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-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 I, then I is a constant function.

-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 I, then I is a constant function.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1b written out on paper please, not typed out.
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
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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