By using the Cauchy-Riemann equations (or otherwise) determine whether the following function of the complex variable zis analytic: f(z) =  z^2.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should remember that a complex function is differentiable if the real and imaginary parts satisfy the Cauchy-Riemann equations.

Considering the function f(z) = z^2 = (x + iy)^2 = x^2 + 2ixy + (iy)^2, you should verify if it satisfies Cauchy-Riemann equations.

`(del(x^2 - y^2))/(del x) = (del(2xy))/(del y) and (del(x^2 - y^2))/(del y) = -(del(2xy))/(del x) `

Substituting -1 for `i^2`  yields:

`f(z) = x^2 - y^2 + i*(2xy)`

`(del(x^2 - y^2))/(del x) = 2x`

`(del(2xy))/(del y) = 2x`

Notice that `(del(x^2 - y^2))/(del x) = (del(2xy))/(del y) = 2x` .

`(del(x^2 - y^2))/(del y) = -2y`

`(del(2xy))/(del x) = 2y`

Notice that `(del(x^2 - y^2))/(del y) = -(del(2xy))/(del x) = 2y` .

Hence, the function  `f(z) = z^2`  satisfies the Cauchy-Riemann equations, thus, the function is analytic complex.

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