Complex Variable - Study Mode

Show All Answers

[#1] If C is a circle |z| = 4 and $${ ext{f}}left( { ext{z}}
ight) = frac{{{{ ext{z}}^2}}}{{{{left( {{{ ext{z}}^2} - 3{ ext{z}} + 2}
ight)}^2}}},$$ xa0 xa0 then $$ointlimits_{ ext{c}} {{ ext{f}}left( { ext{z}}
ight){ ext{dz}}} $$ xa0 is

(A) 1

(B) 0

(C) -1

(D) -2

Correct Answer

(B) 0

[#2] What is the residue of the function $$frac{{1 - {{ ext{e}}^{2{ ext{z}}}}}}{{{{ ext{z}}^4}}}$$ xa0at its pole?

(A) $$frac{4}{3}$$

(B) $$ - frac{4}{3}$$

(C) $$ - frac{2}{3}$$

(D) $$frac{2}{3}$$

Correct Answer

(B) $$ - frac{4}{3}$$

[#3] Let z = x + jy where $${ ext{j}} = sqrt { - 1} .$$ xa0 Then $$overline {cos { ext{z}}} = ?$$

(A) cos z

(B) cos $$overline { ext{z}} $$

(C) sin z

(D) sin $$overline { ext{z}} $$

Correct Answer

(B) cos $$overline { ext{z}} $$

[#4] Consider the function f(z) = z + z ∗ where z is a complex variable and z ∗ denotes its complex conjugate. Which one of the following is TRUE?

(A) f(z) is both continuous and analytic

(B) f(z) is continuous but not analytic

(C) f(z) is not continuous but is analytic

(D) f(z) is neither continuous nor analytic

Correct Answer

(B) f(z) is continuous but not analytic

[#5] Consider the analytic function f(z) = x 2 - y 2 + i2xy of the complex variable z = x + iy, where $${ ext{i}} = sqrt { - 1} .$$ xa0The derivative f'(z) is

(A) 2x + i2y

(B) x 2 + iy 2

(C) x + iy

(D) 2x - i2y

Correct Answer

(A) 2x + i2y