(a) Show that e z is entire. What about e 1/z ? e z ? e x...
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(a) Show that ez is entire. What about e1/z? ez̅? ex(cos ky + i sin ky)? (Use the Cauchy–Riemann equations.)
(b) Find all z such that
(i) ez is real.
(ii) |e-z| < 1.
(iii) ez̅ = e̅z̅.
(c) It is interesting that f(z) = ez is uniquely determined by the two properties f(x + i0) = ex and f'(z) = f(z), where f is assumed to be entire. Prove this using the Cauchy–Riemann equations.
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