GTU 2141905 Complex Variables and Numerical Methods Summer 2025 Question Paper & PDF

Discuss continuity of 𝑓(𝑧) = { 𝑅𝑒(𝑧²) |𝑧|2 ; 𝑧 ≠ 0 0 ; 𝑧 = 0 at z = 0.

Find the Laurent’s series expansion of 𝑓(𝑧) = 1 𝑧(𝑧²−3𝑧+2) about z = 0, for the regions

Evaluate the integral ∫ 𝑑𝑥 1+𝑥2 1 −1 by Gaussian integration two pint formula.

Evaluate ∫ (𝑠𝑖𝑛𝑥 − 𝑙𝑜𝑔𝑥 + 𝑒𝑥)𝑑𝑥 1.4 0.2 with h=0.2 by simpson’s 1/3rd and 3/8th rule.

Use the power method to find the largest eigen value and corresponding eigen vector of the matrix 𝐴 = [ 1 6 1 1 2 0 0 0 3 ]

Use secant method to estimate the root of 𝑐𝑜𝑠𝑥 = 𝑥𝑒𝑥 correct to four significant digits

Find a real root of 𝑥𝑒𝑥 = 2, correct up three decimal places, by using Newton- raphson method

Compute f (9.2) from the following value Newton’s divided difference formula. X 8.0 9.0 9.5 11 F(x) 2.079442 2.197225 2.251292 2.397895

If 𝑓(𝑧) = 𝑢 + 𝑖𝑣 is analytic in domain D then prove that ( 𝜕2 𝜕𝑥2 + 𝜕2 𝜕𝑦2) |𝑅𝑒𝑓(𝑧)|2 = 2|𝑓′(𝑧)|2.

Determine the poles of the function 𝑓(𝑧) = 𝑧2 (𝑧−1)²(𝑧+2) and residue at each pole. Evaluate ∫ 𝑓(𝑧)𝑑𝑧, 𝑐 where c is the circle |𝑧| = 3.

Show that the function 𝑓(𝑧) = √|𝑥𝑦| satisfies the Cauchy- Riemann equations at the origin but 𝑓′(0) fails to exits.

Prove that the nth root of unity are in geometric progression with the common ratio (cos 2𝜋 𝑛 + 𝑖 sin 2𝜋 𝑛 )and show that the continued product of all 𝑛𝑡ℎ roots is (−1)𝑛+1.