TITLE: How did Euler calculate $i^i$?
QUESTION [6 upvotes]: 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?

REPLY [10 votes]: On the numerical value of $i^i$ and Historical notes on the relation $e^{-\pi/2}=i^i$ 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].