Page 426 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 417

2 dv 2 1
¼ .
du v 2
(b) The slope of the tangent to the curve v ¼ 4ð1 þ uÞ at ð3; 4Þ is m 1 ¼ ¼
ð3;4Þ ð3;4Þ

2 dv
¼ u ¼ 3.
du
The slope of the tangent to the curve u ¼ 2v þ 1at ð3; 4Þ is m 2 ¼
ð3;4Þ
Then the angle between the two curves at A is given by
0
1
m 2 m 1 3 2
1
tan ¼ ¼ ¼ 1; and ¼ =4
1 þ m 1 m 2
2
1 þð3Þð Þ
Similarly, we can show that the angle between A C and B C is =4, while the angle between A B 0
0
0
0
0
0
and B C is =2. Therefore, the angles of the curvilinear triangle are equal to the corresponding ones of
0
0
the given triangle. In general, if w ¼ f ðzÞ is a transformation where f ðzÞ is analytic, the angle between
two curves in the z plane intersecting at z ¼ z 0 has the same magnitude and sense (orientation) as the
angle between the images of the two curves, so long as f ðz 0 Þ 6¼ 0. This property is called the conformal
0
property of analytic functions, and for this reason, the transformation w ¼ f ðzÞ is often called a con-
formal transformation or conformal mapping function.
p ffiffiffi
z define a transformation from the z plane to the w plane. A point moves counter-
16.37. Let w ¼
clockwise along the circle jzj¼ 1. Show that when it has returned to its starting position for the
first time, its image point has not yet returned, but that when it has returned for the second time,
its image point returns for the first time.
i
ffiffiffi
Let z ¼ e . Then w ¼ p z ¼ e i =2 . Let ¼ 0correspond to the starting position. Then z ¼ 1 and
w ¼ 1 [corresponding to A and P in Figures 16-13(a) and (b)].

Fig. 16-13

i
When one complete revolution in the z plane has been made, ¼ 2 ; z ¼ 1, but w ¼ e i =2 ¼ e ¼ 1, so
the image point has not yet returned to its starting position.
However, after two complete revolutions in the z plane have been made, ¼ 4 ; z ¼ 1 and
w ¼ e i =2 ¼ e 2 i ¼ 1, so the image point has returned for the first time.
It follows from the above that w is not a single-valued function of z but is a double-valued function of z;
i.e., given z,there are two values of w.Ifwewish to consider it a single-valued function, we must restrict .
We can, for example, choose 0 @ < 2 ,although other possibilities exist. This represents one branch of
ffiffiffi
z.Incontinuing beyond this interval we are on the second branch, e.g.,
p
the double-valued function w ¼
2 @ < 4 . The point z ¼ 0 about which the rotation is taking place is called a branch point. Equiva-
p ffiffiffi
z will be single-valued by agreeing not to cross the line Ox,called a branch
lently, we can insure that f ðzÞ¼
line.
ð p 1
1 x
16.38. Show that dx ¼ ; 0 < p < 1.
0 1 þ x sin p