A Logarithmic Derivative of Theta Function and Implication
Yaling Men^{1}, Jiaolian Zhao^{2}
^{1}School of Mathematics, Xianyang Vocational and Technical College, Xianyang, P. R. China
^{2}School 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 periodsand ,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 replacebyand 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 byand 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