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 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

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