1Evaluate 1 ii 2Find all the roots of the equation sin z c

1.Evaluate (1 + i)^i

2.Find all the roots of the equation sin z = cosh 4

3. ) Find u = Re(tan z) and v =Im(tan z), z = x + yi.

4. Show that if z = x + yi then (a) |sinh y| |sin z| cosh y (b) |sin z| 2 + |cos z| 2 1 and prove that the equality holds iff z is a real number. (Hint: You may show first the identities : |sin z| 2 = sin2 x + sinh2 y and |cos z| 2 = cos2 x + sinh2 y).

Solution

|sin z| 2 = |sin(x + i y)| ^2 = |sin x cos(iy) + cos x sin(iy)| 2 =

|sin x cosh y + i cos x sinh y| ^2 = sin2 x cosh2 y + cos2 x sinh2 y sin2 x cosh2 y + cos2 x cosh2 y =

cosh2 y, since sinh2 y cosh2 y for all y R, and |sin z| cosh y.

Also, |sin z| 2 = sin2 x cosh2 y + sinh2 y cos2 x sin2 x sinh2 y + cos2 x sinh2 y = sinh2 y,

and |sinh y| |sin z|

| cos z| 2 = | cos(x + i y)| 2

= | cos x cos(iy) sin x sin(iy)| 2

= | cos x cosh y isin x sinh y| 2

= cos2 x cosh2 y + sin2 x sinh2 y cos2 x cosh2 y + sin2 x cosh2 y

= cosh2 y, and | cos z| cosh y.

Also, | cos z| 2 = cos2 x cosh2 y + sin2 x sinh2 y cos2 x sinh2 y + sin2 x sinh2 y

= sinh2 y and | cos z| |sinh y|.