Complex Variables - completing the square & graphing circles?

Question

Sketch the families of level curves of the component functions u and v when

f(z) = (z-1) / (z+1)

Now I dont need anyone to sketch a graph because I see how that would be tricky on the keyboard - but I know that

z=x+iy and if you put those values in, I can see perhaps that if I complete the square I can represent the Real and Im parts of the level curves as circles - I am only making an educated guess, I am really stuck, I've tried a bunch of working out with completing the square but can't get it

I know the answer is 2 orthogonal familes of circles with different centres - thats my other question, how would I find the centers??

Thanks :)

Answer 1

z=x+iy so( x-1 +iy)/(x+1+iy)= (x-1)+iy)((x+1)-iy)/(x+1)^2+y^2=

(x^2-1-iy(x-1)+iy(x+1) +y^2)/((x+1)^2+y^2) =(x^2+y^2-1+2iy)/(x+1)^2 +y^2 =u(x,y)+i v(x,y)

u(x,y)= (x^2+y^2-1)/((x+1)^2 +y)^2) v(x,y) = 2y/ ((x+1)^2+y^2)If you put u=k and v=k' You'll get circles.Don't forget that in the general equation of a circunsference,to find out the center. the coefficients of x^2 and y^2 must be 1

Answer 2

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