Page 29 - Calculus with Complex Numbers

Ffgure 2.8

2.8 A pp I ication 2

The inequality Isin .x I s 1 for real .x fails for complex variables. lf we write

z = .x + iy, then we have

I sin cI2 = Isint.x + iy) 12 = sin2 .x coshz y + cos2 .x sinhz y
= sin2 . x(1 + sinhz y) + (1 - sin2 x) sirthz y = sin2 x + sinhz y .
.
.
So if, for example, z = zr/2 + iE, where E7> 0, then I sin zl2 = 1 + sinhz (F > 1.

2.9 A pp I ication 3
The only zeros of sin z for complex z are the real zeros at z = n:r for integral rl.
This is because if z = .x + iy and sin z = 0 then

0 = I sin zI2 = sin2 .x + sirthz y.

Therefore sin.x = sirth y = 0 which gives .x = n:r y = 0 and hence z = nn'.
Similarly we leave it as an exercise for the reader to show that the only zeros
of cos z for complex z are at z = n:r + zr/2 for integral rl.

2. I 0 Identities for hyperbol i c fun ctions

The fundamental formulae (see Section 2.6) can be used to obtain identities for
hyperbolic functions from analogous identities for trigonometric functions. For
example, the trigonometric identity sin2 x + cos2
.x = 1 gives, on substituting ix
.
for .x,