Page 35 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 3. New function is an extension of the real logarithm function: ...................
4. There are many values of log z, and so there can be many values of z . As one might guess,
c
e cLog z is called the ................... of z .
c
3.7 Review Questions

1. Show that exp(z + 2i) = exp(z)

exp(z)
2. Show that  exp(z w).

exp(w)
3. Show that |exp(z)| = e , and arg (exp(z) = y + 2k for any arg (exp(z)) and some integer k.
x
4. Find all z such that exp(z) = –1, or explain why there are none.

5. Find all z such that exp(z) = 1 + i, or explain why there are none.
6. For what complex numbers w does the equation exp(z) = w have solutions? Explain.
7. Find the indicated mesh currents in the network:

8. Show that for all z,

(a) sin(z + 2p) = sin z; (b) cos(z + 2) = cos z;


(c) sin z      cosz.

 2 
9. Show that |sin z| = sin x + sinh y and |cos z| = cos x + sinh y.
2
2
2
2
2
2
10. Find all z such that sin z = 0.
11. Find all z such that cos z = 2, or explain why there are none.
1
12. Is the collection of all values of log(i ) the same as the collection of all values of log i?
1/2
2
Explain.
13. Is the collection of all values of log(i ) the same as the collection of all values of 2log i ?
2
Explain.
14. Find all values of log(z ). (in rectangular form)
1/2
15. At what points is the function given by Log (z + 1) analytic? Explain.
2
16. Find the principal value of
(a) i . i (b) (1 – i) 4i
17. Find all values of |i |.
i

28 LOVELY PROFESSIONAL UNIVERSITY