find the analytic function f(z)=u(x,y)+iv(x,y) such that u+v=(x-y)(x^2+4xy+y^2)

Question

amistre64 · Answer

split the fraction maybe?  or decomp it?

anonymous · Answer

what do you mean by split the fraction ??

anonymous · Answer

@amistre64  what do you mean by split the fraction ???

amistre64 · Answer

just a thought

(a+b)/k = a/k + b/k

anonymous · Answer

ok that's partial fraction how am I supposed to use it here ?? am I missing something ?

amistre64 · Answer

partial would have use factor k, i havent seen a way to factor k so i suggested the splits

zepdrix · Answer

Hmm I don't remember complex very well...
Oldrin could probably explain this well.

If f is analytic, then it must satisfy the Cauchy Reimann equations, yes?
Do we want to make use of those here maybe?
$\large\rm u_x=v_y$

$\large\rm u_y=-v_x$

amistre64 · Answer

is u+v division or multiplication?  these old eyes get tired

zepdrix · Answer

on the right side?
multiplication :)

amistre64 · Answer

if multiplication, then the sum of 2 things is prolly better expressed as the sum of other things and so distributing it all out might be beneficial ... but i got no cauchy reimann tricks to play with :)

amistre64 · Answer

u+v=(x-y)(x^2+4xy+y^2)

(u+v)/(x-y) = x^2+4xy+y^2


       x^2 +4xy +y^2
      ------------------
x-y| [x^3] [+3x^2y]
       (x^3    -x^2y)
      -------------
                4x^2y [-3xy^2]
               (4x^2y -4xy^2)
              ---------------
                            xy^2 [-y^3] 
                           (xy^2-y^3)


u+v = x^3 +3x^2y -3xy^2 -y^3

anonymous · Answer

yes the cauchy reimann equation usually problems go like this it gives me u or v and says that it satisfies cauchy reimann equation then using differentiation I get Ux then by integration I get V but never gave me the sum of U+V before

amistre64 · Answer

i just worked it 'backwards' to determine what u+v needs to be in order to be divided by (x-y) and gives us (x^2 + 4xy +y^2)

amistre64 · Answer

can we split that apart into an appropriate u and v that meet the couchy?

amistre64 · Answer

lol, i spose distribution would have sufficed, but when i got an idea in my head its hard to let it go

anonymous · Answer

I found this by searching the internet but can't make sense out of it but it's supposed to help me

amistre64 · Answer

yeah, that wouldnt have helped me either :/

anonymous · Answer

lol just asked my colleagues on our facebook group one of them hit me with a killer comment

zepdrix · Answer

lol

amistre64 · Answer

heheh, they are still on page 14 arent they :)

anonymous · Answer

I think they don't even know that there is a textbook at all