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Which of the following complex functions is/are analytic on the complex plane?

1. f(z) = jRe(z)
2. f(z) = Im(z)
3. f(z) = e^(|z|)
4. f(z) = z² - z

Correct answer: f(z) = z² - z

Solution

The function f(z) = z² - z is a polynomial, and all polynomials are entire functions, meaning they are analytic everywhere in the complex plane. In contrast, the other options involve operations that do not satisfy the Cauchy-Riemann equations or are not defined for all complex numbers.

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