Case III. Complex conjugate roots are of minor practical importance, and we discuss the derivation of real solutions from complex ones just in terms of a typical example
Step 3. Solution of the Entire Problem
17.4 Conformal Mapping by Other Functions
We shall now cover mappings by trigonometric and hyperbolic analytic functions. So far we have covered the mappings by and (Sec. 17.1) as well as linear fractional transformations (Secs. 17.2 and 17.3).
Sine Function.Figure 391 shows the mapping by
(1) w⫽u⫹iv⫽sin z⫽sin x cosh y⫹i cos x sinh y (Sec. 13.6).
ez zn
Hence (2)
Since is periodic with period the mapping is certainly not one-to-one if we consider it in the full z-plane. We restrict zto the vertical strip in Fig. 391. Since at the mapping is not conformal at these two critical points. We claim that the rectangular net of straight lines and
in Fig. 391 is mapped onto a net in the w-plane consisting of hyperbolas (the images of the vertical lines and ellipses (the images of the horizontal lines
intersecting the hyperbolas at right angles (conformality!). Corresponding calculations are
simple. From (2) and the relations and we
obtain
(Hyperbolas)
(Ellipses).
Exceptions are the vertical lines which are “folded” onto and respectively.
Figure 392 illustrates this further. The upper and lower sides of the rectangle are mapped onto semi-ellipses and the vertical sides onto and
respectively. An application to a potential problem will be given in Prob. 3 of Sec. 18.2.
(v⫽0),
1⬉u⬉cosh 1
⫺cosh 1⬉u⬉ ⫺1 u⭌1 (v⫽0),
u⬉ ⫺1 x⫽ ⫺12px⫽12p,
u2
cosh2 y ⫹ v2
sinh2 y ⫽sin2 x⫹cos2 x⫽1 u2
sin2 x ⫺ v2
cos2 x ⫽cosh2 y⫺sinh2 y⫽1
cosh2 y⫺sinh2 y⫽1 sin2 x⫹cos2 x⫽1
y⫽const) x⫽const)
y⫽const x⫽const
z⫽ ⫾12p, fr(z)⫽cos z⫽0
S:⫺12p⬉x⬉12p 2p,
sin z
v⫽cos x sinh y.
u⫽sin x cosh y,
v
– u
1
–1
–2 –1 1 2
(z-plane) (w-plane)
y
π x 2
π 2
1 –1
Fig. 391. Mapping w⫽u⫹iv⫽sin z
y
x
v
u A
B C
D
E F
1
–1
C*
E*
B*
D* A* F*
1 π –1
2 π
–2
Fig. 392. Mapping by w⫽sin z
Cosine Function. The mapping could be discussed independently, but since (3)
we see at once that this is the same mapping as preceded by a translation to the right through units.
Hyperbolic Sine. Since (4)
the mapping is a counterclockwise rotation through (i.e., followed by the sine mapping followed by a clockwise -rotation
Hyperbolic Cosine. This function (5)
defines a mapping that is a rotation followed by the mapping Figure 393 shows the mapping of a semi-infinite strip onto a half-plane by
Since the point is mapped onto For real is
real and increases with increasing xin a monotone fashion, starting from 1. Hence the positive x-axis is mapped onto the portion of the u-axis.
For pure imaginary we have Hence the left boundary of the strip is mapped onto the segment of the u-axis, the point corresponding to
On the upper boundary of the strip, and since and it
follows that this part of the boundary is mapped onto the portion of the u-axis.
Hence the boundary of the strip is mapped onto the u-axis. It is not difficult to see that the interior of the strip is mapped onto the upper half of the w-plane, and the mapping is one-to-one.
This mapping in Fig. 393 has applications in potential theory, as we shall see in Prob. 12 of Sec. 18.3.
u⬉ ⫺1
cos p⫽ ⫺1, sin p⫽0
y⫽p,
w⫽cosh ip⫽cos p⫽ ⫺1.
z⫽pi 1⭌u⭌ ⫺1
cosh iy⫽cos y.
z⫽iy
u⭌1
z⫽x⭌0, cosh z w⫽1.
z⫽0 cosh 0⫽1,
w⫽cosh z.
w⫽cos Z.
Z⫽iz
w⫽cosh z⫽cos (iz)
w⫽ ⫺iZ*.
90°
Z*⫽sin Z,
90°),
1 2p Z⫽iz
w⫽sinh z⫽ ⫺i sin (iz),
1 2p
sin z w⫽cos z⫽sin (z⫹12p),
w⫽cos z
0 v
u x
y
B* A*
1 A –1
π B
Fig. 393. Mapping by w⫽cosh z
Tangent Function. Figure 394 shows the mapping of a vertical infinite strip onto the unit circle by accomplished in three steps as suggested by the representation (Sec. 13.6)
w⫽tan z⫽ sin z
cos z ⫽ (eiz⫺eⴚiz)>i
eiz⫹eⴚiz ⫽ (e2iz⫺1)>i e2iz⫹1 . w⫽tan z,
Hence if we set and use we have (6)
We now see that is a linear fractional transformation preceded by an exponential mapping (see Sec. 17.1) and followed by a clockwise rotation through an angle
The strip is and we show that it is mapped onto the unit disk in the w-plane. Since we see from (10) in Sec. 13.5 that
Hence the vertical lines are mapped onto the rays respectively. Hence Sis mapped onto the right Z-half-plane. Also if and if Hence the upper half of Sis mapped inside the unit circle and the lower half of Soutside as shown in Fig. 394.
Now comes the linear fractional transformation in (6), which we denote by (7)
For real Zthis is real. Hence the real Z-axis is mapped onto the real W-axis. Furthermore, the imaginary Z-axis is mapped onto the unit circle because for pure imaginary
we get from (7)
The right Z-half-plane is mapped inside this unit circle not outside, because has its image inside that circle. Finally, the unit circle is mapped onto the imaginary W-axis, because this circle is so that (7) gives a pure imaginary expression, namely,
From the W-plane we get to the w-plane simply by a clockwise rotation through see (6).
Together we have shown that maps onto the unit
disk with the four quarters of Smapped as indicated in Fig. 394. This mapping is conformal and one-to-one.
ƒwƒ ⬍1,
S:⫺p>4⬍Rez⬍p>4 w⫽tan z
p>2;
g(ei)⫽ei⫺1
ei⫹1 ⫽ ei>2⫺eⴚi>2
ei>2⫹eⴚi>2 ⫽ i sin (>2) cos (>2) . Z⫽ei,
ƒZƒ ⫽1 g(1)⫽0
Z⫽1
ƒWƒ ⫽1, ƒWƒ ⫽ ƒg(iY)ƒ ⫽ `iY⫺1
iY⫹1` ⫽1.
Z⫽iY
ƒWƒ ⫽1 W⫽g(Z)⫽ Z⫺1
Z⫹1.
g(Z):
ƒZƒ ⫽1, ƒZƒ ⫽1
y⬍0.
ƒZƒ ⬎1 y⬎0
ƒZƒ ⫽eⴚ2y⬍1 Arg Z⫽ ⫺p>2, 0, p>2,
x⫽ ⫺p>4, 0, p>4 Arg Z⫽2x.
ƒZƒ ⫽eⴚ2y, Z⫽e2iz⫽eⴚ2y⫹2ix,
S:⫺14p⬍x⬍14p,
1 2p(90°).
w⫽tan z
Z⫽e2iz. W⫽ Z⫺1
Z⫹1, w⫽tan z⫽ ⫺iW,
1>i⫽ ⫺i, Z⫽e2iz
y
x
v
u
(z-plane) (Z-plane) (W-plane) (w-plane)
Fig. 394. Mapping by w⫽tan z