Page 26 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 2: Complex Functions

Answers: Self Assessment Notes

1. Elementary calculus 2. f : D  C
3. Limit of a function 4. Differentiable


0
5. f'(z )  lim f(z   z) f(z )
0
0
 z 0  z
2.7 Review Questions
1. (a) What curve is described by the function (t) = (3t + 4) + i(t – 6), 0  t  1?

(b) Suppose z and w are complex numbers. What is the curve described by
(t) = (1 – t)w + tz, 0  t  1?
2. Find a function  that describes that part of the curve y = 4x + 1 between x = 0 and x = 10.
3
3. Find a function  that describes the circle of radius 2 centered at z = 3 – 2i .
4. Note that in the discussion of the motion of a body in a central gravitational force field, it
was assumed that the angular momentum  is nonzero. Explain what happens in case
 = 0.
5. Suppose f(z) = 3xy + i(x – y ). Find lim f(z), or explain carefully why it does not exist.
2

z 3 2i
6. Prove that if f has a derivative at z, then f is continuous at z.
7. Find all points at which the valued function f defined by f(z) = z has a derivative.
8. Find all points at which the valued function f defined by
f(z) = (2 ± i)z – iz + 4z – (1 + 7i)
3
2
has a derivative.
9. Is the function f given by

 (z) 2
  2 , z  0
f(z)  
 0 , z  0
 

differentiable at z = 0? Explain.
10. At what points is the function f given by f(z) = x + i(1 – y) analytic? Explain.
3
3
11. Find all points at which f(z) = 2y – ix is differentiable.
12. Suppose f is analytic on a connected open set D, and f’(z) = 0 for all z  D. Prove that f is
constant.

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