Reference no: EM132458102

Complex Variable Analysis Assignment -

Q1. The Jacobian of two real-valued functions u(x, y), v(x, y) of (x, y) is defined by the determinant

If f(z) = u(x, y) + iv(x, y) is an analytic function of z = x + iy, prove that J(x, y) = |f'(z)|.

Q2. Define f(z) = z + (1/z)

(a) Find the image of the unit circle |z| = 1 under f.

(b) On what open sets Ω ⊂ C is f: Ω → C a conformal map?

Q3. Let γ: [0, π] → C with γ(t) = 2e^it be the positively oriented semicircle in the upper half plane with center the origin and radius 2. Prove that

|∫_γe^z/(z²+1) dz| ≤ (2πe²)/3

(Do not try to evaluate the integral exactly.)

Q4. Suppose that a, b, z ∈ C are such that az + b ≠ 0 and |z| = 1. Prove that

|(b^-z+a^-)/(az+b)| = 1.

Q5. Find the radius of convergence of the following power series:

(a) _n=1∑^∞ (3ⁿ/n)zⁿ;

(b) _n=0∑^∞ (2ⁿ/n!) z³ⁿ;

(c) _n=0∑^∞ n! z^n!.