to the variable, and
for
,
From the above equations, the following facts are obvious
(1.6)
With respect to the (quasi) period and
, Jacobi theta functions
satisfy the following relations
and
also have
(1.7)
(1.8)
Where
The following trigonometric series expressions for the logarithmic derivative with respect to of Jacobi Theta functions will be very useful in this paper,
(1.9)
Theorem 1.1. The sum of all the residues of an elliptic function in the period parallelogram is zero.
2. Main Theorem and Proofs
Theorem 2.1. Forand
, we have
Proof. We consider the following function
(2.1)
by the definition of , we can readily verify that
is an elliptic function with periods
and
,The only poles of
is
and
. Furthermore,
is
its simple pole and 0 is its pole with order two. By virtue of the residue theorem of elliptic functions, we have
(2.2)
And applying relation of and
in (1.7-1.8) and L’Hospital’ rule, we can obtain
(2.3)
Next we compute
(2.4)
From Theorem 1.1, substituting (2.3) and (2.4) into (2.2), by performing a little reduction we can complete the proof of Theorem 2.1.
Corollary 2.1. Forand
, we have
Proof. We differentiate the formulae of Theorem2.1 with respect to, and then set
, then
(2.5)
Now we combine with another elementary identity [7, p.467]
(2.6)
From formula (2.5) and (2.6), we can obtain
This completes the proof of Corollary 2.2.
Remark 2.1. The corollary2.1 is often written in terms of the weierstrass elliptic and sigma functions as [7, p.451]
Theorem 2.2. For and
are real, we have
(2.7)
where denotes the imaginary part of the complex number
.
Proof. Firstly, we replaceby
and
by
in Theorem 2.1. it becomes
(2.8)
Sine is real,
is also real valued, then we have
(2.9)
We note that (2.9) is precisely the numerator of (2.7). We now consider its denominator. In Corollary2.1, replace by
and
by
, then obtain
(2.10)
Now from (2.9) and (2.10), the left hand side of (2.7) becomes
(2.11)
Here we can see that it is crucial that and
are both real , Since
. On the other hand, we can derive a different expression for the imaginary part of the above quantity. Since we note that in Corollary2.1, replacing
by
,
by
and
by
, we can deduce (where
).
Substituting above equality into (2.11), we can obtain the result (2.7). This complete the proof of Theorem 2.2.
3. Implications for Square Sum
In this section, we will re-deduce the Lambert series representations forfrom Theorem 2.1 easily and difference methods from [4-6].
Theorem 3.1. For Jacobi Theta function, we have
Proof. We note that , then
In Theorem 2.1, we replace by
, then it becomes
Next,we chooseand
with the facts that
,then above equality becomes
This complete the proof of Theorem 3.1
Theorem 3.2. For Jacobi Theta function, we have
Proof. We set in (12 ), then differentiate it with respect to
and set
, We recall (1.9) for
then
And from (1.9), we can obtain
Hence, we can obtain
This complete the proof of Theorem 3.2
Acknowledgements
This paper was partially supported by The Natural sciences funding project of Shaanxi Province (15JK1264) and the Key disciplines funding of Weinan Teacher’s University (14TSXK02).
References