complex analysis 2
DESCRIPTION
Complex Analysis 2TRANSCRIPT
COMPLEX ANALYSIS
Math 252 Complex Analysis Page 1
Exercises2: Triangle Inequality, Roots of the Complex
Numbers
Question1.
a. Verify that 1 1
3,1 3, 1 , 2,15 10
Solution:
Since 1 2 1 1 2 2 1 2 1 2 1 2 2 1, , , z z x y x y x x y y x y x y
b.
5
1 2 3 2
i
i i i
Solution:
5 5 5 5 5
1 2 3 2 3 1 3 1 3 3 3 10 3 10
5 5 =
10 10 2
i i i i i i i i i
i i i
i i
c. 4
1 4 i
Solution:
4
1 1 1 1 1 1 2 1 1 2 1 2 2 4 i i i i i i i i i
1 1 1 1 1 13,1 3, 1 , 3 3 1 1 , 3 1 1 3 , 10,0 ,
5 10 5 10 5 10
1 1 1 1 = 10 0 , 10 0 2,1
5 10 10 5
COMPLEX ANALYSIS
Math 252 Complex Analysis Page 2
Question2. Show that 2 21 1 2 z z z
Solution: Since and then
Let where , z a ib a b
Thus, 2 21 1 2 z z z
Question3. Verify that each of the two numbers satisfies the equation Solution:
For , we have For , we have
Thus, satisfies the equation Question4. Show that a. Solution:
Let where , z a ib a b
, and Thus,
2 2Re 1 Re 1 2 z z z
2 2Im 1 Im 1 2 z z z
2 21 1 2 , z z z z
2 2 2 2 2 21 1 1 2 1 1 2 2 2 z a ib a a bi b a a b i b ab
22 2 2 2 21 2 1 2 1 2 2 2 1 2 2 2 z z a ib a ib a i b a i ab b a a b i b ab
1 z i2 2 2 0 z z
1 z i 22 2 2 1 2 1 2 1 2 1 2 2 2 0 z z i i i i
1 z i 22 2 2 1 2 1 2 1 2 1 2 2 2 0 z z i i i i
1 z i 2 2 2 0 z z
Im Reiz z
Im iz a iz i a ib ia b Re z a
Im Reiz z
COMPLEX ANALYSIS
Math 252 Complex Analysis Page 3
b. Solution:
Let where , z a ib a b
, and Thus, c. Solution:
Let where x, z x iy y
Since , and then
Re iz b iz i a ib ia b Im z b
Re Im iz z
Re Im iz z
1, 0
1 z z
z
11 zz
1
2 2 2 2,
x yz
x y x y
1 11
1
1 2 2 2 2 2 2
2 2
2 2 2 2
1 1 1, ,
1
, ,
x yz x y
z x y x y x y
z
x yx y x y z
x y x y
COMPLEX ANALYSIS
Math 252 Complex Analysis Page 4
Question5. Solve the equation for by writing and then solving a pair of simultaneous equations in x and y. Solution: Let , then implies that
2 2 1 0 x y x and 1
2 0, 2 1 0, 2
xy y y x x ,
if 0y then 2 1 0 x x that is x is complex but we found x is real. Substitute
1
2 x into the
2 2 1 0 x y x
2
2 2 21 11 1 0
2 2
x y x y ,
3
2 y .
Thus, 1 3
, ,2 2
z x y
2 1 0 z z ,z x y
, , , 1,0 0,0 x y x y x y
,z x y 2 1 0 z z
2 2
2 2
2 2
, , , 1,0 0,0
, , 1,0 0,0
,2 , 1,0 0,0
1,2 0,0
x y x y x y
x y xy xy x y
x y xy x y
x y x xy y