Show 2 ways that ln(x^(2) + y^(2)) is harmonic?

Question

Show in 2 ways that the function ln(x^2+y^2) is harmonic in every domain that does not contain the origin.

My problem with this quesiton is firstly, I don't know two ways, because normally I would do Cauchy-Riemann equations but there is no real/imaginary numbers so I don't know what is u(x,y) and what is v(x,y)

As a function I know that if I differentiate in terms of x I get
2x/(x^2+y^2) which is a continuous function (where x and y don't equal 0)

Same goes for differentiating in terms of y I get 2y/(x^2+y^2)

This is a question from the next book with no answer and I just don't see where it's going, it's in the start of a chapter regarding logs and complex variables, however I dont see 2 methods of proving it is harmonic.

Answer 1

fxx+fyy=0 is the condition for beeing a harmonic function (Laplace's equation)
fx=1/(x^2+y^2) *2x
fxx= 2/(x^2+y^2) -4x^2/(x^2+y^2)^2
fy= 1/(x^2+y^2)*2y
fyy= 2/(x^2+y^2) -4y^2/(x^+y^2)^2
fxx+fyy=
4/(x^2+y^2) -4(x^2+y^2)/(x^2+y^2)^2=

1/(x^2+y^2)^2 *(4x^2+4y^2 -4x^-4y^2)=0
I don´t know a second way

Answer 2

I'll give you some help with the Cauchy Riemann conditions,
note that (x^2 + y^2) = (x+iy)(x-iy)
Now notice how log ab = log a + log b

Answer 3

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Answer 4

for U to be a harmonic function

show that Uxx +Uyy =0

where Uxx refers to second order partial derivative
of U w.r.t.x

here if U = ln(x^(2) + y^(2))

Uxx = 2(y^2 -x^2 ) / { (x^(2) + y^(2))^2 }

Uyy = 2(x^2 -y^2 ) / { (x^(2) + y^(2))^2 }

adding them you get zero.