A Rational Approximation for the Complete Elliptic Integral of the First Kind

• Main Authors: Zhen-Hang Yang, Jing-Feng Tian, Ya-Ru Zhu
• Format: Article
• Language: English
• Published: MDPI AG 2020-04-01
• Series: Mathematics
• Subjects: complete integrals of the first kind; arithmetic-geometric mean; rational approximation
• Online Access: https://www.mdpi.com/2227-7390/8/4/635

Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">K</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">K</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. More precisely, we establish the inequality <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> <mi mathvariant="script">K</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>></mo> <mfrac> <mrow> <mn>5</mn> <msup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mn>126</mn> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>+</mo> <mn>61</mn> </mrow> <mrow> <mn>61</mn> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mn>110</mn> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>+</mo> <mn>21</mn> </mrow> </mfrac> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>=</mo> <msqrt> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </semantics> </math> </inline-formula>. The lower bound is sharp.