Please have detail answers step by step.Thank you! -...

1. Note that you want to find z42 = −1 =⇒ z = (−1)1/42. Now, −1 = cos
(
(2n+ 1)π
)
+ i · sin
(

(2n+ 1)π
)
. Now by
De Moivre’s theorem we have
(−1)1/42 = cos
(
(2n+ 1)π
42
)
+ i · sin
(
(2n+ 1)π
42
)
2. Note that u(x, y) = e−y · (x · cos(x)− y sin(x)) and v(x, y) = e−y · (y cos(x) + x · sin(x)).
ux(x, y) = e
−y ·
(
−x · sin(x) + cos(x)− y · cos(x)
)
vy(x, y) = e
−y · cos(x)−
(
y · cos(x) + x · sin(x)
)
· e−y = e−y ·
(
−x · sin(x) + cos(x)− y ·...