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

1. ## 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.

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

Hey mous99.

What have you tried for this question?

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

3. ## Re: Complex variables : f(z) = zIm(z^2)?

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?

4. ## Re: Complex variables : f(z) = zIm(z^2)?

Originally Posted by mous99
Let z = x+ iy z^2 = (x^2-y^2) + 2ixy or,
im(z^2 = 2ixy F(z) = z (2ixy))

You have a serious mistake.

$\text{Im}(z^2)=2xy$. NOT $2xyi$.

5. ## Re: Complex variables : f(z) = zIm(z^2)?

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

6. ## Re: Complex variables : f(z) = zIm(z^2)?

Originally Posted by mous99
why {Im}(z^2)=2xy?
If indeed you do not know that, then you have no business trying this question.

If $z=x+yi$ then $\text{Re}(z)=x~\&~\text{Im}(z)=y$.

EXAMPLE:
$\text{Re}(xy+x^2i)=xy~\&~\text{Im}(xy+x^2i)=x^2$.

7. ## Re: Complex variables : f(z) = zIm(z^2)?

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

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

8. ## Re: Complex variables : f(z) = zIm(z^2)?

Originally Posted by mous99
Let z = x+ yi
then how to continue for Im (z2) = 2xy???

$z\text{Im}(z^2)=(2x^2y)+i(2xy^2)$

9. ## Re: Complex variables : f(z) = zIm(z^2)?

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?

10. ## Re: Complex variables : f(z) = zIm(z^2)?

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

11. ## Re: Complex variables : f(z) = zIm(z^2)?

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?