b) Compute Re fli) when: $(2) = √2-1² and f(0) = -i
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Question
2b written out on paper please, but not using the method of the second picture (same problem but a different solution please)
![2(b) solution:
flz)= √2-1
=> fli) = √i-T
= f(1)= √√(1) + i.
Take x=-1₁ y = 1
•: 1²1= √√√n²+y² = 121= √√²
.
.: √xtiy
= +
[₁
→ √(1)+i•1
⇒ | + (1) = ±
Iz1+Re(z)
2
= ± [√√√²+
+
i sgnly) √
√√2+ (-12 + 1.1. √√ √2-6-
√2-(-1)
2
121-Pre(2)
= = [
± [√√√₂ = 1 + ₁ √ √₂+1
2
|sgn (1)=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b030cd5-5424-4c4c-9fb9-bb5124dfa56d%2F0fceb0b5-eb92-4ca4-9c1d-33cba9696ab5%2Fjpjudtg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2(b) solution:
flz)= √2-1
=> fli) = √i-T
= f(1)= √√(1) + i.
Take x=-1₁ y = 1
•: 1²1= √√√n²+y² = 121= √√²
.
.: √xtiy
= +
[₁
→ √(1)+i•1
⇒ | + (1) = ±
Iz1+Re(z)
2
= ± [√√√²+
+
i sgnly) √
√√2+ (-12 + 1.1. √√ √2-6-
√2-(-1)
2
121-Pre(2)
= = [
± [√√√₂ = 1 + ₁ √ √₂+1
2
|sgn (1)=1
![1) a) Prove that f(z) = 3x+y +i (3y-x) is analytic in
namely: entire
b) Prove that if f(z) and f(z) are analytic in a
domain &, then I is a constant function.
c) Prove that if f(z) is analytic and purely imaginary
in a domain №r, then I is a constant function
2) a) Find all the branch points and all the discontinuity
points for the following functions:
ii) f(z) = √ (Z+8) ³ (2z - 4) 5
i) f(z) = 3-3√√√2+1
1+ √Z-1
iii) f(z) =
Z+Z-4
--√√22+5
b) Compute Re fli) when: $(2) = √2-1² and f(0) = -i
c) Which of the following functions are analytic in 1Z1 < 2
and /or
IZI> 2:
i) 2-√√24-42²
iii)
2²+62+9
ii)
2²1
Z-√Z+4
√2²2² +22+2
z2-42 + 3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b030cd5-5424-4c4c-9fb9-bb5124dfa56d%2F0fceb0b5-eb92-4ca4-9c1d-33cba9696ab5%2Fn3z9edj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1) a) Prove that f(z) = 3x+y +i (3y-x) is analytic in
namely: entire
b) Prove that if f(z) and f(z) are analytic in a
domain &, then I is a constant function.
c) Prove that if f(z) is analytic and purely imaginary
in a domain №r, then I is a constant function
2) a) Find all the branch points and all the discontinuity
points for the following functions:
ii) f(z) = √ (Z+8) ³ (2z - 4) 5
i) f(z) = 3-3√√√2+1
1+ √Z-1
iii) f(z) =
Z+Z-4
--√√22+5
b) Compute Re fli) when: $(2) = √2-1² and f(0) = -i
c) Which of the following functions are analytic in 1Z1 < 2
and /or
IZI> 2:
i) 2-√√24-42²
iii)
2²+62+9
ii)
2²1
Z-√Z+4
√2²2² +22+2
z2-42 + 3
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