au, av, find v, the conjugate harmonic functions of u. (a) u(x, y) = x - y (c) u(x,y)= sinh 2 cos y (b) u(x,y) = x-3x (d) u(x, y) = r+y2 Solution (a) Use Cauchy-Riemanns equations page 83. u(x, y) = x - y du = dx-dy dv = dx + dy v(x, y) = x+y (b) Use Cauchy-Riemanns equations page 83. (c) u(x, y) = x-3x du=(3x-3y) dx - 6xy dy dv = 6xy dx+(3x - 3y) dy v(x, y) = 3xy- y Use Cauchy-Riemanns equations page 83. (d) u (2,y)=sinh cosy du = cosh .r cos y da sinh r sin y d dv == sinh x sin y dx + cosh x cos y dy v(x, y) = cosh x sin y Use Cauchy-Riemanns equations page 83. u(x, y) = x+ y du = (x+y) 22-y2 -2xy=dx+(2+2) d dy -2xy=dx + x + y (2) dy = (x+12)2 (x+y2)2