Complex Variables and Numerical Methods — Summer 2025

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

Find the analytic function f(z) = u+iv, if u−v = ex(cosy−siny).

Find the Laurent’s series expansion of f(z)=1 about z = 0, for the7 z(z2−3z+2) regions (i) 1 < |z| < 2 (ii) |z| > 2.

1 dx Evaluate the integral ∫ by Gaussian integration two pint formula. −11+x2

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

Use the power method to find the largest eigen value and corresponding eigen vector of the matrix A = [1 2 0]003

Solve the following system of equations using Gauss seidel method: 5x+y−z = 10 ; 2x+4y+z = 14 ; x+y+8z =

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

Use trapezoidal rule to evaluate ∫ 1 x3dx considering five sub intervals.

Using Newton’s forward formula, find the value of f(1.6) if X 1 1.4 1.8 2.2 F(x) 3.49 4.82 5.96 6.5

7 Solve the following system of equation by gauss elimination method with partial pivoting 2x +2x +x = 6; 4x +2x +3x = 4 ; x +x +x = 0

If f(z)= u+iv is analytic in domain Dthen prove that ( 𝜕2 + 𝜕2 )|Ref(z)|2 = 2|f′(z)|2.3 𝜕x2 𝜕y2

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

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

Expand f(z) = z−1 as a Taylor’s series about the point z = 1. z+1

Find the radius of converges of ∑∞ ( 6n+1 ) 2 (z−2i)n. n=1 2n+5

𝜋 𝜋 1+sin +icos Simplify ( 8 8) 𝜋 𝜋 1+sin −icos88

Find the fourth root of -1

Prove that sinh−1x = log{x+√x2+1}.

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

Find the image of infinite strip 0 ≤ x ≤ 1 under the transformation w = iz+1. Sketch the region in the w-plane.

Show that u =x2−y2+x is harmonic, Find the corresponding analytic function f(z) =u+iv

Using Cauchy’s Integral formula, evaluate ∮ 1 dz, where Cis the circle |z| = z2−7z+12 3.5