# THE NOVELTY OF INFINITE SERIES FOR THE COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND

• ROHEDI, A.Y. (DEPARTMENT OF PHYSICS, FACULTY OF MATHEMATICS AND NATURAL SCIENCES, INSTITUT TEKNOLOGI SEPULUH NOPEMBER(ITS)) ;
• YAHYA, E. (DEPARTMENT OF PHYSICS, FACULTY OF MATHEMATICS AND NATURAL SCIENCES, INSTITUT TEKNOLOGI SEPULUH NOPEMBER(ITS)) ;
• PRAMONO, Y.H. (DEPARTMENT OF PHYSICS, FACULTY OF MATHEMATICS AND NATURAL SCIENCES, INSTITUT TEKNOLOGI SEPULUH NOPEMBER(ITS)) ;
• WIDODO, B. (DEPARTMENT OF MATHEMATICS, FACULTY OF MATHEMATICS AND NATURAL SCIENCES, INSTITUT TEKNOLOGI SEPULUH NOPEMBER(ITS))
• Accepted : 2017.08.16
• Published : 2017.09.25

#### Abstract

According to the fact that the low convergence level of the complete elliptic integral of the first kind for the modulus which having values approach to one. In this paper we propose novelty of the complete elliptic integral which having new infinite series that consists of new modulus introduced as own modulus function. We apply scheme of iteration by substituting the common modulus with own modulus function into the new infinite series. We obtained so many new exact formulas of the complete elliptic integral derived from this method correspond to the number of iterations. On the other hand, it has been also obtained a lot of new transformation functions with the corresponding own modulus functions. The calculation results show that the enhancement of the number of significant figures of the new infinite series of the complete elliptic integral of the first kind corresponds to the level of quadratic convergence.

#### Acknowledgement

Supported by : Ministry of Education and Culture of Indonesia (DIKTI)

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