10.6.27 Analytic functions
condition: Present the given function \( \omega=f(z) \), where \( z=x+i y \), in the form \( \omega=u(x, y)+i v(x, y) \); check if it is analytical. If yes, then find the value of its derivative at a given point \( z_{0} \). \[ \omega=e^{-z^{2}}, \quad z_{0}=i \]
Differentiation of analytical functions, finding their real and imaginary parts, finding the number of roots of complex equations using the argument principle, Roucher's theorem and much more in this section.