Conformality implies differentiabilityPreben Alsholm, October 1 2007The proof is as in Zeev Nehari, Conformal Mapping, Dover 1976 (1952), pp. 150-152.Let the Jacobian matrix beJ:=Matrix([[ux,uy],[vx,vy]]);z1 and z2 are the tangent vectors to two curves through (x0, y0) in the xy-planez1:=<x1,y1>;z2:=<x2,y2>;w1 and w2 are the corresponding tangent vectors in the uv-planew1:=J.z1;w2:=J.z2;q is just an auxiliary function:q:=vk->vk[2]/vk[1];We shall use conformality at (x0, y0), so that the angle between z1 and z2 is the same as the angle between w1 and w2.Except for the case when A-B and a-b are right angles we haveexpand(tan(A-B)=tan(a-b));We have subs(tan(A)=q(w2),tan(B)=q(w1),tan(a)=q(z2),tan(b)=q(z1),%);In order to get our result we need only consider the case (x1, y1) = (1, 0) and (x2, y2) = (1, y):subs(y1=0,x1=1, x2=1, y2=y,%);normal((lhs-rhs)(%));collect(numer(%),y,factor);This must be zero for all y, thus the coefficients of the powers of y are zerocoeffs(%,y);solve({%},{uy,vy});These are the Cauchy-Riemann.equations.