We give a formula for \(f(\eta ),\) where \(f:{\mathbb {C}} \rightarrow {\mathbb {C}}\) is a continuously differentiable function satisfying \(f(\bar{z}) = \overline{f(z)},\) and \(\eta \) is a dual quaternion. Note this formula is straightforward or well known if \(\eta \) is merely a dual number or a quaternion. If one is willing to prove the result only when f is a polynomial, then the methods of this paper are elementary.

中文翻译：

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对偶四元数的函数微积分

我们给出\(f(\eta ),\)的公式，其中\(f:{\mathbb {C}} \rightarrow {\mathbb {C}}\)是满足\(f(\bar)的连续可微函数{z}) = \overline{f(z)},\)和\(\eta \)是对偶四元数。请注意，如果\(\eta \)只是一个对偶数或四元数，则该公式是简单的或众所周知的。如果只愿意证明f是多项式时的结果，那么本文的方法就是初等的。