The analytic region of the following function f (x) is pointed out and its derivative is obtained （1）Z^3+2iZ ;

The analytic region of the following function f (x) is pointed out and its derivative is obtained （1）Z^3+2iZ ;

z=x+iy
Substituting: F (z) = (x + iy) & # 179; + 2I (x + iy)
=x³+3ix²y-3xy²-iy³+2ix-2y
=x³-3xy²-2y+i(3x²y-y³+2x)
Then: u = x & # 179; - 3xy & # 178; - 2Y, v = 3x & # 178; Y-Y & # 179; + 2x
The analytic requirement satisfies the Cauchy Riemann condition
∂u/∂x=∂v/∂y,∂u/∂y=-∂v/∂x
∂ U / & 8706; X = 3x & 178; - 3Y & 178;, & 8706; V / & 8706; y = 3x & 178; - 3Y & 178; are equal
∂ U / & # 8706; y = - 6xy-2, & # 8706; V / x = 6xy + 2 are opposite numbers and satisfy the Cauchy Riemann condition, so the function is analytic everywhere in the complex plane
f '(z)=3z²+2i
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