1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

84 CHAPTER 2 • COMPLEX FUNCTIONS

The beauty of this structure is that it makes this "full square root func-
tion" continuous for all z =F O. Normally, the principal square root function
would be discontinuous along the negative real axis, as points near - 1 but above
that axis would get mapped to poin ts close to i, and points near - 1 but below
the axis would get mapped to points close to - i. As Figure 2.20(c) indicates,
however, between the point A and the point B, the domain switches from the
edge of D 1 in the upper half-plane to the edge of Di in the lower half-plane.
The corresponding mapped points A' and B
1
are exactly where they should be.
The surface works in sucli a way that going directly between the edges of D1 in
the upper and lower half-planes is impossible (likewise for Dz}. Going counter-
clockwise, the only way to get from the point A to the point C, for example, is
to follow the path indicated by t he arrows in Figure 2.20(c).

-------~EXERCISES FOR SECTION 2.4

1. Let Ji (z) and h (z) be the two branches of the square root function given by
   Equations (2-28) and (2-29), respectively. Use the polar coordinate formulas in
   Section 2.2 to find the image of

(a) quadrant II, x < 0 and y > 0, under the mapping w = fi (z).

(b) quadrant II, x < 0 a.nd y > 0, under the mapping w = fz (z).

(c) the right half-plane Re(z) > 0 under the mapping w = fi (z).

(d) the right half-plane Re(z) > 0 under the mapping w = fz (z).

2. Let c:t = 0 in Equation (2-30). Find the range of the function w = j,, (z).

3. Let c:t = 271' in Equation (2-30). Find the range of the function w = /"' (z).

4. Find a branch of the square root that is continuous along the negative x-axis.


5. Let fi (z) = lz lk ei~ = r! cos ~+ir! sin~. where lzl = r # 0, and 6 = Arg(z).
   ft denotes the principal cube root function.
   1
   (a) Show that !1 is a branch of the multivalued cube root f (z) = z•.
   (b) What is the range of /1?
   ( c) Where is fi continuous?


6. Let h (z) = r! cos (^8 "4;^2 ~) + ir! sin (^8 "4;^2 ~ ), where r > 0 and -71' < () :S 71'.

(a) Show that h is a branch of the multivalued cube root f (z) = z!.

(b) What is the range of h?

(c) Where is h continuous?

(d) What is the branch point associated with/?

7. Find a branch of the multivalued cube root function that is different from those in
   Exercises 5 and 6. State the domain and range of the branch you find.


8. Let f (z) = z!. denote the multivalued nth root, where n is a positive integer.