ISSN 1119-4618
 

Original Article 


ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS

Swem Samuel Swem.

Abstract
Abstract
In this paper, we have applied Weierstrass and Hadamard factorization theorems to obtain the product representation of sin‚Ā°z and cos‚Ā°z and compared the suitability of the two theorems in their applications. It was found that the main challenge in applying Weierstrass theorem is in the identification of the series that results in the course of determining the unknown arbitrary entire function in the theorem. In the case of Hadamardís theorem the unknown function is a polynomial function of degree at most the order of the entire function whose product representation is being sought. This additional information about the unknown function in the case of Hadamardís theorem made its determination extremely easy. Hadamards theorem is therefore preferable when factorizing entire functions of finite order. However, Weierstrass theorem is a more general theorem than that of Hadamard. In course of application of Hadamardís theorem, we also found the maximum modulus of sin‚Ā°z and cos‚Ā°z in the disc |z|‚ȧR and the points at which these maximum occurs. It was also found that the maximum modulus of these functions tends to infinity with R as expected of entire functions. In addition, the two functions were both found to be of order and genus 1, respectively. Further research in this field should focus on applying the two theorems in obtaining the product representation of other entire functions such as sinh‚Ā°z and cosh‚Ā°z and making similar comparisons.

Key words: Keywords and Phrases: Entire Function, Infinite Product, Convergence of Infinite Product, Zeros of entire Functions, Weierstrass theorem, Maximum Modulus of a Function, Order of an entire Function, Hadamard theorem.


 
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Pubmed Style

Swem SS, . ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS. JPAS. 2021; 21(1): 166-172. doi:10.5455/sf.108128


Web Style

Swem SS, . ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS. http://www.atbuscienceforum.com/?mno=108128 [Access: May 11, 2021]. doi:10.5455/sf.108128


AMA (American Medical Association) Style

Swem SS, . ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS. JPAS. 2021; 21(1): 166-172. doi:10.5455/sf.108128



Vancouver/ICMJE Style

Swem SS, . ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS. JPAS. (2021), [cited May 11, 2021]; 21(1): 166-172. doi:10.5455/sf.108128



Harvard Style

Swem, S. S. & (2021) ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS. JPAS, 21 (1), 166-172. doi:10.5455/sf.108128



Turabian Style

Swem, Swem Samuel, and . 2021. ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS. Science Forum (Journal of Pure and Applied Sciences), 21 (1), 166-172. doi:10.5455/sf.108128



Chicago Style

Swem, Swem Samuel, and . "ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS." Science Forum (Journal of Pure and Applied Sciences) 21 (2021), 166-172. doi:10.5455/sf.108128



MLA (The Modern Language Association) Style

Swem, Swem Samuel, and . "ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS." Science Forum (Journal of Pure and Applied Sciences) 21.1 (2021), 166-172. Print. doi:10.5455/sf.108128



APA (American Psychological Association) Style

Swem, S. S. & (2021) ON THE PRODUCT REPRESENTATION OF ENTIRE FUNCTIONS USING WEIERSTRASS AND HADAMARD FACTORIZATION THEOREMS. Science Forum (Journal of Pure and Applied Sciences), 21 (1), 166-172. doi:10.5455/sf.108128