Question

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1. Find the region of analyticity of functions F (z) = 2?", F. (z) = z22 and F3 (2) = sin(V2) and in each case compute the derivative.

Answer 1

on since giver function Fi(z) = 222, Falzzz2Z & F312) = Sin(JZ) are entire function. Means It is analytic in whole complex plane c. i.l. Here F(z) = 22² is exponential function which is analytic in whole complex plane C F2 (2) = z2z is a polynomial function which is also analytic in whole complex plane and F3 (2)=siniz) is trigonometrical function, which is also analytic in c. - It means in whole complex plane derivative ise.f'czy exist for each function. Hence the Region of analyticity of each given function is c. Now Derivatives is given by Fi(Z) = 2 2² then f '(z) = 2²² 22 F2 (2) = 222 then F (2) take log bathil an

Let w=222 taking log both side loge = 22 loge Now differentiate wort z we get ücret = 22.4 +log2.2 {ollegz) = { du = w [22 - 2 +2 logz] Ed 12 (22) = 2 = 222 [2+2 10gz], By putting wazaz f3 (2)=sin( ) then fac2) = cos(Iz). Izza Coscuz). Z Ž = zvz cos (JZ) Answer 12