Question

1. (a) Find real numbers a and b such that a + bi = p.v.(-86]1/3. [4] (b) Consider the following statement. "Log(-x)2 = Logza because (-2)2 = 22. Therefore, 2 Log(-2) = 2 Logz." Explain whether or not the statement is true. [4] (c) Consider the following statement. "The rational function (2), where p and q are co-prime non-constant polynomials, is holomorphic everywhere except at the set of zeros of q." Does this explain if any primitive of me is also holomorphic ev- erywhere except at the zeros of q? Explain why. [4]

solve it ,i need urgent, no need to write neat and clean.. thanks!

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Answer 1

0 Answed- Given data- Now to find real numbers "a" and "6" TO prove that atbis [87]' Where a=is Real parts co-effitient imaginady Parts (0-effitient. The deal part of the co-effitient is Q=-2PV (TT)'3 b=0 b = → J Similodly the imag; nady co-effitients are a=0 bi = -2PV T 13 b = -2PV JT la - The imaginozy number say P.VX+X2i SD a = -24, (n)%3 b = -282 b) The statement is L09(-2) ?- 109Z2 because (-2) ?= 72 :. 2009(Z) = 2logz. Now To apply logoth:m:c Popesty Jogz? = 210912)

Similodly 10972 - 2109 1Z 109 (-2)2 - 2/09 (-21 We know that IZ) |-z1 2109 (-2) = 2109(Z) is not Toue. It is not Cordect statement Question, The rational function PLZ) {(z) Where "P" and "a" are co-prime non- constant Poly nomals. → The Primitive of a function (U= f(W+C where- (= Any Deal constant. If "c" is deal, SO for the holomorphic, will get attached with u(x,y), Any Primitive Of PCBD is also 200) holomosphic. f(x+iy) = u (y) tiv (Sy)

3 Where U(Xy) and Vaxy) ade real valued functions. For example, A(Z) = Uiv The complexed valued function is f(Z) = Z2. Z=xtiy, → SO We can see that f(Z) = (xtiy) 2 - 2 + 2xy; Hy;) ? f(x) =x2y2 + 2Xy; We denote its real and imaginary Parts uGxy) = x²-42 -deal part -Smagi nody port, توی)۷ no