Question: (a) Show that e z is entire. What about e 1/z ? e z ? e x (cos ky + i sin ky)? (Use the

(a) Show that e^z is entire. What about e^1/z? e^z̅? e^x(cos ky + i sin ky)? (Use the Cauchy–Riemann equations.)
(b) Find all z such that
(i) e^z is real.
(ii) |e^-z| < 1.
(iii) e^z̅= e̅^z̅.

(c) It is interesting that f(z) = e^z is uniquely determined by the two properties f(x + i0) = e^x and f'(z) = f(z), where f is assumed to be entire. Prove this using the Cauchy–Riemann equations.

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