1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

186 CHAPTER 5 • ELEMENTARY FUNCTIONS

The complex quantity Z defined by

Z = R + i ( wL - w~)

is called the complex impedance. Substituting this last expression into Equation
(5- 44 ) gives

E* =Zl*,

which is the complex extension of Ohm's law.

-------... EXERCISES FOR SECTION 5.4

1. Establish t hat f. cos z = - sin z for all z.

2. Demonstrate that, for all z, sin^2 z + cos^2 z = l, as follows.

(a) Define the function g (z) = sin^2 z + cos^2 z. Explain why g is entire.

(b) Show that g is constant. Hint: Look at g' (z).

(c) Use part (b) to establish that, for all z, sin^2 z + cos^2 z = 1.

3. Show that Equation (5-38) simplifies t o Equation (5-39). Hint: Use t he facts that
   cosh^2 y - sinh^2 y = 1 and sinh2y = 2coshysinhy.


4. Explain why the diagrams in Figures 5.8 and 5.9 came out the way they did.


5. Show that, for all z,

(a) sin (7r - z) =sin z.

(b) sin(~ - z) = cosz.

(c) sinh(z+i7r) = - sinhz.

( d) tanh (z + i?r) = tanh z.

(e) sin (iz) = isinhz.

(f) cosh (iz) = cosz.

6. Express the following quantities in u +iv form.

(a) cos (1 + i).

(b) sin c·~·;).

(c) sin2i.

(d) cos(- 2 +i).

(e) tan ( "!^2 ').

(f) tan(~).

(g) sinh (1 + i7r).