Preprint Article Version 1 NOT YET PEER-REVIEWED

# The Novelty of Infinite Series for the Complete Elliptic Integral of the First Kind

Version 1 : Received: 26 August 2016 / Approved: 26 August 2016 / Online: 26 August 2016 (12:57:10 CEST)

How to cite: Rohedi, A.; Pramono, Y.; Widodo, B.; Yahya, E. The Novelty of Infinite Series for the Complete Elliptic Integral of the First Kind. Preprints 2016, 2016080215 (doi: 10.20944/preprints201608.0215.v1). Rohedi, A.; Pramono, Y.; Widodo, B.; Yahya, E. The Novelty of Infinite Series for the Complete Elliptic Integral of the First Kind. Preprints 2016, 2016080215 (doi: 10.20944/preprints201608.0215.v1).

## Abstract

According to the fact that the low convergence level on 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 consists of the new infinite series. We apply the 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 iteration order. On the other hand, it has been also obtained a lot of new modulus functions rather than common used previously. The calculation results show that the number of significant figures of the new infinite series of the complete elliptic integral of the first kind is increased more and more. It means more fastly convergent would be obtained comparison values with the previous infinite series.

## Subject Areas

integral form; infinite series; modulus function; transformation function