10.6.30 Analytic functions
condition: Recover the analytic function \( f(z) \) in a neighborhood of the point \( z_{0} \) from the known real part \( u(x, y) \) or imaginary part \( v(x, y) \) and the value \( f\left(z_{0}\right) \). \[ v=y-\frac{y}{x^{2}+y^{2}}, f(1)=2 \]
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.