Show that u sin rsinhy and cos x cosh y satisfy the Cauchy Riemann equations Do you recognize the analytic function f u iv Determine its complex form Solution We have du sin r sinh cos 2 sinh y r x du y y sin z sinh y sin xz cosh y y 4 ez y e iz y 2 cos a cosh y r Hence u and satisfy Cauchy Riemann equations and now we calculate v y cos rcosh y f 6 i e 2i ex y eiz y e e iz iy eix y elz y e ix y e iz y 4 i el x y e i x y 2 v es 6 2 i cos z