1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

10.l • BASIC PROPERTIES OF CONFORMAL MAPPINGS 401

the familiar expression

g '( wo ) = Li m g(w)-g(wo)

w-+wo W -wo

Li z-zo 1 1

= z-+O:.. f (z) -f (zo) = f I (zo) = f^1 (g (wo)) •

-------~EXERCISES FOR SECTION 10.1

1. State where the following mappings are conformal.

(a) w = expz.

(b) w = sinz.

(c) w = z^2 + 2z.

(d) w=exp(z^2 +1).

1

(e) w = -.

z

(f) w = z + 1.

z-1

For Exercises 2- 5, find the angle of rotation a= Argf' (z) and the scale factor
I f ' ( z) I of the mapping w = f ( z) at the indicated points.

2. w = .!. at the points 1, 1 + i, and i.
   z


3. w =In r + ifJ, where - 2 " < () <^3 ; at the points 1, 1 + i , i, and -1.


4. w = r4 cos~+ irt sin~, where -7r < () < 7r, at the points i, l, -i, and 3 + 4i.


5. w =sin z at the points ~ + i, 0, and -; + i.


6. Consider the mapping w = z^2 • If a # 0 and b # 0, show that the lines x = a and
   y = b are mapped onto orthogonal parabolas.


7. Consider the mapping w = z!, where z~ denotes the principal branch of the
   square root function. If a > 0 and b > 0, show that the lines x = a and y = b are
   mapped onto orthogonal curves.


8. Consider the mapping w = expz. Show that the lines x = a and y =bare mapped
   onto orthogonal curves.


9. Consider the mapping w = sin z. Show that the line segment -; < x < ~, y = 0,
   and the vertical line x = a, where lal < i, are mapped onto orthogonal curves.


10. Consider the mapping w = Logz, where Logz denotes the principal branch of the
    logarithm function. Show that the positive x-axis and the vertical line x = 1 are
    mapped onto orthogonal curves.