Question

Найдите функцию Im(exp (z^2)). Вверху над z стоиn черточка

1) exp (x^2−y^2)cos (2xy)
2) exp (x^2−y^2)sh(2xy)
3) −exp (x^2−y^2)sin (2xy)

Answer 1

Положим $z = x + iy$, тогда $overline{z}=x-iy$.

$ImBigg(expBig{overline{z}^2Big}Bigg)=ImBigg(expBig{(x-iy)^2Big}Bigg)=ImBigg(expBig{x^2-y^2-2xyiBig}Bigg)=\ \ \ =expBig{x^2-y^2Big}cdot ImBigg(expBig{-2xyiBig}Bigg)=expBig{x^2-y^2Big}cdot ImBigg(expBig{cos 2xy\ \\ -isin 2xyBig}Bigg)=expBig{x^2-y^2Big}cdot Big(-sin 2xyBig)=-expBig{x^2-y^2Big}sin2xy$

Ответ: 3)