There is a well-know quotation of Euler in a letter from 1746 to Goldbach:



> 
> Letztens habe ich gefunden, dass diese expressio $\sqrt{-1}^{\sqrt{-1}}$ einen valorem realem habe, welcher in fractionibus decimalibus
> $=0,2078795763$, welches mir merkwürdig zu seyn scheinet.
> 
> 
> 


For the principal value of the logarithm the *expresssion* is $i^i=\exp(i \log(e^{i\pi/2}))= e^{-\pi/2}$.


My question is, how did Euler calculate this with such an accuracy?



How did Euler calculate $i^i$?
[On the numerical value of $i^i$](https://www.jstor.org/stable/2972387%0A) and [Historical notes on the relation $e^{-\pi/2}=i^i$](https://doi.org/10.2307/2972388) describe how these accurate computations can be performed with logarithmic tables.


Euler described how he arrived at the identity in a paper read at the Berlin Academy in 1746, giving more decimals (13) than in the letter to Goldbach. A later calculation by Gauss computed 35 decimal places. Euler did not present his computation, but Gauss did [[source](https://archive.org/details/werkecarlf03gausrich/page/n439/mode/2up)].


