Question

21) Find the maximum value of the function f (z)=e^3z on the curve C:|z|=2.
Explain the problem with step by step process?

Answer 1

Let$z_j =x_j +i_j,$ where $xj,yj$ are real number=1,2,3.

 (a) We have

$z1 +z2 =(x1 +x2)+i(y1 +y2)=(x2 +x1)+i(y2 +y1)=z2 +z1.$ $z1z2 =(x1x2 −y1y2)+i(x1y2 +x2y1)=(x2x1 −y2y1)+i(y2x1 +y1x2)=z2z1.$

is valid for real numbers and so it holds for the real and imaginary parts of$z_1z_2,z_3.$

Consequently we have

$(z_1 +z_2)+z_3 =z_1 +(z_2 +z_3).$