Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

中文翻译：

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3D Clifford 代数中一般多向量的指数

对于 n = p + q = 3，给出了计算 Clifford 几何代数 (GA) Clp;q 中一般多向量 (MV) 指数的封闭形式表达式。应用所获得的指数公式来找到精确的 GA 三角函数和双曲函数MV 论据。我们已经验证了所提出的精确公式与 MV 双曲和三角函数的级数展开一致。在信号和图像处理、自动控制和机器人技术中，指数可用于求解 GA 微分方程。