Complex Analysis - Maths KU

212 Chapter 4 Elementary Functions
– π
2
–2 –1
y
π
2
–
(a) The square S0 π
2
2
1
–1
–2
v
1 2
(b) The image E of S 0
π
2
Figure 4.13 The mapping w = sin z
S 0
E 0
Figure 4.14 The cut elliptical regions E0
and E1
5
0
–5
–2
–2
0
0
2
2
E 1
Figure 4.15 A Riemann surface for
w = sin z
S 1
u
x
(iii) Since the complex sine function is periodic, the mapping w =sinz
isnot one-to-one on the complex plane. Constructing a Riemann
surface, for thisfunction, asdescribed in the Remarksat the end
of Section 2.4 and Section 4.1, will help usvisualize the complex
mapping w =sinz. In order to construct a Riemann surface consider
the mapping on the square S0 defined by −π/2 ≤ x ≤ π/2,
−π/2 ≤ y ≤ π/2. From Example 3, we find that the square S0
shown in color in Figure 4.13(a) maps onto the elliptical region E
shown in gray in Figure 4.13(b). Similarly, the adjacent square S1
defined by π/2 ≤ x ≤ 3π/2, −π/2 ≤ y ≤ π/2, also maps onto E. A
Riemann surface is constructed by starting with two copies of E, E0
and E1, representing the images of S0 and S1, respectively. We then
cut E0 and E1 open along the line segmentsin the real axisfrom
1 to cosh (π/2) and from −1 to− cosh (π/2). Asshown in Figure
4.14, the segment shown in color in the boundary of S0 ismapped
onto the segment shown in black in the boundary of E0, while the
dashed segment shown in color in the boundary of S0 ismapped
onto the dashed segment shown in black in the boundary of E0. In
a similar manner, the segments shown in color in the boundary of
S1 are mapped onto the segments shown in black in the boundary of
E1. Part of the Riemann surface consists of the two elliptical regions
E0 and E1 with the segments shown in black glued together and the
dashed segments glued together. To complete the Riemann surface,
we take for every integer n an elliptical region En representing the
image of the square Sn defined by (2n − 1)π/2 ≤ x ≤ (2n +1)π/2,
−π/2 ≤ y ≤ π/2. Each region En iscut open, asE0 and E1 were,
and En isglued to En+1 along their boundariesin a manner analogousto
that used for E0 and E1. ThisReimann surface, placed in
xyz-space, is illustrated in Figure 4.15.
EXERCISES 4.3 Answers to selected odd-numbered problems begin on page ANS-14.
4.3.1 Complex Trigonometric Functions
In Problems 1–8, express the value of the given trigonometric function in the form
a + ib.
1. sin (4i) 2. cos (−3i)
�
π
�
3. cos (2 − 4i) 4. sin + i
4
5. tan (2i) 6. cot (π +2i)
�
π
�
7. sec − i
8. csc (1 + i)
2
In Problems 9–12, find all complex values z satisfying the given equation.
9. sin z = i 10. cos z =4
11. sin z = cos z 12. cos z = i sin z