What is the numerical value of $(-3)^{\pi}$

Question

As the title suggest what is the numerical value of $(-3)^{\pi}$?

could we derive an answer using numerical analysis something along the lines of well if its basically $(-3) \cdot(-3) \cdot (-3) \cdot(-3)^{\pi-3}$?

Answer 1

Mathematicians generally define powers of negative real numbers using the principal value of the complex logarithm, that is

$$(-3)^\pi:= e^{\pi\ln(-3)}=e^{\pi(\ln|-3|+i\arg(-3))}=e^{\pi\ln 3+i\pi^2}=3^\pi\cdot e^{i\pi^2}$$

where $e^{i\pi^2}$ is the complex number defined by a vector on the complex plane of length $1$ such that the angle with the real line is $\pi^2$, that is

$$e^{i\pi^2}=\cos\pi^2+i\sin\pi^2$$

With WolframAlpha I get this numerical approximation:

$$(-3)^\pi\approx -28.47456 -i\, 13.57354$$