The derivative of the function f(z) is: f′(z)=2z(ez−1)+z2ez=ez(z2+2z)−2z.
f(0)=0. A necessary conditions for the function to be analytic are: ∂zˉ∂f(z,zˉ)=0. The latter holds. Rewrite the function f(z) as: f(z)=(x+iy)2(ex+iy−1)=(x2−y2+2xiy)(exeiy−1)=(x2−y2)exeiy−(x2−y2)+2xiyexeiy−2xyi=(x2−y2)ex(cos(y)+isin(y))+2xyiex(cos(y)+isin(y))−2xiy
In case we split real and imaginary part as: f(z)=u(x,y)+iv(x,y) it is obvious that partial derivatives ∂x∂u, ∂y∂u, ∂x∂v, ∂y∂v are continuous. The latter are sufficient conditions for the function to be analytic. Thus, the function is analytic. In particular, at point 0.
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