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1. (a) Assume that f(z) is an everywhere holomorphic function which is periodic with period , so that f(z + ) = f(z). Show that if f is bounded on the strip {a+bi|/2b/2}thenf isaconstant.

(b) Show that the sum f(z) := 1 converges for all z, and show that (z+n)2

it is periodic with period . (c) You may assume f(z) defined above is holomorphic. Show that f is

bounded for z satisfying |Im(z)| 1. (d) We will see later (when we study Laurent series) that f(z) 1 is

sin(z)2 everywhere holomorphic (or rather, can be extended to an everywhere holomor-

phic function). Taking this on faith, show that f(z) = constant c). (Hint: it will be helpful to see that 1

|Im(z)| 1.)

1. (a) Assume that f(z) is an everywhere holomorphic function which is periodic with period 1, so that f(z + 7) = f(z). Show that if f is bounded on the strip {a + bi | -7/2 1. (d) We will see later (when we study Laurent series) that f(z) sinta) 2 is everywhere holomorphic (or rather, can be extended to an everywhere holomor- phic function). Taking this on faith, show that f(x) = sini,)2 + c (for some constant c). (Hint: it will be helpful to see that sin tx)2 is also bounded for Im(2)| > 1.) 1. (a) Assume that f(z) is an everywhere holomorphic function which is periodic with period 1, so that f(z + 7) = f(z). Show that if f is bounded on the strip {a + bi | -7/2 1. (d) We will see later (when we study Laurent series) that f(z) sinta) 2 is everywhere holomorphic (or rather, can be extended to an everywhere holomor- phic function). Taking this on faith, show that f(x) = sini,)2 + c (for some constant c). (Hint: it will be helpful to see that sin tx)2 is also bounded for Im(2)| > 1.)

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