1 1)Given the
function: u(x,y) = x

^{3}– 3xy^{2}
¶
u/ ¶
x =
3 x

^{2}– 3y^{2}
¶
u/ ¶
y =
- 6xy

And
also: ¶

^{2}u/ ¶ x^{2}= 6x
And: ¶

^{2}u/ ¶ y^{2}= - 6x
Since: ¶

^{2}u/ ¶ x^{2}+ ¶^{2}u/ ¶ y^{2}= 6x + (-6x) = 0
It’s clear u(x,y) and its derivatives are everywhere
continuous on the whole complex plane.

2) Given the function: u(x.y) = exp(-x) [x sin y – y cos y]

¶
u/ ¶
x =
exp(-x) (sin y) + (-exp(-x)) [x sin
y – y cos y]

= exp(-x) (sin y) - x
exp(-x) sin y + y (exp(-x) ) cos y

¶

y (exp(-x) ) cos y]

^{2}u/ ¶ x^{2}= ¶ / ¶ x [exp(-x) (sin y) - x exp(-x) sin y +y (exp(-x) ) cos y]

= -2 exp(-x) sin y +
x exp(-x) sin y - y (exp(-x) ) cos y

AAlso: ¶u/ ¶ y = exp (-x) (x cos y + y sin y – cos y) =

x exp (-x) cos y + y (exp(-x) sin y - exp(-x)
cos y

¶

^{2}u/ ¶ y^{2}= ¶ / ¶ y [x exp (-x) cos y + y (exp(-x) sin y - exp(-x) cos y]
= - x exp(-x) sin
y + 2 exp(-x) sin y + y (exp(-x) ) cos y

So that: ¶

^{2}u/ ¶ x^{2}+ ¶^{2}u/ ¶ y^{2}= 0
And the function u(x,y) is harmonic

(To be continued.)

## No comments:

Post a Comment