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.

Printable View

- December 6th 2012, 07:00 PMmous99Complex 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. - December 6th 2012, 07:40 PMchiroRe: 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). - December 19th 2012, 07:15 AMmous99Re: 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? - December 19th 2012, 07:36 AMPlatoRe: Complex variables : f(z) = zIm(z^2)?
- December 19th 2012, 07:59 AMmous99Re: Complex variables : f(z) = zIm(z^2)?
why {Im}(z^2)=2xy?

- December 19th 2012, 08:07 AMPlatoRe: Complex variables : f(z) = zIm(z^2)?
- December 19th 2012, 08:19 AMmous99Re: 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??? - December 19th 2012, 08:36 AMPlatoRe: Complex variables : f(z) = zIm(z^2)?
- December 19th 2012, 08:50 AMmous99Re: 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? - December 19th 2012, 08:54 AMPlatoRe: Complex variables : f(z) = zIm(z^2)?
- December 23rd 2012, 05:26 AMmous99Re: 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?