1. Find the derivatives of the following functions.
sinz sinhz
(a) tanz = – (b) tanhz = -h- (c) secz = l/cosz
cosz cos z
2. Show that u = sin x sinh y and v = cos x cosh y satisfy the Cauchy-
Riemann equations. Do you recognize the analytic function I =
u + iv? (Determine its complex form.)
3. Show that if I and J are both analytic on a domain D, then I is
constant.
4. Show that if I is analytic on a domain D, and if III is constant,
then I is constant. Hint. Write J= 1112 11.
5. If I = u + iv is analytic, then IV”u I = IV”v I = III
6. If 1= u + iv is analytic on D, then V”v is obtained by rotating V”u
by 90°. In particular, V”u and V”v are orthogonal.
7. Sketch the vector fields V”u and V”v for the following functions I =
u + iv. (a) iz, (b) z2, (c) liz.
8. Derive the polar form of the Cauchy-Riemann equations for u and v:
au 1 av au av
Or ;: 00′ 00 = -rOr .
Check that for any integer m, the functions u(re i8 ) = rm cos(mO)
and v(re i8 ) = rm sin(mO) satisfy the Cauchy-Riemann equations.
