1550251515-Classical_Complex_Analysis__Gonzalez_

284 Chapter^5

4. Find the image of the square

Q = {z: z = x + iy, 0 S x S %7r, 0 Sy S^1 / 2 7r}

under w = ez.

5. Find the image Q' of the square

Q = {z: z = x + iy, Ix - al S €, IYI S €}

where a is real and 0 < € < 7r, under w :== ez. Evaluate the limit of

the ratio area Q' /area Q as € ---t 0.

6. Show that w = e^2 u maps Q = {z: 0 < x < 1, 0 Sy <-1} into the ring

G = { w: 1 < lwl < e^2 1l"} and that the mapping is one-to-one.

7. Find the image of the line L = {z: z = (1 + ib)t, b-=/- 0 real, -oo <

t < +oo} under w = ez.

8. Show that ez = e".


9. Prove that ez = limn__. 00 (1 + z/n)n.

10. Evaluate the roots of the equation ez = z that lie in lzl < 2.

11. Show: ·
    (a) sinz = sinz

(c) tanz = tanz

12. Evaluate:
    (a) sin ( i + i)

(

(c) cos 7r '4 + i7r)

2

13. Prove that

(b) cos z = cos z

(b) tanh i

(d) sinh4i

I cos zl^2 = cos^2 x + sinh^2 y

= cosh^2 y - sin^2 x

14. Prove that:
    (a) I sinh y I S I sin z I S cosh y
    (b) lsinhyl S icoszl S coshy
    and deduce that I sin z I and I cos z I are not bounded in the complex
    plane.


15. Show that:

(a) I sin(x + iy)I = I sinx +sin iyl

(b) lsinh(x+iy)I = lsinhx+sinhiyl

16. If z = x + iy, express I sinhzl^2 and I coshzl^2 as functions of x and y.

17. Show that

tan^2 x + tanh^2 y

I tan(x + iy)l^2 = -------=-

1 + tan^2 x tanh^2 y

assuming that x -=f. %(2n + l)7r.