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how do I find if the following function is analytic and compute derivative...?
the function is 1/[(z^3-1)(z^2+2)]
another one is [1/(z-1)]^10 (is this just the power rule?)
1 Answer
- Anonymous1 decade agoFavorite Answer
Let
f(z)=1/(z-1)=>
f(z)=1/[(x-1)+iy]=>
f(z)=(x-1)/[(x-1)^2+y^2]-
iy/[(x-1)^2+y^2]=>
Now let
U(x,y)=(x-1)/[(x-1)^2+y^2]
V(x,y)=-y/[(x-1)^2+y^2]
Ux=[y^2-(x-1)^2]/[(x-1)^2+y^2]^2
Vy=(y^2-(x-1)^2]/[(x-1)^2+y^2]^2=>
Ux=Vy
Uy=-2y(x-1)/[(x-1)^2+y^2]^2
Vx=2y(x-1)/[(x-1)^2+y^2]^2=>
Uy=-Vx
So, f(z)=1/(z-1) is analytic and so is f(z)^10.
f '(z)=-10/(z-1)^9
Similar reasoning applies to 1/[(z^3-1)(z^2+2)].
You may check whether it is analytic or not yourself.