Reference no: EM132728602
Complex Analysis
1.The Jacobian of two real-valued functions u(x, y), v(x, y) of (x, y) is defined by the determinant
J=(∂(u,v))/(∂(x,y))=|(ux&uy @vx &vy)|
If f (z) = u(x, y) + iv(x, y) is an analytic function of z = x + iy, prove that
J (x, y) = |f ‘(z)|2.
2. Define
f(z)=z+1/z
(a) Find the image of the unit circle |z| = 1 under f .
(b) On what open sets ? ⊂ C is f : ? → C a conformal map?
3. Let γ: [0, π] → C with γ(t) = 2eit be the positively oriented semicircle in the upper half plane with center the origin and radius 2. Prove that
∫_γ?e^z/(z^2+1) dz?≤(?2πe?^2 )/3
(Do not try to evaluate the integral exactly.)
4. Suppose that a, b, z ∈ C are such that az + b ≠ 0 and |z| = 1. Prove that
|(b ¯z+a ¯)/(az+b)|=1
5. Find the radius of convergence of the following power series:
(a)∑_(n=1)^∞ 3^n/n z^n; (b) ∑_(n=0)^∞ 2^n/n! z^3n ; (c) ∑_(n=0)^∞ n! z^n!