(1) Parametrize the directed parabolic segment, I, y = x that joins 2 + 4i to...

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(1) Parametrize the directed parabolic segment, I, y = x² that joins 2 + 4i to -1- i and evaluate Real(z) dz. (2) Let a > 0 be a real number and consider the rectangle R whose vertices are +a+4i described in ccd. Evaluate by using the Cauchy's Integral Theorem sin(z) fr (3z-51)²(2-3) dz R if (a) a = 1/4. (b) a = 1. n! (3) In the power series E-0 (2)! (2-1)³n+1, what is ao, a₁, a2? (4) Find the radius of convergence R of the power series n=0 5n - 2 (z −2+i)³n+¹. 47n+2 n=0 Define a complex-valued function f on the open disc |z −2+i|<R by 5n - 2 f(2)=> 3n+1 (z-2+i)³+¹, (z-2+i)| < R. 47m+2 n=0 Write down the power series expansion of f'. What is the radius of convergence (1) Parametrize the directed parabolic segment, I, y = x² that joins 2 + 4i to -1- i and evaluate Real(z) dz. (2) Let a > 0 be a real number and consider the rectangle R whose vertices are +a+4i described in ccd. Evaluate by using the Cauchy's Integral Theorem sin(z) fr (3z-51)²(2-3) dz R if (a) a = 1/4. (b) a = 1. n! (3) In the power series E-0 (2)! (2-1)³n+1, what is ao, a₁, a2? (4) Find the radius of convergence R of the power series n=0 5n - 2 (z −2+i)³n+¹. 47n+2 n=0 Define a complex-valued function f on the open disc |z −2+i|<R by 5n - 2 f(2)=> 3n+1 (z-2+i)³+¹, (z-2+i)| < R. 47m+2 n=0 Write down the power series expansion of f'. What is the radius of convergence