Determine the disc of convergence of the power series n=1 (3x-2). 1 (b) Let f(z)...

 

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Determine the disc of convergence of the power series Στις n=1 (3x-2)". 1 (b) Let f(z) = 2+1 Use Taylor's Theorem to determine the Taylor series about 2i for f, up to the term in (z - 2i)2, and find an open disc centred at 2i on which the series converges. (c) Find the Taylor series about 0 for each of the following functions. (i) f(z) = (1+2)1/3 sinh 3z (up to the term in z³) (ii) f(z) =sin(Log(1 + z)) (up to the term in z¹) (d) Locate the zeros of the function f(z) = (³ 1)(z - 1)³, and find their orders. (e) Prove that if f is analytic on the region R = {z: Rez > 0} and f(et) log 2+ it, for π/2<t<π/2, then f(z) Log(22), for z E R. Determine the disc of convergence of the power series Στις n=1 (3x-2)". 1 (b) Let f(z) = 2+1 Use Taylor's Theorem to determine the Taylor series about 2i for f, up to the term in (z - 2i)2, and find an open disc centred at 2i on which the series converges. (c) Find the Taylor series about 0 for each of the following functions. (i) f(z) = (1+2)1/3 sinh 3z (up to the term in z³) (ii) f(z) =sin(Log(1 + z)) (up to the term in z¹) (d) Locate the zeros of the function f(z) = (³ 1)(z - 1)³, and find their orders. (e) Prove that if f is analytic on the region R = {z: Rez > 0} and f(et) log 2+ it, for π/2<t<π/2, then f(z) Log(22), for z E R.