UNIT 4

Complex variable

1. Show that the function are analytic i) sinhz., sinz
2. Find the analytic function f(z)=u+iv   u=2+y2)-ytan-1(y/x)+sinx copshy.
3. If f(z)=u+iv is analytic and u-v=ex(cosy-siny)
4. If v =3x2y+6xy-y3 show that v is harmonic and find the corresponding analytic function
5. Find the orthogo0nal tgrajectory orf the family of curves x3y-xy3
6. Evaluate dz wher5e c is the circle z=4
7. Evaluate dz where c is |z-1|=1
8. Expand f(z)= using Laurent all possible expansion
9. Using Cauchy residue theorem evaluate dz wher5e c is the circle| z-2|=4
10. Evaluate