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Question Number 190779 by 07049753053 last updated on 11/Apr/23

$$\mathbf{R}\mathit{e}{\left(\frac{1}{1-\mathit{a}}\right)}^{\mathit{a}-1}$$

Answered by Frix last updated on 11/Apr/23

$$f\left(a\right)={\left(\frac{1}{1-a}\right)}^{a-1}\in \mathbb{R}\mathrm{\forall }a<1$$$$\underset{a\to 1}{\lim }f\left(a\right)=1$$$$\mathrm{\forall }a>1:f\left(a\right)=\frac{1}{{(a-1)}^a}((1-a)\cos \pi a+(1-a)\mathrm{i}\sin \pi a)$$$$\mathrm{\forall }a\in {\mathbb{N}}^{★}:f\left(a\right)=-{(-1)}^a{(a-1)}^{1-a}\in \mathbb{R}$$$$\mathrm{\forall }a>1\wedge a\notin \mathbb{N}:\mathrm{Re}\left(f\left(a\right)\right)=-{(a-1)}^{1-a}\cos \pi a$$

Commented by 07049753053 last updated on 12/Apr/23

$$\mathbf{thanks}$$

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