## Thursday, October 10, 2013

### Math Solutions (Part I)

1     1)Given the function: u(x,y) = x3 – 3xy2

Show the function is harmonic on the entire complex plane.

The Cauchy- Riemann equations are:

u/ x  =   3 x2 – 3y2

u/ y     =  - 6xy

And also:   2 u/ x2  =   6x

And:  2 u/ y2  =   - 6x

Since:    2 u/ x2  +  2 u/ y2    =  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

2 u/ x2    =   / 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/ y2    =  / 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/ x2  +  2 u/ y2    =   0

And the function u(x,y) is harmonic

(To be continued.)