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Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions

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  • Ji Eun Kim

    (Department of Mathematics, Dongguk University, Gyeongju 38066, Korea)

Abstract

We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i , which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j + k 2 direction is determined using the rule between bases i , j , and k defined in the quaternion. We present examples in which the definition of the j -step derivative and ( j , k ) -step derivative are applied to elementary functions e z , sin z , and cos z .

Suggested Citation

  • Ji Eun Kim, 2021. "Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions," Mathematics, MDPI, vol. 9(6), pages 1-14, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:668-:d:521344
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    References listed on IDEAS

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    1. Ji Eun Kim & Su Jin Lim & Kwang Ho Shon, 2014. "Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-8, March.
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