Complex variables : f(z) = zIm(z^2)?

Complex variables: f(z) = z Im(z^2)?

If f(z) = z Im(z^2), determine the points in the complex plane such that f is differentiable.

Hey mous99.

What have you tried for this question?

(Hint: Something is differentiable if it satisfies the Cauchy-Riemann equations).

Let z = x+ iy z^2 = (x^2-y^2) + 2ixy or,
im(z^2 = 2ixy F(z) = z (2ixy))

Hence F(z) is differentiable at all points in complex plain except at (x,y)=(0,0) that is origin.

Is this answer right or wrong?

Quote:

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Originally Posted by mous99

Let z = x+ iy z^2 = (x^2-y^2) + 2ixy or,
im(z^2 = 2ixy F(z) = z (2ixy))

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You have a serious mistake.

. NOT .

why {Im}(z^2)=2xy?

Quote:

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Originally Posted by mous99

why {Im}(z^2)=2xy?

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If indeed you do not know that, then you have no business trying this question.

If then .

EXAMPLE:
.

Let z = x+ yi
Im (z2) = Im (x+yi) ^2
= Im (x^2- y ^ 2 +2xyi)


then how to continue for Im (z2) = 2xy???

Quote:

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Originally Posted by mous99

Let z = x+ yi
then how to continue for Im (z2) = 2xy???

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yes, i have same answer with you.
but how to determine the points 2x^2 y + 2xy^2 i in the complex plane such that f is differentiable?

Quote:

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Originally Posted by mous99

yes, i have same answer with you.
but how to determine the points 2x^2 y + 2xy^2 i in the complex plane such that f is differentiable?

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(Hint: Something is differentiable if it satisfies the Cauchy-Riemann equations).

f(z) = z Im (z^2) = (x+iy) (2xy) = 2(x^2)y + 2x(y^2)i



determine the points in the complex plane such that f is differentiable.

df/dx=2xy+2y(x+iy), so i*df/dx=2ixy+2iy(x+iy)

And df/dy=2ixy+3x(x+iy)

This tell us f(z) is differentiable when x=iy.


is it right?