Question

2. Every 2 x 2 real matrix M = (C) determines a complex function M(x + iy) = um(X,Y) + ivm (2+iy), where real-valued functions up and um are determined by the following equation. um(2,y) UM(x,y) (a) Show that there are constants w; and w2 EC such that fm(2) = w12+ w22. What are these constants in terms of a, b,c,d? [8] (b) Determine an equivalent condition on M such that fm is an entire function. U

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

......b0nGrr.......

Answer 1

(2) sol:- from the given data we have a 6 Every 2x2 real Matrix M = :) determines a с d complex funetion FM (*+) - UM (214) tum latiy), → where real-Valued functions Um and Vm are determined by the following UM = axtby, Vm = cxtdy Um (x,y) @ in (2614) equation '딧 consecteurs) = () (3) y a) Show that there are consts w, and W2 € C such that fm (z) =W, Z TW z ž 'What are these constants in terms of a,b,cidi > using eq ③ in eq ③ we get Catic) FM (2) = (axtby) + y tenga Cx+dy) fm (2) = X + (btidly fm (2) Catic) (2+²) +(6+9d) (2-2) 2 atic -i (btid) atic tirbtid) , wherez=xtiy 2 21 z t 2 - 2 2 WiZt W2 ::FM (2) (a-d) +(6+) (a+d) -(6-0) 1 Wza E C where W,= ay

Condition on M such that 6 Determining an equivalent fm is entere function : an suppose fm is an entene function. Then the Cauchy- Riemann (C-R) equation must satisfy dum JUM = Jam dy 2 dum dac da dy b=-c a=+d b ME ) is (- if fm is an eukene fonction b a :. The requered condition on Miff is an entere fimation care a=d, b=-c ... If you Impress with Answer, please give me like